446.    7.77%

First, to convert to a decimal, divide 777 by 10,000. This is equivalent to moving the decimal point 4 places to the left.

Now, change this decimal to a percent by moving the decimal point two places to the right and attaching a percent sign (%).

447.    100.1%

First, change to a mixed number.

To convert to a decimal, divide 1,001 by 1,000. This is equivalent to moving the decimal point 3 places to the left.

Now, change this decimal to a percent by moving the decimal point two places to the right and attaching a percent sign (%).

448.    10

50% equals , so divide 20 by 2.

449.    15

25% equals , so divide 60 by 4.

450.    40

20% equals , so divide 200 by 5.

451.    13

10% equals , so divide 130 by 10.

452.    33

equals , so divide 99 by 3.

453.    24

1% equals , so divide 2,400 by 100.

454.    9

18% of 50 equals 50% of 18, which is much easier to evaluate. 50% equals , so divide 18 by 2.

455.    8

32% of 25 equals 25% of 32, which is much easier to evaluate. 25% equals , so divide 32 by 4.

456.    4

12% of equals of 12, which is much easier to evaluate. equals , so divide 12 by 3.

457.    3.44

Convert 8% to the decimal 0.08 and then multiply by 43.

458.    6.97

Convert 41% to the decimal 0.41 and then multiply by 17.

459.    6.88

Convert 215% to the decimal 2.15 and then multiply by 3.2.

460.    0.81

Convert 7.5% to the decimal 0.075 and then multiply by 10.8.

461.    75

Turn the problem into an equation:

Substitute multiplication by 0.01 for %.

Simplify by multiplying 0.01 by 40.

Now, divide both sides by 0.40.

Therefore, 75% of 40 is 30.

462.    12.5

Turn the problem into an equation.

Substitute multiplication by 0.01 for %.

Simplify by multiplying 0.01 by 160.

Now, divide both sides by 1.60.

Therefore, 20 is 12.5% of 160.

463.    288

Turn the problem into an equation.

Change 25% into the decimal 0.25.

Now, divide both sides by 0.25.

Therefore, 72 is 25% of 288.

464.    300

Turn the problem into an equation.

Change 85% into the decimal 0.85.

Now, divide both sides by 0.85.

Therefore, 85% of 300 is 255.

465.    8,500

Turn the problem into an equation.

Change 71% into the decimal 0.71.

Now, divide both sides by 1.08.

Therefore, 71% of 6,035 is 8,500.

466.    16,300

Turn the problem into an equation.

Change 108% into the decimal 1.08.

Now, divide both sides by 1.08.

Therefore, 108% of 16,300 is 17,604.

467.    96

Turn the problem into an equation:

Change the % into multiplication by 0.01:

Next, multiply the two decimals on the left side of the equation:

Now, divide both sides of the equation by 0.025:

Therefore, 2.5 is 96% of 2.4.

468.    1,000

Turn the problem into an equation.

Change 9.95% into the decimal 0.0995.

Now, divide both sides by 0.0995.

Therefore, 99.5 is 9.95% of 1,000.

469.    150

Turn the problem into an equation:

Change the % into multiplication by :

Next, multiply the two fractions on the right side of the equation:

Now, multiply both sides of the equation by 300:

Therefore, is 150% of .

470.    

Turn the problem into an equation.

Change the mixed number into the improper fraction , and change 75% into the fraction .

Now, multiply both sides by .

Simplify.

Now, change the improper fraction to a mixed number.

Therefore, is 75% of .

471.    2:3

Make a fraction of dogs to cats and then reduce it to lowest terms, as follows:

The fraction is equivalent to the ratio 2:3.

472.    4 to 5

Make a fraction of boys to girls and then reduce it to lowest terms, as follows:

The fraction is equivalent to the ratio 4 to 5.

473.    7:5

Make a fraction of married people to single people and then reduce it to lowest terms, as follows:

The fraction is equivalent to the ratio 7:5.

474.    16 to 21

Make a fraction of Karina’s earnings to Tamara’s and then reduce it to lowest terms, as follows:

The fraction is equivalent to the ratio 16 to 21.

475.    7 to 11

Make a fraction of the distances from both yesterday and today, increasing the terms of the fraction so both the numerator and denominator are integers, as follows:

Now reduce the fraction to lowest terms.

The fraction is equivalent to the ratio 7 to 11.

476.    3:5

Make a complex fraction of the time fulfilled on Saturday and Sunday.

Now, evaluate this complex fraction as fraction division.

Change this to multiplication by taking the reciprocal of the second fraction.

The fraction is equivalent to the ratio 3:5.

477.    2:7

Make a fraction of the number of managers and the total number of staff.

Now, reduce the fraction to lowest terms.

The fraction is equivalent to the ratio 2:7

478.    5:6:4

The ratio of sophomores to juniors to seniors is 10:12:8. All three of these numbers are even, so you can divide each by 2 to reduce the ratio.

479.    4:15

Make a fraction of the number of seniors and the total number of students:

Now reduce the fraction to lowest terms.

The fraction is equivalent to the ratio 4:15.

480.    2 to 3

Make a fraction of the number of juniors and the combined number of sophomores and seniors:

Now reduce the fraction to lowest terms.

The fraction is equivalent to the ratio 2 to 3.

481.    2:4:3

If one person moves from the first floor to the second floor, the ratio of first-floor ­residents to second-floor residents to third-floor residents becomes 4:8:6. All three of these numbers are even, so you can divide each by 2 to deduce the ratio.

482.    4 to 1

Originally, Ann was using 2,400 watts, but then she reduced her usage by 1,800 watts, so her usage went down to 2,400 – 1,800 = 600. Make a fraction of her usage before and after as follows:

This fraction is equivalent to the ratio 4 to 1.

483.    6:7

Make a fraction of the building height and the combined height of the buiding and the tower:

Now, reduce the fraction to lowest terms.

The fraction is equivalent to the ratio 6:7.

484.    4

Make a proportion with nonregistered voters in the numerator and registered voters in the denominator; then plug in the number of registered members:

Now, multiply both sides of this equation by 28 to cancel out the fraction on the left side.

Therefore, the organization has four nonregistered members.

485.    18

Make a proportion with windows in the numerator and doors in the denominator; then plug in the number of doors:

Now, multiply both sides of this equation by 4 to cancel out the fraction on the left side.

Therefore, the house has 18 windows.

486.    36

Make a proportion with purchasers in the numerator and entrants in the denominator; then plug in the number of entrants:

Now, multiply both sides of this equation by 120 to cancel out the fraction on the left side:

Therefore, 36 people made purchases.

487.    1,815

The diet requires a 6:4:1 ratio of protein to fat to carbohydrates. Thus, its ratio of fat to total calories is 4 to (6 + 4 + 1), which is a 4:11 ratio. Make a proportion with the total in the numerator and fat in the denominator; then plug in the number of fat calories:

Now, multiply both sides of this equation by 660 to cancel out the fraction on the left side.

Thus, the diet permits 1,815 total calories.

488.    14

The project manager estimates that her newest project will require a 2:9 ratio of team leaders to programmers. Thus, this is a 2 to (2 + 9) ratio of team leaders to total members, which is a 2:11 ratio.

Make a proportion with the team leaders in the numerator and the total members in the denominator; then plug in the total number:

Now, multiply both sides of this equation by 77 to cancel out the fraction on the left side.

Thus, the project manager will need 14 team leaders.

489.    104

Make a proportion with dinner customers in the numerator and lunch customers in the denominator; then plug in the number of lunch customers:

Now, multiply both sides of this equation by 40 to cancel out the fraction on the left side:

Thus, the diner has an average of 64 dinner customers, so the total number of customers for both lunch and dinner is 40 + 64 = 104.

490.    1,140

Make a proportion with fiction books in the numerator and nonfiction books in the denominator; then plug in the number of nonfiction books:

Now, multiply both sides of this equation by 900 to cancel out the fraction on the left side.

Thus, the bookmobile has 240 fiction books, so the total number of fiction and nonfiction books is 240 + 900 = 1,140.

491.    150

The organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and New Hampshire. Thus, it has a 5:2 ratio of members from Massachusetts to members from New Hampshire. Make a proportion with Massachusetts in the numerator and New Hampshire in the denominator; then plug in the number of members from New Hampshire:

Now, multiply both sides of this equation by 60 to cancel out the fraction on the left side.

Thus, the organization has 150 members from Massachusetts.

492.    98

The organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and New Hampshire. Thus, the ratio of members from Vermont to members from Massachusetts or New Hampshire is 3 to (5 + 2), which is 3:7. Make a proportion with Massachusetts plus New Hampshire in the numerator and Vermont in the denominator; then plug in the number of members from Vermont:

Now, multiply both sides of this equation by 42 to cancel out the fraction on the left side.

Thus, the organization has 98 members from either Massachusetts or New Hampshire.

493.    72

The organization has a 5:3:2 ratio of members from, respectively, Massachusetts, Vermont, and New Hampshire. Thus, the ratio of members from Vermont to the total number of members is 3 to (5 + 3 + 2), which is 3:10. Make a proportion with Vermont in the numerator and the total in the denominator; then plug in the total number of members:

Now, multiply both sides of this equation by 240 to cancel out the fraction on the left side.

Thus, the organization has 72 members from Vermont.

494.    90

The ratio of Jason’s laps to Anton’s laps is 9 to 5, so the ratio of Jason’s laps to the total laps is 9 to 14. Make a proportion with Jason in the numerator and the total laps in the denominator; then plug in 140 for the total number of laps:

Multiply both sides by 140 to get rid of the fraction on the left side:

Therefore, Jason swam 90 laps.

495.    $50,000

The ratio of domestic sales to foreign sales is 6 to 1, so the ratio of foreign sales to total sales is 1 to 7.

Make a proportion with foreign sales in the numerator and total sales in the denominator; then plug in $350,000 for the total amount of revenue:

Now, multiply both sides of this equation by 350,000 to cancel out the fraction on the left side.

496. 56

The restaurant sells a 5 to 3 ratio of red wine to white wine. So, in terms of the ratio, its total sales are 5 + 3 = 8, and the difference between its red wine sales and its white wine sales is 5 – 3 = 2. Thus, the restaurant has an 8 to 2 ratio regarding the total sales and the difference in red and white wine sales, which simplifies to a 4 to 1 ratio. Make a proportion and then fill in the difference in sales as follows:

Now, multiply both sides of the equation by 14 to cancel out the fraction on the left side.

497.    50 to 53

The portfolio began with 100% of funds and rose to 106% of value, so make a proportion of these values:

Cancel the percentages; then reduce.

498.    $60

Make a proportion of dollars to francs; then reduce:

Thus, the ratio of dollars to francs in any exchange is 10:9. Now, using this ratio, make an equation and plug in 54 for the number of francs that Karl returned with.

Multiply both sides of this equation by 54 to cancel out the fraction on the left side.

499.    $1,000

Charles spends 20% of his income on rent and 15% on transportation, so he spends the remaining 65% on everything else. Thus, the proportion of rent to everything else is 20:65, which simplifies to 4:13.

Make a proportion of rent to everything else; then plug in 3,250 for everything else:

Now, multiply both sides of the equation by 3,250 to get rid of the fraction on the left side.

Therefore, his rent is $1,000.

500.    4

Multiplication in the alternative universe is proportional to our multiplication. In the alternative universe , but in our universe, . So, make a proportion of these two values as follows:

Simplify this proportion by multiplying both the numerator and denominator by 2.

In our universe, , so plug this value into the preceding equation:

Now, multiply both sides of the equation by 3 to get rid of the fractions.

Therefore, in the alternative universe, .

501.    

To find the total fraction of candy that was bought, add the two fractions. Because the two fractions both have 1 in the numerator, you can add them quickly: Add the two denominators (8 + 6 = 14) to find the numerator of the answer, then multiply the two denominators () to find the denominator of the answer.

Reduce the fraction by dividing both the numerator and the denominator by 2.

502.     mile

To find the difference between the distances the girls ran, subtract the smaller fraction from the larger one. Subtract minus using cross-multiplication techniques:

503.    

The word of in a fraction word problem means multiplication, so multiply by :

504.    

To find the amount of land in each subdivision, divide the fraction by 4. To divide by 4, multiply it by its reciprocal, which is :

505.     miles

To find half of miles, first convert from a mixed number to an improper fraction:

Now, divide by 2:

506.    

Divide to find each child’s portion of cookies. To divide 14 by 3, make an improper fraction with 14 in the numerator and 3 in the denominator; then turn it into a mixed number:

507.    

The word of in a fraction word problem means multiplication, so multiply the four fractions:

508.    

First, calculate what part of the distance Arnold and Marion drove together:

Next, calculate how much farther they had to drive by subtracting this amount from 1:

509.    12 hours

Jake practices for hours 5 days a week, and for hours 2 times a week, so calculate as follows:

Convert both mixed numbers to improper fractions:

Solve:

Therefore, Jake practices basketball for 12 hours every week.

510.    5

The pizza had 16 slices. Jeff took of these, so he took 4 slices, leaving 12. Molly took 2 more slices, leaving 10. Tracy took half of the remaining slices, so she took 5 and left 5.

511.     miles

Calculate by converting all three mixed numbers to improper fractions and then adding:

Change each fraction to a common denominator of 20:

Convert the result back to a mixed number:

512.    

First, add the lengths that Esther has already found:

Now, subtract this result from the amount she needs to build the shelves:

Therefore, she needs an additional feet of wood.

513.     gallon

Nate drank of the gallon on Monday, so he left of the gallon. Then on Tuesday, he drank of what was left, which was:

Thus, on Tuesday, he drank of a gallon from a container that held of a gallon, so he left behind:

Thus, he left behind of a gallon.

514.     pounds

First, figure out how many batches you need to make by dividing the number of cookies you need (150) by the number in each batch (25):

Now, multiply the amount of butter in each batch ( pounds) by 6:

Reduce this fraction; then change it to a mixed number:

515.     gallon

First, convert gallons to an improper fraction ( gallons); then divide it by both 5 and 4:

Next, subtract to find the difference:

516.     hours

To find how many words Harry can write in an hour, divide the number of words by the number of hours:

Calculate by changing the mixed number to an improper fraction and then changing division to multiplication:

You can simplify this calculation by canceling a factor of 13 in both the numerator and denominator:

Thus, Harry can write 200 words per hour. To calculate how many hours he needs to write 750 words, divide 750 by 200:

Therefore, Harry needs hours to write a 750-word article.

517.    

Craig ate of the apple pie, so he left of it. His mom ate of the blueberry pie, so she left of it. So, add the two parts that they didn’t eat as follows:

Change this improper fraction into a mixed number:

518.    

David’s piece was of the cake, which left of the cake untouched. Then, Sharon cut of what was left, so calculate this amount as follows:

Thus, Sharon also ate of the cake. So you can calculate what David and Sharon ate as follows:

So, David and Sharon ate of the cake, leaving . Armand ate of this, so he ate of the cake and left behind .

519.    

An hour is 60 minutes, which is 10 times as long as 6 minutes, so multiply by 10:

Reduce and then convert the improper fraction into a mixed number:

520.    7

The trick here is to think of easier numbers and then see what happens when you double them: For example, suppose you knew that 1 chicken could lay 1 egg in 1 day. Then, if you had 2 chickens, they could lay 2 eggs in the same amount of time — that is, in 1 day.

Now, apply this same thinking to the problem: If chickens can lay eggs in days, then if you had 3 chickens, they could lay 3 eggs in the same amount of time — that is, days. Or, similarly, if you had chickens, they could lay eggs, again, in the same amount of time — days.

So now, if you double the amount of time to 3 days, those same chickens would double their output to 7 eggs.

521.    5.6 kilos

To begin, add up the number of kilos of chocolate that Connie bought:

2.7 + 4.9 + 3.6 = 11.2

Then, divide this amount by 2:

Therefore, Connie ended up with 5.6 kilos of chocolate.

522.    0.87 m

Calculate by subtracting Blair’s height, 0.97, from his father’s height, 1.84:

1.84 – 0.97 = 0.87

523.    60.9 m

Calculate by multiplying the number of meters in a step, 0.7, by the number of steps, 87:

524.    82 seconds

Divide the total number of gallons, 861, by the rate at which the water is filling the tank, 10.5:

525.    1.3 miles

Ed ran a total of miles, and Heather ran a total of miles. Calculate how much farther Heather ran by subtracting their total distances:

11.5 – 10.2 = 1.3

Therefore, Heather ran 1.3 miles farther than Ed.

526.    32.5

To find out how many miles per gallon Myra got, divide the total number of miles she drove, 403, by the total number of gallons of gas she used, 12.4:

527.    1.85

Calculate by dividing the total number of pages, 111, by the total amount of time, 1 hour or 60 minutes:

528.    4.55

Calculate by multiplying the number of liters in each can, 1.3, by the number of cans, 3.5:

529.    $1,824.60

Tony paid $356.10 per month for 36 months, so he paid a total of . Subtract the sticker price of $10,995 from this amount:

$12,819.60 – $10,995 = $1,824.60

530.    1.2 seconds

First, calculate Ronaldo’s total time by adding:

12.6 + 12.3 + 13.1 = 38.0

Next, calculate Keith’s time:

11.8 + 12.4 + 12.6 = 36.8

Subtract Ronaldo’s time from Keith’s time:

38.0 – 36.8 = 1.2

531.    $31.25

First, divide $187.50 by 3 to find the cost of one day:

Now, divide this result by 2 to find the cost of half a day:

Therefore, Dora should pay $31.25.

532.    $59.50

Calculate the total amount that Stephanie would have paid if she had paid $6.50 for each of the 29 days she went to the pool by multiplying:

Find how much she saved by subtracting what she paid for the pass, $129, from the preceding result:

$188.50 – 129 = $59.50

Therefore, she saved $59.50.

533.    $240.00

The cost for a child between 6 and 12 is , and the cost for a child under 6 is .

Calculate the cost for 2 adults as follows:

Calculate the cost for 3 children between 6 and 12 as follows:

Calculate the cost for 2 children under 6 as follows:

Add up these three results:

$115.20 + $86.40 + $38.40 = $240.00

534.    37.5 mph

Secretariat ran 1.5 miles in 2 minutes and 24 seconds, which equals 144 seconds (because ), so calculate how many seconds it would take him to run one mile as follows:

Thus, Secretariat ran at a rate of 1 mile in 96 seconds. An hour contains 3,600 seconds (because ), so calculate how many miles he could have run in one hour as follows:

Thus, Secretariat ran the Belmont Stakes at an average rate of 37.5 miles per hour.

535.    3.05 miles

On Monday, Anita swam 0.8 miles. On Tuesday, she swam miles farther than on Monday, so she swam 0.8 + 0.2 = 1 mile. On Wednesday, she swam miles farther than on Tuesday, so she swam 1 + 0.25 = 1.25 miles. Therefore, she swam 0.8 + 1 + 1.25 = 3.05 miles.

536.    6 hours

Angela spent 15 hours in total, and 40% of this time working with her flash cards, so you want to calculate 40% of 15:

Therefore, Angela spent 6 hours working with her flash cards.

537.    0.99 kilos

Ten percent of 1.1 is 0.11 (), so subtract this amount from the weight of the competitor’s laptop:

1.1 – 0.11 = 0.99

538.    35%

Make a fraction of the two numbers and then reduce:

Convert this number to a decimal by dividing; then convert to a percent:

539.    20%

Beth received a raise of $13.80 – $11.50 = $2.30. Calculate the percentage by making a fraction with $2.30 in the numerator and $11.50 in the denominator and reducing:

This fraction equals 0.2, which equals 20%.

540.    297.5 miles

The trip was 850 miles, and Geoff drove 35% of it the first day, so you want to calculate 35% of 850:

Therefore, Geoff drove 297.5 miles the first day.

541.    231

The book was 420 pages, and Nora read 55% of it the first day, so you want to calculate 55% of 420:

Therefore, Nora read 231 pages.

542.    12

Kenneth mowed the lawn 25 times, and 52% of this work was in May and June. Therefore, 48% was from July to September. You can calculate 48% of 25 easily as 25% of 48, as follows:

Therefore, Kenneth mowed the lawn 12 times from July to September.

543.    19.5 minutes

The 60-minute show has 32.5% commercials, so calculate 32.5% of 60:

Therefore, the show has 19.5 minutes of commercials.

544.    20%

Jason spent 3 hours and 45 minutes in total. Three hours is equal to 180 minutes (because ), so he spent 180 + 45 = 225 minutes altogether. He spent 45 minutes of this on the windows, so make the fraction and convert it to a ­percentage as follows:

545.    18.75%

Eve received a total of $8,000, of which $1,500 was from the scholarship, so make a fraction of these two numbers and reduce it as follows:

Now, convert this fraction to a decimal and then a percent:

546.    72.5%

Janey’s goal is 400 hours, of which she has completed 290. Thus, make a fraction of these two numbers and reduce it as follows:

Now, convert this fraction to a decimal and then a percent:

547.    300 hours

Steven studied Italian for 45 hours, which represented 15% of his preparation time. Thus, you want to solve the percent problem, “15% of what number is 45?” Turn the problem into an equation:

Change the percent to a decimal:

Now, divide both sides by 0.15:

Therefore, 15% of 300 hours is 45 hours.

548.    125 m

The atrium is 6.25 meters, which represents 5% of the height of the building. Thus, you want to solve the percent problem, “5% of what number is 6.25?” Turn the problem into an equation:

Change the percent to a decimal:

Now, divide both sides by 0.05:

Therefore, 5% of 125 is 6.25.

549.    $6,200

Karan’s mortgage payment is $1,736, which represents 28% of her monthly income. Thus, you want to solve the percent problem, “28% of what number is 1,736?” Turn the problem into an equation:

Change the percent to a decimal:

Now, divide both sides by 0.28:

Therefore, 28% of $6,200 is $1,736.

550.    $60,000

Madeleine earns $135,000, which represents 225% of her previous earnings. Thus, you want to solve the percent problem, “225% of what number is 135,000?” Turn the problem into an equation:

Change the percent to a decimal:

Now, divide both sides by 2.25:

Therefore, 225% of $60,000 is $135,000.

551.    $13,200

A percent increase of 10% is equivalent to 110% of the original amount, so you want to calculate 110% of $12,000:

552.    $637.50

A percent decrease of 15% is equivalent to 85% of the original amount, so you want to calculate 85% of $750:

553.    $31

A percent increase of 18% is equivalent to 118% of the original amount, so you want to calculate 118% of $26.00:

This amount rounds up to $31.

554.    $222,000

A percent decrease of 3% is equivalent to 97% of the original amount, so you want to calculate 97% of $229,000:

This amount rounds down to $222,000.

555.    $9.43

A percent increase of 15% is equivalent to 115% of the original amount, so you want to calculate 115% of $8.20:

556.    $4,866.25

A percent increase of 14.5% is equivalent to 114.5% of the original amount, so you want to calculate 114.5% of $4,250:

557.    3.225 g

A percent increase of 7.5% is equivalent to 107.5% of the original amount, so you want to calculate 107.5% of 3:

558.    $17,690.40

Marian received a 9% discount on an $18,000 car, so calculate the before-tax price as 91% of $18,000:

Then, 8% of this price was added on, so calculate the after-tax price as 108% of $16,380:

559.    8%

Dane invested $7,200 and walked away with $6,624. Make a fraction of these two numbers:

To turn this fraction into a percent, divide the numerator by the denominator; then convert the resulting decimal to a percent:

This result of 92% represents an 8% decrease from the original 100%.

560.    $27.50

A percent increase of 18% is equivalent to 118% of the original amount. Thus, 118% of some number is $32.45, so set up the equation as follows:

Change the percent to a decimal:

Now, divide both sides by 1.18:

Therefore, 118% of $27.50 is $32.45

561.    

Begin by multiplying 1,776 by (recall that , so this multiplication doesn’t change the value of the number):

Now, move the decimal point one place to the left and add 1 to the exponent until the decimal portion of the number is between 1 and 10:

562.    

Begin by multiplying 900,800 by :

Now, move the decimal point one place to the left and add 1 to the exponent until the decimal portion of the number is between 1 and 10:

563.    

Begin by multiplying 881.99 by :

Now, move the decimal point one place to the left and add 1 to the exponent until the decimal portion of the number is between 1 and 10:

564.    

Begin by multiplying 987,654,321 by :

Now, move the decimal point one place to the left and add 1 to the exponent until the decimal portion of the number is between 1 and 10 — that is, 8 places to the left:

565.    

Ten million is 10,000,000. Begin by multiplying 10,000,000 by :

Now, move the decimal point one place to the left and add 1 to the exponent until the decimal portion of the number is between 1 and 10, but not 10 — that is, 7 places to the left:

566.    

Begin by multiplying 0.41 by :

Now, move the decimal point one place to the right and subtract 1 from the exponent until the decimal portion of the number is between 1 and 10:

567.    

Begin by multiplying 0.000259 by :

Now, move the decimal point one place to the right and subtract 1 from the exponent until the decimal portion of the number is between 1 and 10 — that is, 4 places to the right:

568.    

Begin by multiplying 0.001 by :

Now, move the decimal point one place to the right and subtract 1 from the exponent until the decimal portion of the number is between 1 and 10 — that is, 3 places to the right:

569.    

Begin by multiplying 0.0000009 by :

Now, move the decimal point one place to the right and subtract 1 from the exponent until the decimal portion of the number is between 1 and 10 — that is, 7 places to the right:

570.    

One-millionth written as a number is 0.000001. Begin by multiplying 0.000001 by :

Now, move the decimal point one place to the right and subtract 1 from the exponent until the decimal portion of the number is between 1 and 10 — that is, 6 places to the right:

571.    2,400

Move the decimal point 3 places to the right and subtract 3 from the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 2,400

572.    345,000

Move the decimal point 5 places to the right and subtract 5 from the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 345,000

573.    150,000,000 km

Move the decimal point 8 places to the right and subtract 8 from the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 150,000,000

574.    14.6 billion years

Move the decimal point 10 places to the right and subtract 1 from the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 14,600,000,000

This value is equal to 14.6 billion.

575.    31 trillion

Move the decimal point 13 places to the right and subtract 13 from the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 31,000,000,000,000

This value is equal to 31 trillion.

576.    0.075

Move the decimal point 2 places to the left and add 2 to the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 0.075

577.    3 thousandths

Move the decimal point 3 places to the left and add 3 to the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 0.003

This value is equivalent to 3 thousandths.

578.    0.0000254

Move the decimal point 5 places to the left and add 5 to the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 0.0000254

579.    

Move the decimal point 10 places to the left and add 10 to the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 0.0000000008

580.    One ten-millionth

Move the decimal point 7 places to the left and add 7 to the exponent:

Now, drop the 100 entirely, because 100 equals 1:

= 0.0000001

The digit 1 is in the ten millionths place.

581.    

Multiply the decimal portions of the two values and multiply the powers of 10 by adding the exponents.

582.    

Multiply the decimal portions of the two values and multiply the powers of 10 by adding the exponents.

583.    

Multiply the decimal portions of the two values and multiply the powers of 10 by adding the exponents.

584.    

Multiply the decimal portions of the two values and add the exponents:

585.    

Multiply the decimal portions of the two values and multiply the powers of 10 by adding the exponents.

Now, move the decimal point one place to the left and add 1 to the exponent:

586.    

Multiply the decimal portions of the two values and multiply the powers of 10 by adding the exponents.

Now, move the decimal point one place to the left and add 1 to the exponent:

587.    

Multiply the decimal portions of the two values and add the exponents:

Now, move the decimal point one place to the left and add 1 to the exponent:

588.    

Multiply the decimal portions of the two values and add the exponents:

Now, move the decimal point one place to the left and add 1 to the exponent:

589.    

Multiply the decimal portions of the three values and add the exponents:

Now, move the decimal point one place to the left and add 1 to the exponent:

590.    

Multiply the decimal portions of the three values and add the exponents:

Now, move the decimal point two places to the left and add 2 to the exponent:

591.    156

Convert 13 feet into inches by multiplying by 12:

592.    1,080

Convert 18 hours into minutes by multiplying by 60:

593.    240

Convert 15 pounds into ounces by multiplying by 16:

594.    220

Convert 55 gallons into quarts by multiplying by 4:

595.    190,080

First, convert 3 miles into feet by multiplying by 5,280:

Next, convert 15,480 feet into inches by multiplying by 12:

596.    416,000

First, convert 13 tons into pounds by multiplying by 2,000:

Next, convert 26,000 pounds into ounces by multiplying by 16:

597.    604,800

A week contains 7 days. To convert 7 days to hours, multiply 7 by 24:

To convert 168 hours to minutes, multiply 168 by 60:

To convert 10,080 minutes to seconds, multiply by 60:

598.    2,176

First, convert 17 gallons into quarts by multiplying by 4:

Next, convert 68 quarts into cups by multiplying by 4:

Finally, convert 272 cups into fluid ounces by multiplying by 8:

599.    46,112

First, convert 26.2 miles into feet by multiplying by 5,280:

Next, convert 138,336 feet into yards by dividing by 3:

600.    166,368,000

First, convert 5,199 tons into pounds by multiplying by 2,000:

Next, convert 10,398,000 pounds into ounces by multiplying by 16:

601.    2,522,880,000

A year contains 365 days. To convert 80 years to days, multiply 80 by 365:

To convert 29,200 days to hours, multiply 29,200 by 24:

To convert 700,800 hours to minutes, multiply by 60:

To convert 42,048,000 minutes to seconds, multiply by 60:

602.    11,520

A raindrop is fluid ounces, so a fluid ounce contains 90 raindrops. Multiply 90 by 8 to find the number of raindrops in a cup:

Now, multiply 720 by 4 to find the number of raindrops in a quart:

Finally, multiply 2,880 by 4 to find the number of raindrops in a gallon:

603.    33

First, convert yards into feet by multiplying by 3:

Next, divide 5,280 by 160:

604.    25,000

A liter contains 1,000 milliliters, so 25 liters contains 25,000 milliliters.

605.    800,000,000

A megaton contains 1,000,000 tons, so 800 megatons contains 800,000,000 tons.

606.    30,000,000,000

A second contains 1,000,000,000 (one billion) nanoseconds, so 30 seconds contains 30,000,000,000 (30 billion) nanoseconds.

607.    1,200,000

A kilometer contains 1,000 meters, so 12 kilometers has 12,000 meters. And a meter contains 100 centimeters, so multiply 12,000 meters by 100:

608.    17,000,000,000

A megagram contains 1,000,000 grams, so 17 megagrams has 17,000,000 grams. And a gram contains 1,000 milligrams, so multiply 17,000,000 grams by 1,000:

609.    

A gigawatt contains 1,000,000,000 watts, so 900 gigawatts has 900,000,000,000 watts. But it takes 1,000 watts to make up a kilowatt, so divide 900,000,000,000 watts by 1,000:

To change this number to scientific notation, move the decimal point 8 places to the left and multiply by :

610.    

A megadyne contains 1,000,000 dynes, so 88 megadynes has 88,000,000 dynes. And a dyne has 1,000,000 microdynes, so multiply 88,000,000 dynes by 1,000,000:

To change this number to scientific notation, move the decimal point 10 places to the left and multiply by :

611.    

A terameter contains 1 trillion meters (), so multiply 333 by :

A meter contains 1,000 millimeters (), so multiply this result by :

Convert to scientific notation by moving the decimal point 2 places to the left and adding 2 to the exponent:

612.    

A microsecond is one-millionth of a second, which is equivalent to . Thus, multiply this amount by 567,811:

To convert to scientific notation, move the decimal point five places to the right and add 5 to the exponent:

613.    

A nanogram contains 1,000 () picograms, and a gram contains 1 billion () nanograms, so multiply these two numbers to get the number of picograms in a gram:

A teragram contains 1 trillion () grams, so multiply this number by the preceding result to get the number of picograms in a teragram:

Finally, a petagram contains 1,000 () teragrams, so multiply this number by the preceding result to get the number of picograms in a petagram:

614.    5

A kilobyte is 1,000 bytes, so a computer that can download 5 kilobytes of information in a nanosecond can download 5,000 bytes in a nanosecond. And a second contains 1 billion nanoseconds, so the number of bytes the computer can download in one second is

However, there are 1 trillion bytes in a terabyte, so divide 5,000,000,000,000 by 1,000,000,000,000:

615.    

Use the formula for converting Celsius to Fahrenheit:

Evaluate:

= 90 + 32 = 122

616.    

Use the formula for converting Fahrenheit to Celsius:

Evaluate:

617.    

Use the formula for converting Fahrenheit to Celsius:

Evaluate:

618.    

Use the formula for converting Fahrenheit to Celsius:

Evaluate:

619.    

Use the formula for converting Celsius to Fahrenheit:

Evaluate:

2,763 + 32 = 2,795

620.    

Use the formula for converting Celsius to Fahrenheit:

Evaluate:

–491.67 + 32 = –459.67

621.    10 miles

1 kilometer equals approximately one-half mile, so 1 mile equals approximately 2 kilometers. Multiply 20 by 2:

622.    48 liters

1 liter equals approximately gallon, so 1 gallon equals approximately 4 liters. Multiply 12 by 4:

623.    90 kilograms

1 kilogram equals approximately 2 pounds, so 1 pound equals approximately kilogram. Multiply 180 by :

624.    2,484 feet

1 meter is approximately equal to 3 feet, so multiply 828 by 3:

625.    20 meters

1 meter equals approximately 3 feet, so 1 foot equals approximately meter. Multiply 60 by :

626.    10,000 pounds

1 kilogram equals approximately 2 pounds, so multiply 5,000 by 2:

627.    140 kilometers

To begin, calculate the total distance in miles for 5 miles a day, 7 times per week, for 2 weeks:

Thus, the total distance is 70 miles. 1 kilometer is approximately equal to ¹⁄₂ mile, so multiply 70 by 2:

628.    95 gallons

First, calculate how many liters of gasoline the commuter puts in her car in 4 weeks by multiplying 95 by 4:

One liter is approximately equal to gallon, so mulitiply 380 by :

629.    60 meters

Begin by finding the length of the swimming pool in kilometers. To do this, multiply 2 (the number of kilometers in a mile) by :

Thus, the swimming pool is kilometer in length. One kilometer is equal to 1,000 meters, so multiply by 1,000:

So rounded to the nearest 10 meters, the pool is approximately 60 meters.

630.    1.28 fluid ounces

To begin, convert 40 milliliters to liters by dividing 40 by 1,000:

One liter equals approximately 1 quart, so 0.04 liter equals approximately 0.04 quart. To convert 0.04 quart to cups, multiply by 4:

To convert 0.16 cup to fluid ounces, multiply 0.16 by 8:

631.    140

The measures of two angles that result in a straight line always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 40 = 140

632.    130

When two lines intersect, the resulting vertical (opposite) angles are always equivalent. Therefore, n = 130.

633.    63

The measures of two angles that result in a straight line always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 117 = 63

634.    61

The measures of three angles that result in a straight line always add up to 180 degrees. A right angle has 90 degrees, so to find n, subtract as follows:

n = 180 – 90 – 29 = 61

635.    14

The measures of three angles that result in a straight line always add up to 180 degrees. A square has four right angles, and a right angle measures 90 degrees. Thus, to find n, subtract as follows:

n = 180 – 90 – 76 = 14

636.    65

The measures of three angles of a triangle always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 73 – 42 = 65

637.    91

The measures of two angles that result in a straight line always add up to 180 degrees. A square has four right angles, and a right angle measures 90 degrees. Thus, to find p, subtract as follows:

p = 180 – 158 = 22

The measures of the three angles of a triangle always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 67 – 22 = 91

638.    14.5

The measures of the two smaller angles of a right triangle always add up to 90 degrees. Thus, to find n, subtract as follows:

n = 90 – 75.5 = 14.5

639.    61.6

A rectangle has four right angles, each of which measures 90 degrees. Thus, to find n, subtract as follows:

n = 90 – 28.4 = 61.6

640.    75.4

The measures of the four angles of a quadrilateral (four-sided polygon) always total 360 degrees. A right angle measures 90 degrees, so to find n, subtract as follows:

n = 360 – 90 – 108.2 – 86.4 = 75.4

641.    57.75

When two lines are parallel, all corresponding angles are equivalent. Thus, you can determine the following:

The measures of two angles that result in a straight line always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 122.25 = 57.75

642.    86.6

The measures of the five angles of a pentagon (five-sided polygon) always total 540 degrees. A right angle measures 90 degrees, so to find n, subtract as follows:

n = 540 – 90 – 118.3 – 83.9 – 161.2 = 86.6

643.    88.2

The measures of two angles that result in a straight line always add up to 180 degrees. Thus, to find p, subtract as follows:

p = 180 – 134.1 = 45.9

An isosceles triangle has two equivalent angles, so you can draw the following:

The measures of the three angles of a triangle always add up to 180 degrees. Thus, to find n, subtract as follows:

n = 180 – 45.9 – 45.9 = 88.2

644.    70.9

When a triangle is inscribed in a circle such that one side of the triangle is a diameter of that circle, the opposite angle of that triangle is a right angle. Thus, ABC is a right triangle, so its two smaller angles add up to 90 degrees. Thus, to find n, subtract as follows:

n = 90 – 19.1 = 70.9

645.    69.75

BCDE is a parallelogram, so and are parallel. Thus, angle BCE and angle BEA are equivalent, so angle BEA = 40.5.

, so triangle BEA is isosceles. Thus the two remaining angles in this triangle are equivalent, so both measure n degrees. And the measures of the three angles in a triangle always add up to 180. Therefore, to find n, use the following equation:

180 = 40.5 + 2n

139.5 = 2n

69.75 = n

646.    36 square inches

Use the formula for the area of a square:

647.    28 meters

Use the formula for the perimeter of a square:

648.    10,201 square miles

Use the formula for the area of a square:

649.    13.6 centimeters

Use the formula for the perimeter of a square:

650.    21 feet

Use the formula for the perimeter of a square, plugging in 84 for the perimeter; then solve for s.

651.    48 feet

Begin by using the formula for the area of a square to find the side of the square. Plug in 144 for the area and solve for s.

Now, plug in 12 for s into the formula for the perimeter of a square:

652.    240.25 square feet

Begin by using the formula for the perimeter of a square to find the side of the square. Plug in 62 for the perimeter and solve for s.

Now, plug 15.5 for s into the formula for the area of a square:

653.    60 feet

Begin by plugging in 25 as the area into the formula for the area of a square () and solve for s.

Therefore, the side of the room is 5 yards. Convert yards to feet by multiplying by 3.

5 yards = 15 feet

Now, plug 15 into the formula for the perimeter of a square:

Therefore, the perimeter of the room is 60 feet.

654.    250,905,600 square feet

Begin by using the formula 1 mile = 5,280 feet to convert from miles to feet.

Thus, the side of the square field is 15,840 feet. Plug this into the formula for the area of a square:

655.    0.4 kilometers

The perimeter of the park is 10 times greater than its area, so:

P = 10A

The perimeter of a square is 4s, so substitute this value for P into the equation above:

4s = 10A

The area of a square is , so substitute this value for A into the preceding equation:

To solve for s, begin by dividing both sides by s.

4 = 10s

Now, divide both sides by 10.

, or 0.4.

656.    24 square centimeters

Use the formula for the area of a rectangle:

657.    36 meters

Use the formula for the perimeter of a rectangle.

Simplify.

= 32 + 4 = 36

658.    11.61 square feet

Use the formula for the area of a rectangle.

659.     inches

Use the formula for the perimeter of a rectangle.

Evaluate by canceling factors of 2:

Convert this improper fraction to a mixed number:

660.    155.25 square inches

Use the formula for the area of a rectangle.

661.     inches

Use the formula for the area of a rectangle.

Simplify by factoring.

662.    50 feet

Begin by using the formula for the area of a rectangle, plugging in 100 for the area and 5 for the width:

Divide both sides by 5.

20 = l

Thus, the length is 20. Now, use the formula for the perimeter of a rectangle, plugging in 20 for the length and 5 for the width.

Evaluate.

= 40 + 10 = 50

663.     inches

Begin by using the formula for the area of a rectangle, plugging in 30 for the area and 8 for the length.

Divide both sides by 8.

Thus, the width is . Now, use the formula for the perimeter of a rectangle, plugging in 8 for the length and for the width.

Evaluate:

664.    61 inches

Begin by using the formula for the area of a rectangle, plugging in 156 for the area and 24 for the length (because 2 feet = 24 inches).

Divide both sides by 24.

6.5 = w

Thus, the width is 6.5. Now, use the formula for the perimeter of a rectangle, plugging in 24 for the length and 6.5 for the width.

Evaluate.

= 48 + 13 = 61

665.    24

If the area of a rectangle is 72 and both the length and width are whole numbers, you can write down all the possible lengths and widths as factor pairs of 72.

To begin, find all the factors of 72.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Now, plug each of these pairs into the formula for the perimeter of a rectangle () until you find one that produces a perimeter of 54.

Therefore, the length and width are 24 and 3.

666.    45

Use the formula for a parallelogram.

667.    3,102.7

Use the formula for a parallelogram.

668.    

Use the formula for a parallelogram.

Evaluate by converting both mixed numbers to improper fractions and then multiplying.

669.    20

Use the formula for the area of a trapezoid.

Simplify the fraction.

670.    85.32

Use the formula for the area of a trapezoid.

Simplify the fraction.

671.    

Use the formula for the area of a trapezoid.

To simplify, begin by multiplying the two fractions.

Next, add the two fractions in the numerator.

Now, evaluate this fraction by turning it into fraction division.

672.    13.5 centimeters

Use the formula for a parallelogram, plugging in 94.5 for the area and 7 for the base.

Divide both sides by 7.

13.5 = h

673.    12

Begin by plugging the area and bases into the formula for a trapezoid.

Simplify the fraction.

Now, divide both sides by 15.

12 = h

674.    

Use the formula for a parallelogram, plugging in for the area and for the base.

Multiply both sides by .

675.    25.5

Begin by plugging the area, height, and base into the formula for a trapezoid.

Divide both sides by 3, then multiply both sides by 2.

Now, subtract 4.5 from both sides.

676.    36 square inches

Use the formula for the area of a triangle () to solve the problem.

677.    34.5 square meters

Use the formula for the area of a triangle () to solve the problem.

678.    

Use the formula for the area of a triangle () to solve the problem.

Cancel common factors in the numerator and denominator.

679.    110.5

Use the formula for the area of a triangle () to solve the problem.

680.    99

In a right triangle, the lengths of the two legs (that is, the two short sides) are the base and height. Use the formula for the area of a triangle () to solve the problem.

681.    24 square centimeters

In a right triangle, the lengths of the two legs (that is, the two short sides) are the base and height. Use the formula for the area of a triangle () to solve the problem.

682.    30 meters

Use the formula for the area of a triangle () to solve the problem, plugging in 60 for the area and 4 for the height.

To solve for the base b, first multiply by 4 on the right side of the equation; then solve for b.

683.    13

Use the formula for the area of a triangle () to solve the problem, plugging in 78 for the area. Be sure to convert the base to inches: 1 foot = 12 inches.

To solve for the height h, first multiply by 12 on the right side of the equation, then solve for h by dividing both sides by 6.

684.     inches

Use the formula for the area of a triangle () to solve the problem, plugging in for the base and for the area.

To solve for the height h, first multiply by on the right side of the equation.

Now, multiply both sides of the equation by .

To finish, reduce the fraction and change it to a mixed number.

685.    13

To begin, use the formula for the area of a triangle (), plugging in 84.5 for the area:

Multiply both sides by 2 to get rid of the fraction.

169 = bh

The base and height are the same, so you can use the same variable h for both of these values. Therefore, , so you can substitute for bh in the preceding equation:

Solve for h by taking the square root of both sides.

686.    5 feet

Use the Pythagorean Theorem () to find the hypotenuse:

687.    26 centimeters

Use the Pythagorean Theorem () to find the hypotenuse:

To finish, take the square root of both sides of the equation:

688.    

Use the Pythagorean Theorem () to find the hypotenuse:

Take the square root of both sides of the equation.

Simplify by factoring as follows:

689.    

Use the Pythagorean Theorem () to find the hypotenuse:

Take the square root of both sides of the equation.

690.    

Use the Pythagorean Theorem () to find the hypotenuse:

Take the square root of both sides of the equation.

691.    

Use the Pythagorean Theorem () to find the hypotenuse:

Evaluate the left side of the equation.

Take the square root of both sides of the equation.

692.    1

Use the Pythagorean Theorem () to find the hypotenuse:

Evaluate the left side of the equation using the following steps:

693.    

Use the Pythagorean Theorem () to find the hypotenuse:

Evaluate the left side of the equation.

Take the square root of both sides of the equation.

694.    40

Use the Pythagorean Theorem () to find the length of the longer leg:

Subtract 5,625 from both sides, then take the square root of both sides of the equation.

695.    

Use the Pythagorean Theorem () to find the length of the longer leg:

Subtract 49 from both sides, then take the square root of both sides of the equation.

Simplify by factoring as follows:

696.    16

Use the formula for the diameter of a circle:

697.    

Use the formula for the area of a circle:

698.    

Use the formula for the circumference of a circle:

699.    

Use the formula for the area of a circle:

700.    

Use the formula for the circumference of a circle:

701.    

The formula for the diameter of a circle is D = 2r, and the formula for the circumference is . Notice that the only difference between the diameter of a circle (D = 2r) and its circumference () is a factor of . So, a quick way to change the diameter to the circumference is simply to multiply by .

Therefore, if a circle has a diameter of 99, its circumference is .

702.    

The formula for the diameter of a circle is D = 2r, and the formula for the circumference is . Notice that the only difference between the diameter of a circle (D = 2r) and its circumference () is a factor of . So, a quick way to change the diameter to the circumference is simply to multiply by .

Therefore, if a circle has a diameter of , its circumference is .

703.    

A circle with a diameter of 100 has a radius of 50 (because D = 2r). Plug this value into the formula for the area of a circle:

704.    9

Use the formula for the area of a circle, plugging in for the area:

Divide both sides of the equation by .

Now, take the square root of each side.

705.    33

Use the formula for the circumference of a circle, plugging in for the circumference:

Divide both sides of the equation by and then by 2.

66 = 2r

33 = r

706.    

To begin, find the radius using the formula for the circumference of a circle, plugging in for the circumference:

Divide both sides of the equation by and then by 2.

10.8 = 2r

5.4 = r

Now, use the area formula, plugging in 5.4 for the radius:

707.    

To begin, find the radius using the formula for the area of a circle, plugging in for the area:

Divide both sides of the equation by .

Now, take the square root of both sides.

Now, use the circumference formula, plugging in for the radius:

708.    

Use the formula for the area of a circle, plugging in 16 for the area:

Divide both sides of the equation by .

Now, take the square root of both sides.

709.    

To begin, find the radius using the formula for the circumference of a circle, plugging in 18.5 for the circumference:

Divide both sides of the equation by 2 and then by .

Now, use the area formula, plugging in for the radius:

To finish, first evaluate the power.

Now, cancel a factor of in both the numerator and denominator.

710.    1,728 cubic inches

Use the formula for the volume of a cube:

Evaluate as follows:

711.    421.875

Use the formula for the volume of a cube:

Evaluate as follows:

712.    100 inches

Use the formula for the volume of a cube, plugging in 1,000,000 for the volume:

To find the value of s, you want to find a number which, when multiplied by itself 3 times, equals 1,000,000. A little trial and error makes this obvious:

713.    600 cubic inches

Use the formula for the volume of a box:

714.    327.25 cubic inches

Use the formula for the volume of a box:

715.     cubic inches

Use the formula for the volume of a box:

716.    5 centimeters

Use the formula for a box, plugging in the value 20,000 for the volume, 80 for the length, and 50 for the width:

Simplify and divide both sides by 4,000.

717.    0.0456 inches

Use the formula for a box, plugging in the value 45.6 for the volume, 10 for the length, and 100 for the height:

Simplify and divide both sides by 1,000.

718.     cubic feet

Use the formula for the volume of a cylinder:

Simplify.

719.    

Use the formula for the volume of a cylinder:

Evaluate.

720.     cubic meters

Use the formula for the volume of a cylinder:

Simplify:

721.     cubic inches

Use the formula for the volume of a cylinder:

Simplify.

722.    6.5 feet

Use the formula for the volume of a cylinder, plugging in for the volume and 3 for the radius:

Divide both sides by and then by 9.

723.    

Use the formula for the volume of a sphere:

Cancel a factor of 3 in both the numerator and denominator and then simplify.

724.    

Use the formula for the volume of a sphere:

Evaluate the power, cancel factors where possible, and then multiply:

725.     cubic meters

Use the formula for the volume of a sphere:

Evaluate the power.

Now, multiply by 1.728. You can do this in two steps: First multiply by 4 and then divide by 3:

726.     foot

Using the formula for a sphere, plug in as the volume:

Divide both sides of the equation by .

Now, multiply both sides of the equation by .

The radius r is a number which, when multiplied by itself 3 times, equals . This number is , because:

727.    32 cubic inches

Use the formula for the volume of a pyramid:

Evaluate the power and cancel a factor of 3 in the numerator and denominator.

728.    4 meters

Use the formula for the volume of a pyramid, plugging in 80 for the volume and 15 for the height:

Cancel a factor of 3 in the numerator and denominator; then divide both sides of the equation by 5.

Now, take the square root of both sides of the equation.

729.     cubic inches

Use the formula for the volume of a cone:

Evaluate the power and cancel a factor of 3 in the numerator and denominator.

730.    11

Use the formula for the volume of a cone, plugging in for the volume and 6 for the radius:

Evaluate the power, cancel a factor of 3 in the numerator and denominator, and then divide both sides of the equation by .

731.    Kent

Brian collected $300 and Kent collected $500, so Kent collected $200 more than Brian.

732.    $1,800

Arianna collected $600, Eva collected $800, and Stella collected $400. Therefore, together they collected $600 + $800 + $400 = $1,800.

733.    

Stella collected $400. The total amount collected was $600 + $300 + $800 + $500 + $400 + $1,000 = $3,600. Make a fraction of these two amounts as follows:

734.    2:5

Stella collected $400 and Tyrone collected $1,000. To find the ratio, make a fraction of these two numbers and reduce it:

Therefore, Stella and Tyrone collected funds in a 2:5 ratio.

735.    Kent

Eva collected $800, so if she had collected $300 less, she would have collected $500. Kent collected $500.

736.    44%

Arianna collected $600 and Tyrone collected $1,000, so together they collected $1,600. The total amount collected was $3,600 (see Answer 733). Make a fraction of these two amounts:

Now, divide to convert this fraction into a repeating decimal and then into a percent:

737.    Biochemistry and Economics

Biochemistry accounts for 35% of Kaitlin’s study time, and Economics accounts for 15%. Together, these account for 50% of her time.

738.    Calculus, Economics, and Spanish

Calculus accounts for 20% of Kaitlin’s study time, Economics 15%, and Spanish 20%. Together, these account for 55% of her time.

739.    4 hours

Kaitlin spends 20% of her time studying for Spanish. Thus, if she spent 20 hours last week studying, she spent 20% of 20 hours studying for Spanish:

Therefore, she spent 4 hours studying for Spanish.

740.    30 hours

Kaitlin spent 20% of her time studying for Calculus and 15% of her time studying for Economics. Thus, she spent 5% more time studying for Calculus than Economics. So if 5% of her time represented 1.5 hours, multiplying this value by 20 would represent 100% of her studying time (because 5% times 20 = 100%):

Therefore, Kaitlin spent 30 hours studying.

741.    30 hours

Kaitlin spends 10% of her time studying for Physics. Thus, if she spent 3 hours studying for this class, she spent 10 times more than that studying for all of her classes. Therefore, she spent 30 hours studying for all of her classes.

742.    4 hours and 40 minutes

Kaitlin spends 15% of her time studying for Economics. If this accounted for 2 hours, then 1/3 of this time – that is, 40 minutes – would account for 5% of her time. Then, multiplying this amount of time by 7 (40 minutes × 7 = 280 minutes) would account for 35% of her time. Thus, Kaitlin spent 280 minutes studying for biochemistry, which equals 4 hours and 40 minutes.

743.    October

Net profit was $2,800 in February and the same in October.

744.    $8,800

Net profit for January, February, and March was $2,400 + $2,800 + $3,600 = $8,800.

745.    March

Between March and April, net profit increased by $4,400 – $3,600 = $800. This is equivalent to the profit shown in March when compared with February ($800), but greater than the increase in profit shown in February ($400), May (decrease in profit), June ($400), July ($400), August ($400), September and October (decrease in profit), November ($400) or December ($400).

746.    August and September

In August and September, the combined net profit was $5,200 + $3,600 = $8,800.

747.    January

To begin, calculate the total net profit for the year:

$2,400 + $2,800 + $3,600 + $4,400 + $4,000 + 4,400 + $4,800 + $5,200 + $ 3,600 + $2,800 + $3,200 + $3,600 = $44,800

Now, calculate 5% of $44,800:

The nearest net profit to $2,240 was $2,400, in January.

748.    22,000

Plattfield is the largest town in Alabaster County. Its population is equivalent to 11 stick figures, each of which represents 2,000 people, so its population is .

749.    Talkingham

To begin, find the total population of the county:

9,000 + 12,000 + 15,000 + 22,000 + 6,000 + 14,000 = 78,000

Now, calculate of 78,000:

Talkingham has a population of 14,000, which is slightly more than 13,000.

750.    19%

As calculated in Answer 749, the entire county has a population of 78,000. Morrissey Station has a population of 15,000. Make a fraction of these two numbers as follows:

Change this fraction to a decimal by dividing , and then change the decimal to a percent as follows:

751.    Barker Lake and Talkingham

Plattfield has a population of 22,000 people. Barker Lake has a population of 9,000 and Talkingham has a population of 14,000. Therefore, together, Barker Lake and Talkingham have a combined population of 9,000 + 14,000 = 23,000, which is 1,000 greater than the population of Plattfield.

752.    20%

The population of Talkingham is 14,000. If it increased by 2,000 (one stick figure), then its population would be 16,000. And if all the other towns remained constant in their population, then the population of the county would also rise by 2,000 people, from 78,000 (see Answer 749) to 80,000.

Make a fraction from these two numbers:

753.    69%

The two top candidates were Bratlafski with 41% and McCullers with 28%, so together they received 69%.

754.    Farelese and McCullers

Faralese received 7% of the vote and McCullers received 28%, so together they received 7% + 28% = 35%.

755.    3,000

Faralese received 7% of the vote, and Williamson received 4%. If 100,000 votes were cast, Faralese received 7,000 votes (0.07 × 100,000) and Williamson received 4,000 (0.04 × 100,000). Therefore, Faralese received 3,000 more votes than Williamson.

756.    200,000

Jordan received 17% of the vote, so if she had received 34,000 votes, each percentage point would count for:

Thus, if each percentage point counted for 2,000 votes, 100% of the vote would be 200,000 votes.

757.    39,200

Bratlaski received 41% of the vote and Pardee received 3%. Thus, Bratlaski received 38% more votes than Pardee. If 38% of the vote represented 53,200 votes, each percentage point would count for:

Thus, if each percentage point counted for 1,400 votes, McCullers’ share of 28% of the vote would be:

758.    2,500

Seven hundred and fifty trees were planted in Edinburgh County and 1,750 in Manchester County, so together there were 750 + 1,750 = 2,500 trees.

759.    8.000

The total number of trees was as follows:

1,500 + 500 + 2,250 + 750 + 1,250 + 1,750 = 8,000

760.    Dublin and Manchester

In Answer 759, the total number of trees among the six counties is calculated at 8,000. Thus, 50% of the trees is 4,000. Dublin accounts for 2,250 trees and Manchester accounts for 1,750, so together this accounts for 2,500 + 1,750 = 4,000.

761.    Birmingham

The total number of trees was 8,000 (see Answer 759). Thus, 18.75% of the trees is

Fifteen hundred trees were planted in Birmingham County.

762.    

The total number of trees was 8,000 (see Answer 759), of which 500 were in Calais County. If 1,000 additional trees had been planted in Calais County, then 1,500 trees would have been planted there out of a total of 9,000. Make a fraction from these two numbers and reduce:

763.    See below.

i. Q

ii. S

iii. R

iv. P

v. T

764.    6

Q = (1, 6). To go from (0, 0) to (1, 6), you need to go

up 6, over 1

Translate these words as follows:

+ 6 / 1

Thus, the slope of the line that passes through both the origin and Q is

765.    

S = (–3, –1). To go from (–3, –1) to (0, 0), you need to go

up 1, over 3

Translate these words as follows:

+ 1 / 3

Thus, the slope of the line that passes through both the origin and S is

766.    –1

P = (3, 4) and Q = (1, 6). To go from (1, 6) to (3, 4), you need to go

down 2, over 2

Translate these words as follows:

– 2 / 2

Thus, the slope of the line that passes through both P and Q is

767.    

R = (–2, 5) and T = (5, –3). To go from (–2, 5) to (5, –3), you need to go

down 8, over 7

Translate these words as follows:

– 8 / 7

Thus, the slope of the line that passes through both R and T is

768.    

S = (–3, –1), T = (5, –3). To go from (–3, –1) to (5, –3), you need to go

down 2, over 8

Translate these words as follows:

– 2 / 8

Thus, the slope of the line that passes through both R and T is

769.    5

To begin, draw a right triangle with the line that you want to measure as the hypotenuse:

Now, notice that the horizontal leg of this triangle has a length of 3, and the vertical leg has a length of 4. Thus, this is a 3-4-5 right triangle, so the distance between the origin and P is 5.

770.    

To begin, draw a right triangle with the line that you want to measure as the hypotenuse:

Now, notice that the horizontal leg of this triangle has a length of 1, and the vertical leg has a length of 6. Use the Pythagorean theorem to measure the length of the hypotenuse:

Thus, the distance between R and S is .

771.    8

To find the average, use the following formula:

Simplify.

772.    26

To find the average, use the following formula:

Simplify.

773.    1,411

To find the average, use the following formula:

Simplify.

774.    48.2

To find the average, use the following formula:

Simplify.

775.    9.1

To find the average, use the following formula:

Simplify.

776.    307.418

To find the average, use the following formula:

Simplify.

777.    

To begin, find the sum of and .

Now, plug this result into the numerator of the formula for the mean, with 2 as the denominator (because you’re finding the average of two items).

Now, simplify the complex fraction by turning it into fraction division.

Simplify by factoring out 2 from both the numerator and denominator; then multiply.

778.    

To begin, find the sum of , , and . To do this, turn all three mixed numbers into improper fractions.

Now, increase the terms of all three fractions to a common denominator of 30.

Next, plug this result into the numerator of the formula for the mean, with 3 as the denominator (because you’re finding the average of 3 items).

Now, simplify the complex fraction by turning it into fraction division.

Turn this improper fraction into a mixed number by dividing 421 by 90.

779.    $65

Kathi worked three days, averaging $60 for the three days. So she earned a total of from Monday to Wednesday. She earned $40 on Monday and $75 on Tuesday, so subtract these amounts from $180 to find what she earned on Wednesday.

Therefore, Kathi earned $65 on Wednesday.

780.    9 miles

Antoine hiked for 4 days at an average of 7 miles per day, so he hiked miles altogether. Subtract the distances that he hiked on the first three days from 28.

28 – 8 – 4.5 – 6.5 = 9

Therefore, Antoine hiked 9 miles on the last day.

781.    

The caterpillar crawled an average of inches in 5 minutes, so multiply to find its total distance.

So, it traveled a total of inches in 5 minutes. It crawled inches in the first four minutes, so subtract to find out how far it traveled in the last minute.

Therefore, it crawled inches in the last minute.

782.    8 hours and 40 minutes

Eleanor studied an average of 9 hours per day for 7 days, so she studied a total of hours over the 7 days.

On the last day, she studied for 4 hours. On the 3 days before this, she studied for an average of 11 hours per day, so she studied for hours. So, subtract these two values from 63.

63 – 4 – 33 = 26

Thus, she studied for a total of 26 hours on the first 3 days of the week. To find the average for these three days, divide 26 by 3.

Thus, she studied an average of hours over the first three days of the week. This equals 8 hours and 40 minutes.

783.    17

To calculate the weighted mean, first calculate the sum of products for the five classes.

Now use this result as the numerator in the formula for the mean and divide by 5.

Therefore, the average class size is 17 students.

784.    8.75 minutes

To calculate the weighted mean, first calculate the sum of products for the eight speeches.

Now use this result as the numerator in the formula for the mean and divide by 8.

Therefore, the average speech length was 8.75 minutes.

785.    $316

To calculate the weighted mean, first calculate the sum of products for the 10 weeks.

Now use this result as the numerator in the formula for the mean and divide by 10.

Therefore, Jake’s average weekly income was $316.

786.    $783

To calculate the weighted mean, first calculate the sum of products for the 12 months.

Now use this result as the numerator in the formula for the mean and divide by 12.

Therefore, the average savings is about $783.

787.    8 minutes and 3 seconds

Plug Angela’s total time into the formula for the mean and divide by the total number of laps she ran, which was 10.

Evaluate.

Therefore, Angela’s average time was 8 minutes and 3 seconds.

788.    8.5

To calculate the weighted mean, first calculate the sum of products for the 12 tests.

Now use this result as the numerator in the formula for the mean and divide by 12.

Therefore, Kevin’s average score was 8.5.

789.    9.4 feet

To calculate the weighted mean, first calculate the sum of products for the 20 floors.

Now use this result as the numerator in the formula for the mean and divide by 20.

Therefore, the average height is 9.4 feet.

790.    350

To calculate the weighted mean, first calculate the sum of products for the puzzles.

Now add up the number of days that she took to do these puzzles.

3 + 7 + 6 = 16

Use these results as the numerator and denominator in the formula for the mean.

Therefore, she put together an average of 350 pieces per day.

791.    65 mph

To calculate the weighted mean, first calculate the sum of products for the 4 legs of the trip (be sure to convert minutes to hours).

Now use this result as the numerator in the formula for the mean and divide by the total time for the trip (0.75 hours + 1.5 hours + 1.25 hours + 1 hour = 4.5 hours).

Therefore, Gerald’s average rate was 65 miles per hour.

792.    15

The median number of any data set with an odd number of values is the middle number (when the numbers are in order). In this case, the median is 15.

793.    41

The median number of any data set with an even number of values is the mean of the two middle numbers (when the numbers are in order). In this case, the middle numbers are 37 and 45, so find the mean as follows:

.

Simplify.

Therefore, the median is 41.

794.    16

The mode of a data set is the value that occurs most frequently. In this case, 16 occurs three times, so this is the mode.

795.    0.5

The median number of any data set with an even number of values is the mean of the two middle numbers when the numbers are listed in ascending (or descending) order. In this case, the middle numbers are 5 and 6, so find the mean as follows:

.

Simplify.

Therefore, the median is 5.5. The mode is the value that occurs most frequently in the data set, so the mode is 5. Thus, the difference between the median and the mode is 5.5 – 5 = 0.5.

796.    13

Calculate the mean using the formula.

Simplify:

Therefore, the mean is 15. The median is the middle number, which is 12. The two modes are the numbers that occur most frequently in the data set, which are 11 and 14. Therefore, 13 (the only integer between 11 and 15 that has not been ruled out) is not the mean, the median, or a mode of the data set.

797.    36

To calculate the number of combinations, multiply the number of possible outcomes for each die. Because there are six sides on each die, there are six different outcomes for each.

798.    1,920

To calculate the number of combinations, multiply the number of possible outcomes for each die.

799.    56

To calculate the number of combinations, multiply the number of suits, shirts, and ties.

Therefore, Jeff had 56 possible combinations of suit, shirt, and tie.

800.    96

To calculate the number of combinations, multiply the number of types of eggs, meat, potatoes, and beverages.

Therefore, 96 breakfast combinations are possible.

801.    1,024

There are ten questions, each of which can be answered either yes or no (two possible ways each), so calculate as follows.

802.    17,576

Each of the 3 letters could be any of the 26 letters, so calculate as follows:

803.    1,679,616

Each of the four symbols could be any of the 10 digits or 26 letters, so there are 36 symbols in all. Calculate as follows:

804.    24

The first letter can be any of the four possible letters (A, B, C, or D). The second can be any of the three remaining letters. The third can be either of the two remaining letters. Finally, the last letter can only be the one remaining letter tile. Multiply these four numbers together to find the total number of possible outcomes.

Therefore, there are 24 different ways to pull four different letters from a bag.

805.    120

The first person to arrive could be any of the five people. The second person could be any of the four remaining. The third could be any of the three remaining. The fourth could be either of the two remaining. And the fifth must be the one person remaining. Calculate the total number of possible outcomes by multiplying.

806.    720

The first topping could be any of the six. The second could be any of the remaining five. The third could be any of the remaining four. The fourth could be any of the remaining three. The fifth could be either of the remaining two. And the sixth must be the one remaining topping. Calculate the total number of possible outcomes by multiplying.

807.    40,320

The first book could be any of the eight. The second could be any of the remaining seven. The third could be any of the remaining six. The fourth could be any of the remaining five. The fifth could be any of the remaining four. The sixth could any of the remaining three. The seventh could be either of the remaining two. And the eighth must be the one remaining book. Calculate the total number of possible outcomes by multiplying.

808.    6,840

The pitcher could be any of the 20 children. The catcher could be any of the remaining 19 children. And the runner could be any of the remaining 18 children. Calculate the total number of possible outcomes by multiplying.

809.    15,600

The first letter could be any of the 26 letters. The second could be any of the remaining 25 letters. And the third could be any of the remaining 24 letters. Calculate the total number of possible outcomes by multiplying.

810.    132,600

The first card could be any of the 52 cards. The second could be any of the remaining 51 cards. And the third could be any of the remaining 50 cards. Calculate the total number of possible outcomes by multiplying.

811.    43,680

The president can be any of the 16 members. The vice-president can be any of the remaining 15 members. The treasurer can be any of the remaining 14 members. And the secretary can be any of the remaining 13 members. Calculate the total number of possible outcomes by multiplying.

812.    27,216

The first digit can be any of the nine digits, 1 through 9. The second digit can be any of the nine remaining digits from 0 through 9. The third can be any of the eight remaining digits. The fourth can be any of the seven remaining digits. And the fifth can be any of the six remaining digits. Calculate the total number of possible outcomes by multiplying.

813.    2,160

The first letter must be one of the three vowels. The second can be any of the remaining six letters. The third can be any of the remaining five letters. The fourth can be any of the remaining four letters. The fifth letter can be any of the remaining three letters. The sixth letter can be either of the remaining two letters. And the seventh letter must be the one remaining letter. Calculate the total number of possible outcomes by multiplying.

814.    432

The first letter must be one of the three vowels. The second letter must be one of the remaining two vowels. The third letter must be one of the four consonants. The fourth letter must be one of the remaining three consonants. The fifth letter can be any of the remaining three letters. The sixth letter can be either of the remaining two letters. And the seventh letter must be the one remaining letter. Calculate the total number of possible outcomes by multiplying.

815.    36

The first arrival was one of the three women, the second was one of the two remaining women, and the third was the one remaining woman. Then, the fourth arrival was one of the three men, the fifth was either of the two remaining men, and the sixth was the one remaining man. Multiply these six numbers together to calculate the total number of possible outcomes.

816.    36

Each man arrived just after a woman, so the women arrived first, third, and fifth, and the men arrived second, fourth, and sixth. The first arrival was one of the three women, the second was one of the three men, the third was one of the two remaining women, the fourth was one of the two remaining men, the fifth arrival was the one remaining woman, and the sixth was the one remaining man. Multiply these six numbers together to calculate the total number of possible outcomes.

817.    

When you pull one ticket from a bag containing ten tickets, there are a total of ten possible outcomes. In this case, there is only one target outcome: pulling the ticket with the number 1. Plug this information into the formula for probability.

Therefore, the probability is .

818.    

When you pull one ticket from a bag containing ten tickets, there are a total of ten possible outcomes. In this case, there are five target outcomes: pulling the tickets with the numbers 2, 4, 6, 8, or 10. Plug this information into the formula for probability.

Therefore, the probability is .

819.    

When you pull one ticket from a bag containing ten tickets, there are a total of ten possible outcomes. In this case, there are four target outcomes: pulling the tickets with the numbers 7, 8, 9, or 10. Plug this information into the formula for probability.

Therefore, the probability is .

820.    

When you pull one ticket from a bag containing ten tickets, there are a total of ten possible outcomes. Then, when you pull a second ticket from the bag, there are nine possible outcomes. Therefore, there are a total of possible outcomes.

In this case, there are five target outcomes for the first pull (the numbers 1, 3, 5, 7, or 9) and four target outcomes for the second pull (any of four odd numbers that remain after the first pull. Therefore, there are target outcomes.

Plug this information into the formula for probability.

Therefore, the probability is .

821.    

When you roll a six-sided die, there are a total of six possible outcomes. In this case, there is only one target outcome: rolling the number 2. Plug this information into the formula for probability.

Therefore, the probability is .

822.    

When you roll a six-sided die, there are a total of six possible outcomes. In this case, there are four target outcomes: rolling the numbers 3, 4, 5, and 6. Plug this information into the formula for probability.

Therefore, the probability is .

823.    

When you roll a six-sided die, there are a total of six possible outcomes. In this case, there are five target outcomes: rolling the numbers 1, 3, 4, 5, and 6. Plug this information into the formula for probability.

Therefore, the probability is .

824.    

When you roll two six-sided dice, there are a total of six possible outcomes for the first die and six possible outcomes for the second die. Thus, the total number of outcomes is .

In this case, there is one target outcome: rolling 6 on the first die and 6 on the second.

Plug this information into the formula for probability.

Therefore, the probability is .

825.    

When you roll two six-sided dice, there are a total of six possible outcomes for the first die and six possible outcomes for the second die. Thus, the total number of outcomes is .

In this case, there are three target outcomes: rolling 4 on the first die and 6 on the second, rolling 5 on the first die and 5 on the second, and rolling 6 on the first die and 4 on the second.

Plug this information into the formula for probability.

Therefore, the probability is .

826.    

When you roll two six-sided dice, there are a total of six possible outcomes for the first die and six possible outcomes for the second die. Thus, the total number of outcomes is .

To count the number of target outcomes, first count the number of 11s and then the number of 7s.

There are two target outcomes that add up to 11: rolling 5 on the first die and 6 on the second, and rolling 6 on the first die and 5 on the second.

There are six target outcomes that add up to 7: rolling 1 on the first die and 6 on the second, rolling 2 on the first die and 5 on the second, rolling 3 on the first die and 4 on the second, rolling 4 on the first die and 3 on the second, rolling 5 on the first die and 2 on the second, and rolling 6 on the first die and 1 on the second.

Therefore, there are 8 target outcomes and 36 total outcomes. Plug this information into the formula for probability.

Therefore, the probability is .

827.    

When you roll three six-sided dice, there are a total of six possible outcomes for the first die, six possible outcomes for the second die, and six possible outcomes for the third die. Thus, the total number of outcomes is .

There are six target outcomes that add up to 16:

6 + 6 + 4

6 + 4 + 6

6 + 5 + 5

5 + 6 + 5

5 + 5 + 6

4 + 6 + 6

Therefore, there are 6 target outcomes and 216 total outcomes. Plug this information into the formula for probability.

Therefore, the probability is .

828.    

When you pick a card from a deck of 52 cards, there are a total of 52 possible outcomes. In this case, there are four target outcomes: pulling one of the four aces. Plug this information into the formula for probability.

Therefore, the probability is .

829.    

When you pick a card from a deck of 52 cards, there are a total of 52 possible outcomes. In this case, there are 13 target outcomes: picking one of the 13 hearts. Plug this information into the formula for probability.

Therefore, the probability is .

830.    

When you pick a card from a deck of 52 cards, there are a total of 52 possible outcomes. In this case, there are 12 target outcomes: picking one of the four kings, one of the four queens, or one of the four jacks. Plug this information into the formula for probability.

Therefore, the probability is .

831.    

When you pick 2 cards from a deck of 52 cards, there are a total of 52 possible outcomes for the first card and 51 possible outcomes for the second card. Thus, total outcomes are possible.

In this case, there are four target outcomes for the first card (picking one of the four aces) and three target outcomes for the second card (picking one of the three remaining aces). Thus, target outcomes are possible.

Plug this information into the formula for probability.

Therefore, the probability is .

832.    

When you pick 4 cards from a deck of 52 cards, there are a total of 52 possible outcomes for the first card, 51 for the second card, 50 for the third card, and 49 for the fourth card. Thus, total outcomes are possible.

In this case, there are four target outcomes for the first card (picking one of the four aces), three for the second card (picking one of the three remaining aces), two for the third (picking one of the two remaining aces), and one for the fourth (picking the one remaining ace). Thus, target outcomes are possible.

Plug this information into the formula for probability.

Therefore, the probability is

833.    

The first person to arrive was one of six people, so the total number of possible outcomes was six. Of these, there are three target outcomes (each of the three women arriving first). Plug this information into the formula for probability.

Therefore, the probability is .

834.    

There are a total of six possible outcomes for the first person, five possible outcomes for the second person, and four possible outcomes for the third person. Thus, the total number of outcomes is .

There are three possible target outcomes for the first person (one of the three women arrives first), two for the second person (one of the remaining two women arrives second) and one for the third (the one remaining woman arrives third). Thus, the number of target outcomes is .

Therefore, there are 6 target outcomes and 120 total outcomes. Plug this information into the formula for probability.

Therefore, the probability is .

835.    

There are a total of six possible outcomes for the first person, five possible outcomes for the second person, four for the third person, three for the fourth person, two for the fifth person, and one for the sixth person. Thus, the total number of outcomes is .

Each man arrived just after a woman, so the women arrived first, third, and fifth, and the men arrived second, fourth, and sixth. The first arrival was one of the three women, the second was one of the three men, the third was one of the two remaining women, the fourth was one of the two remaining men, the fifth arrival was the one remaining woman, and the sixth was the one remaining man. Multiply these six numbers together:

So the number of total outcomes is 720, and the number of target outcomes is 36. Plug these numbers into the formula for probability.

Therefore, the probability is .

836.    {1, 3, 5, 6, 7, 8, 9}

is the union of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The union includes every element in either set.

837.    {7}

is the intersection of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The intersection includes every element in both sets.

838.    {1, 3, 5, 9}

P – Q is the relative complement of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The relative complement includes only elements of the first set (P) that are not in the second set (Q).

839.    {6, 8}

Q – P is the relative complement of Q = {6, 7, 8} and P = {1, 3, 5, 7, 9}. The relative complement includes only elements of the first set (Q) that are not in the second set (P).

840.    {3, 6, 7, 8, 9}

is the union of Q = {6, 7, 8} and S = {3, 6, 9}, which includes every element in either set.

841.    

is the intersection of R = {1, 2, 4, 5} and S = {3, 6, 9}, which includes every element in both sets. The two sets have no element in common, so the intersection of these sets is the empty set.

842.    {1, 5}

Begin by finding . This is the union of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The union includes every element in either set, so

= {1, 3, 5, 6, 7, 8, 9}

Now, find the intersection of this set and R = {1, 2, 4, 5}. The intersection includes every element in both sets, so

= {1, 5}

843.    {1, 3, 5, 7, 9}

Begin by finding . This is the intersection of Q = {6, 7, 8} and R = {1, 2, 4, 5}. The intersection includes every element in both sets, so

Now, find the union of the empty set and P = {1, 3, 5, 7, 9}. The union includes every element in either set, so

­ = {1, 3, 5, 7, 9}

844.    {1, 5}

Begin by finding . This is the union of Q = {6, 7, 8} and S = {3, 6, 9}, which includes every element in either set, so

= {3, 6, 7, 8, 9}

Now, find the relative complement of P = {1, 3, 5, 7, 9} and this set — that is, the elements of P that are not in :

= {1, 5}

845.    {1, 3, 5, 6, 9}

Begin by finding P – Q. This is the relative complement of P = {1, 3, 5, 7, 9} and Q = {6, 7, 8}. The relative complement includes only elements of the first set (P) that are not in the second set (Q), so

P – Q = {1, 3, 5, 9}

Now, find the union of this set and S = {3, 6, 9}. The union includes every element in either set, so

­= {1, 3, 5, 6, 9}

846.    

Begin by finding Q – S. This is the relative complement of Q = {6, 7, 8} and S = {3, 6, 9}. The relative complement includes only elements of the first set (Q) that are not in the second set (S), so

Q – S = {7, 8}

Now, find the intersection of this set and R = {1, 2, 4, 5}. The intersection includes every element in both sets, so

­

847.    {1, 5, 7}

Begin by finding and :

= {1, 2, 4, 5, 6, 7, 8}

= {1, 5, 7}

Now, find the intersection of these two sets:

= {1, 5, 7}

848.    {…, –4, –2, 0, 2, 4, …}

The set of integers is {…, –2, –1, 0, 1, 2, …}, and the set of even integers is {…, –4, –2, 0, 2, 4, …}. The intersection includes every element in both sets.

849.    {1, 3, 5, 7, …}

The set of positive integers is {1, 2, 3, 4, …}, and the set of even integers is {…, –4, –2, 0, 2, 4, …}. The relative complement includes only elements of the first set that are not in the second set.

850.    {…, –3, –1, 2, 4, 6, …}

The set of odd negative integers is {…, –7, –5, –3, –1}, and the set of even positive integers is {2, 4, 6, 8, …}. The union includes elements that are in either set.

851.    

The set of odd negative integers is {…, –7, –5, –3, –1}, and the set of even positive integers is {2, 4, 6, 8, …}. The intersection includes all elements that are in both sets.

852.    {…, 3, 4, 5, 6, 7}

The complement of a set includes every element that is in the universal set but not in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the set of integers greater than 7 is {8, 9, 10, 11, 12, …}. Therefore, the complement of this set is all the integers less than or equal to 7, {…, 3, 4, 5, 6, 7}.

853.    {…, –4, –2, 0, 2, 4, …}

The complement of a set includes every element that is in the universal set but not in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the set of odd integers is {…, –3, –1, 1, 3, 5, …}. Thus, the complement of the set of odd integers is the set of even integers {…, –4, –2, 0, 2, 4, …}.

854.    {…, –2, –1, 0, 1, 2, …}

The complement of a set includes every element that is in the universal set but not in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the set itself is the empty set, . Because the empty set has no elements, no elements need to be removed from the universal set to form its complement. Therefore, the complement of is {…, –2, –1, 0, 1, 2, …}.

855.    {…, –5, –4, –3, –2, –1}

The complement of a set includes every element that is in the universal set but not in the set itself. The universal set in this case is {…, –2, –1, 0, 1, 2, …}, and the set of non-negative integers is {0, 1, 2, 3, 4 ,…}. Therefore, the complement of this set is {…, –5, –4, –3, –2, –1}.

Another way to think about it: The complement of the set of non-negative integers is the set of negative integers.

856.    29

The diagram shows that 6 students are seniors only, 3 are honors students only, 8 are both seniors and honors students, and 12 are neither. Thus, the club has 6 + 3 + 8 + 12 = 29 members.

857.    27

The diagram shows that 20 people are surnamed Kinney only, 9 live out of state only, and 6 are neither surnamed Kinney nor live out of state. This accounts for 20 + 9 + 6 = 35 of the 42 attendees. Thus, 7 people both are surnamed Kinney and live out of state. So, a total of 20 + 7 = 27 people are surnamed Kinney.

858.    2

The diagram shows that 10 people were in 12 Angry Men but not in Long Day’s Journey Into Night. And 12 Angry Men had 13 people, so 3 were in both plays. Long Day’s Journey Into Night had 5 people, so 2 were in this play but not 12 Angry Men.

859.    3

The board includes 2 officers who have served more than one term. Thus, of the 7 officers, the other 5 are serving their first term. And of the 10 people who have served more than one term, 8 are nonofficers. This accounts for 15 board members, so the remaining 3 are nonofficers who are serving their first term. The following Venn diagram shows this information:

860.    11

Three of the children own neither a cat nor a dog, so 21 students own at least one of these animals. Of these, 15 own a cat and 10 own a dog. 15 + 10 = 25, which is 4 greater than 21. Therefore, exactly 4 students own both a cat and a dog.

You can see this breakdown in the following Venn diagram:

So, of the 15 students who own at least one cat, 11 own at least one cat but no dog.

861.    7

Substitute 9 for x and 4 for y and simplify as follows:

862.    51

Substitute 5 for x and –2 for y and simplify as follows:

863.    83

Substitute –6 for x and –1 for y and simplify as follows:

864.    –65

Substitute –2 for x and 3 for y and simplify as follows:

865.    –0.75

Substitute 0.5 for x and –0.5 for y and simplify as follows:

866.    0.1805

Substitute 0.1 for x and 3 for y and simplify as follows:

867.    –11.62

Substitute 7 for x and 9 for y and simplify as follows:

868.    

Substitute 5 for x and –8 for y and simplify as follows:

869.    –1,458

Substitute –1 for x and 2 for y and simplify as follows:

870.    

Substitute –2 for x and 3 for y and simplify as follows:

871.    7x + 2y

Simplify by combining the two x terms and the two y terms.

872.

Simplify by combining each pair of like terms.

873.    

Simplify by combining the two x terms, the three constant terms, and the two xy terms.

874.    

Simplify by combining the two x terms and the two terms.

875.    

Multiply the coefficients (); then multiply the x variables by adding the exponents (3 + 4 = 7).

876.    

Multiply the coefficients (); then multiply the like variables by adding the exponents of the x variables (2 + 1 = 3) and y variables (2 + 4 = 6).

877.    

Multiply the coefficients (); then multiply the like variables by adding the exponents of the x variables (2 + 3 = 5), y variables (1 + 1 + 1 = 3), and z variables (1 + 4 = 5).

878.    

Apply the rule for simplifying exponents: Take the coefficient (9) to the power of 2, and multiply the exponent of x (3) by 2. To show why this works, I do this in two steps:

879.    

Apply the rule for simplifying exponents: Take the coefficient (6) to the power of 3, and multiply the exponents of x, y, and z by 3:

880.    

Begin by expanding the exponents.

Now, multiply the coefficients, and then add the exponents of the x variables and the y variables.

881.    

Cancel the common factor of the coefficients (2) in both the numerator and denominator.

Next, simplify the variables by subtracting the exponents in the numerator minus the corresponding exponents in the denominator.

882.    

Begin by applying the rule for simplifying exponents in both the numerator and denominator:

Now, cancel the common factor of the coefficients (8) in both the numerator and denominator. Then simplify the variables by subtracting the exponents in the numerator minus the exponents in the denominator.

883.    y

Begin by applying the rule for simplifying exponents in both the numerator and denominator; then simplify.

Now, cancel the common factor of the coefficients (64) in both the numerator and denominator. Then simplify the variables by subtracting the exponents in the numerator minus the corresponding exponents in the denominator.

884.    

To simplify, first remove the parentheses; then combine like terms.

885.    

To simplify, first negate all the terms inside the first set of parentheses; then remove both sets of parentheses and combine like terms.

886.    

To simplify, first distribute 3x among all terms inside the first set of parentheses and negate all terms inside the second set of parentheses; then remove both sets of parentheses.

Now, combine like terms.

887.    

To simplify, first distribute –6xy among all terms inside the first set of parentheses and –yz among all terms inside the second set of parentheses; then remove both sets of parentheses.

Now, combine like terms.

888.    

To simplify, distribute to remove all three sets of parentheses.

Simplify by combining like terms.

889.    

Multiply the two expressions by using the “FOIL” method.

Simplify by combining like terms.

890.    

Multiply the two expressions by using the “FOIL” method.

Simplify by combining like terms.

891.    

Multiply the two expressions by using the “FOIL” method.

892.    

Begin by distributing 4x over (x – 6).

Multiply the two resulting expressions by using the “FOIL” method.

Simplify by combining like terms:

893.    

Multiply the first two expressions by using the “FOIL” method.

Simplify the first expression by combining like terms.

Now, multiply each term in the first expression by each term in the second expression.

Simplify by combining like terms.

894.    

You can factor an x out of both terms.

895.    

You can factor an out of both terms.

896.    

You can factor an out of all three terms.

897.    

The greatest common factor of 12, 6, and 4 is 2. And the greatest common factor of the x variables has the smallest exponent among the three terms, 3. Thus, you can factor a out of all three terms.

898.    

The greatest common factor of 24, 15, and 9 is 3. And the greatest common factor of the x variables has the smallest exponent among the three terms, 4. Thus, you can factor a out of all three terms.

899.    

The greatest common factor of the x variables has the smallest exponent among the three terms, 2, and the greatest common factor of the y variables has an exponent of 1. Thus, you can factor an out of all three terms.

900.    

The greatest common factor of 8, 20, and 40 is 4. The greatest common factor of the x variables has the smallest exponent among the three terms, 6. And the greatest common factor of the y variables has the smallest exponent of all three terms, 8. Thus, you can factor a out of all three terms.

901.    

The greatest common factor of 36, 24, and 90 is 6. And the greatest common x, y, and z exponents are, respectively, 1, 1, and 3. Thus, you can factor a out of all three terms.

902.    

Both terms are perfect squares, so you can use the rule for factoring the difference of two squares.

903.    

Both terms are perfect squares, so you can use the rule for factoring the difference of two squares.

904.    

Both terms are perfect squares, so you can use the rule for factoring the difference of two squares.

905.    

Both terms are perfect squares, so you can use the rule for factoring the difference of two squares.

906.    

Begin by generating a list of all possible factor pairs of integers (both negative and positive) that multiply to 14 (the constant).

Identify the factor pair whose sum is 9 (the coefficient of the x term).

So, factor using the numbers 2 and 7, as follows:

907.    

Begin by generating a list of all possible factor pairs of integers (both negative and positive) that multiply to 18 (the constant).

Identify the factor pair whose sum is –11 (the coefficient of the x term).

So, factor using the numbers –2 and –9, as follows:

908.    

Begin by generating a list of all possible factor pairs of integers (both negative and positive) that multiply to –20 (the constant).

Identify the factor pair whose sum is 1 (the coefficient of the x term).

So, factor using the numbers –4 and 5, as follows:

909.    

Begin by generating a list of all possible factor pairs of integers (both negative and positive) that multiply to –24 (the constant).

Identify the factor pair whose sum is –10 (the coefficient of the x term).

So, factor using the numbers 2 and –12, as follows:

910.    

Begin by factoring x out of the denominator.

Now, cancel a factor of x from both the numerator and denominator.

911.    x

Begin by factoring x out of the numerator.

Now, cancel a factor of x – 1 from both the numerator and denominator.

= x

912.    

Begin by factoring out of the numerator and 3 out of the denominator.

Now, cancel a factor of from both the numerator and denominator.

913.    

Begin by factoring out of the numerator and out of the denominator.

Now, cancel a factor of from both the numerator and denominator.

Additionally, you can cancel a factor of from both the numerator and the denominator.

914.    

Begin by factoring 2x out of the numerator and 5 out of the denominator.

Now, cancel a factor of from both the numerator and denominator.

915.    x – 2

Begin by factoring the numerator as the difference of squares.

Now, cancel a factor of x + 2 from both the numerator and denominator.

= x – 2

916.    

Begin by factoring the numerator as the difference of squares.

Next, factor out the GCF, 2, in the denominator.

Now, cancel a factor of x + y from both the numerator and denominator.

917.    

Begin by factoring the numerator as the difference of squares.

Now, factor out the GCF, 4, in the denominator.

Finally, cancel a factor of 2x + 5 from both the numerator and denominator.

918.    

To begin, factor out the GCF, 16x, from the denominator.

Next, factor in the denominator as the difference of squares.

Now, cancel a factor of x – 2 from both the numerator and denominator.

919.    

To begin, factor the numerator as the difference of squares.

Next, factor the quadratic expression in the denominator.

Now, cancel a factor of x – 6 from both the numerator and denominator.

920.    

To begin, factor the quadratic expression in the numerator.

Next, factor the quadratic expression in the denominator.

Now, cancel a factor of x + 2 from both the numerator and denominator.

921.    i. 8, ii. 12, iii. 9, iv. 14, v. 11

i. 6 + 8 = 14

ii. 21 – 12 = 9

iii. 7(9) = 63

iv. 14 ÷ 1 = 14

v. 99 ÷ 11 = 9

922.    i. 49, ii. 112, iii. 45, iv. 76, v. 247

i. 117 – 68 = 49

ii. 29 + 83 = 112

iii. 585 ÷ 13 = 45

iv. 3,116 ÷ 41 = 76

v. 19 × 13 = 247

923.    12

Begin by testing x = 10.

Because 104 < 122, you know that x = 10 is a little low, so try x = 11.

This is still a little low, so try x = 12.

924.    29

Begin by testing x = 25.

This is low, so try x = 30.

This is just a little high, so try x = 29.

Therefore, x = 29.

925.    5

Begin by adding 3 to each side of the equation.

6x – 3 = 27

+3 +3

6x = 30

Now, divide both sides by 6.

Therefore, x = 5.

926.    7

Begin by subtracting 9n from each side of the equation.

9n + 14 = 11n

– 9n – 9n

14 = 2n

Now, divide both sides by 2.

Therefore, n = 7.

927.    –3

Begin by subtracting v from both sides of the equation.

v + 18 = –5v

–v –v

18 = –6v

Now, divide both sides by –6.

Therefore, v = –3.

928.    

Begin by subtracting 3k from both sides of the equation.

9k = 3k + 2

–3k –3k

6k = 2

Now, divide both sides by 6.

929.    9

Begin by subtracting 2y from both sides of the equation.

2y + 7 = 3y – 2

–2y –2y

7 = y – 2

Now, add 2 to both sides.

7 = y – 2

+ 2 + 2

9 = y

Therefore, y = 9.

930.    16

Begin by subtracting m from both sides of the equation.

m + 24 = 3m – 8

– m –m

24 = 2m – 8

Now, add 8 to both sides.

24 = 2m – 8

+8 +8

32 = 2m

Finally, divide both sides by 2.

Therefore, m = 16.

931.    

Begin by subtracting –7a from both sides of the equation.

Now, subtract 27 from both sides; then divide by 6.

932.    –5

Begin by simplifying the equation by combining like terms.

Now, isolate h and solve.

933.    11

Begin by simplifying the equation by combining like terms.

Now, isolate x and solve.

934.    5

Begin by subtracting 2.3w from both sides of the equation.

Now, divide both sides by 1.4.

935.    3.5

Begin by adding 1.9p to both sides of the equation.

Now, add 7 to both sides and then divide by 4.

936.    

Begin by simplifying the equation by combining like terms.

Now, add 1.6j to both sides and then subtract 0.7 from both sides.

Finally, divide both sides by 11.

937.    

To begin, simplify each side of the equation, removing parentheses by distributing.

Next, combine like terms on each side of the equation; then isolate and solve for x.

938.    –6

To begin, simplify each side of the equation, removing parentheses by distributing.

Next, combine like terms on each side of the equation; then isolate and solve for u.

939.    

To begin, simplify each side of the equation, removing parentheses by distributing.

Next, combine like terms on each side of the equation; then isolate and solve for k.

940.    

To begin, simplify each side of the equation, removing parentheses by distributing.

Next, subtract from each side of the equation; then combine like terms.

Isolate x.

941.    2

To begin, distribute on the left side of the equation.

Simplify and solve for v.

942.    1.9

To begin, simplify each side of the equation, removing parentheses by distributing.

Isolate y.

943.    –15.75

To begin, simplify the left side of the equation, removing the parentheses by distributing.

Simplify and solve for m.

944.    –3.2

To begin, simplify each side of the equation, removing parentheses by distributing.

Isolate n.

945.    –2

To begin, simplify each side of the equation, removing parentheses by distributing.

Simplify and solve for s.

946.    42

Multiply both sides of the equation by 6.

947.    44

Multiply both sides of the equation by .

948.    

Multiply both sides of the equation by .

949.    

To begin, cross-multiply to remove the fractions.

Simplify and solve for q.

950.    

To begin, cross-multiply to remove the fractions.

Simplify and solve for c.

951.    

To begin, cross-multiply to remove the fractions.

“FOIL” the left side of the equation; then subtract from both sides.

Combine like terms and isolate t.

952.    

To begin, cross-multiply to remove the fractions.

“FOIL” both sides of the equation; then subtract from both sides.

Simplify and isolate z.

953.    

To begin, cross-multiply to remove the fractions.

“FOIL” the left side of the equation and distribute the right side; then subtract from both sides.

Simplify and isolate b.

954.    18

To begin, add the two terms on the left side of the equation.

Multiply both sides by 9 and solve for p.

955.    6

To begin, use cross-multiplication techniques to add the fractions on the left side of the equation.

Multiply both sides of the equation by 6 and then isolate d.

956.    

To begin, use cross-multiplication techniques to add the fractions on the left side of the equation.

Multiply both sides of the equation by 8 and then simplify and isolate s.

957.    

To begin, use cross-multiplication techniques to add the fractions on the left side of the equation.

Multiply both sides of the equation by 30 and then simplify.

Isolate r.

958.    3

To begin, increase the terms of the first fractions by 2 (so that the common denominator is 4); then multiply both sides of the equation by 4 to eliminate the fractions.

Isolate j to solve.

3 = j

959.    

To begin, change all three fractions so that they have denominators of 8; then multiply both sides of the equation by 8 to eliminate the fractions.

Isolate k to solve.

960.    

To begin, increase the terms of all three fractions to change all denominators to 12. Then multiply both sides of the equation by 12 to eliminate the fractions.

Isolate a to solve.

961.    6

To begin, increase the terms of all three fractions to change all denominators to 60; then multiply both sides of the equation by 60 to eliminate the fractions.

Isolate h to solve.

11h = 66

h = 6

962.    –11

Begin by changing all terms to a denominator of 9; then multiply both sides of the equation by 9 to eliminate the fractions.

Isolate k.

963.    

Begin by changing all terms to a denominator of 8; then multiply both sides of the equation by 8 to eliminate the fractions.

Distribute to remove parentheses:

Isolate y:

14 – 4y = 2y + 6

8 – 4y = 2y

964.    7

Divide both sides of the equation by .

Now, divide both sides by 5.

x = 7

965.    

Divide both sides of the equation by .

Next, divide both sides by 45.

Now, take the square root of both sides.

966.    5 and –5

Isolate x and solve.

967.    7 and –9

Begin by factoring the left side of the equation.

Now, split this equation into two separate equations and solve them.

x + 9 = 0 x – 7 = 0

x = –9 x = 7

968.    1 and –8

Begin by moving all terms to one side of the equation.

Now, factor the left side of the equation.

(x + 8)(x – 1) = 0

Now, split this equation into two separate equations and solve them.

x + 8 = 0 x – 1 = 0

x = –8 x = 1

969.    6 and 7

Begin by distributing on both sides of the equation to remove the parentheses; then move all terms to one side.

Factor the left side of the equation.

(x – 6)(x – 7) = 0

Now, split this equation into two separate equations and solve them.

x – 6 = 0 x – 7 = 0

x = 6 x = 7

970.    –3 and –5

Begin by cross-multiplying to remove the fractions.

Now, distribute on both sides of the equation to remove the parentheses; then move all terms to one side.

Now, factor the left side of the equation.

(x + 3)(x + 5) = 0

Now, split this equation into two separate equations and solve them.

x + 3 = 0 x + 5 = 0

x = –3 x = –5

971.    2d + 1,000

The amount d doubles to 2d, and then increases by 1,000 to 2d + 1,000.

972.    3c – 60

The day begins with c chairs. Then 20 chairs are removed, bringing the number to c – 20. After that, this number is tripled, which brings the number to

3(c – 20) = 3c – 60

973.    p – 234

Penny starts with p pennies. She then removes 300 pennies, bringing the total to p – 300 pennies. The next day, she adds back in 66 pennies, so the total becomes

p – 300 + 66

You can simplify this amount as follows:

= p – 234

974.    t – 2

The temperature begins at t degrees and then changes as follows:

t + 5 + 2 – 3 – 6 = t – 2

975.    6w + 12

The puppy’s weight begins at w. It triples to 3w, then increases by 6 pounds to 3w + 6, and finally doubles to

2(3w + 6) = 6w + 12

976.    2k + 57

Kyle has k baseball cards. Randy has half as many, so Randy has cards. And Jacob has 57 more cards than Randy, so Jacob has cards. Add these up as follows:

You can further simplify this by combining the three k terms.

= 2k + 57

977.    0.72s + 425

The school currently has s students. The number of graduating students is 0.28s. When these students leave, the number of remaining students will be

s – 0.28s = 0.72s

Additionally, 425 new students will be at the school, so this number will increase to 0.72s + 425.

978.    5m + 10

Millie walked m miles the first day, m + 1 miles the second day, m + 2 miles the third day, m + 3 miles the fourth day, and m + 4 miles the fifth day. The sum of these numbers is

m + m + 1 + m + 2 + m + 3 + m + 4

Combine like terms to simplify.

= 5m + 10

979.    4n + 12

Every consecutive odd number is exactly two greater than the preceding one. So, you can represent the four numbers as n, n + 2, n + 4, and n + 6. Thus, the sum of these numbers is:

n + n + 2 + n + 4 + n + 6

Simplify as follows:

= 4n + 12

980.    4

Let x equal the number. Then, set up and solve the following equation:

981.    4

Let x equal the number. Then, set up and solve the following equation:

982.    8

Let x equal the number. Then, set up and solve the following equation:

983.    –2

Let x equal the number. Then, set up and solve the following equation:

984.    7

Let x equal the number. Then, set up and solve the following equation:

985.    17

Let x equal the number. Then, set up and solve the following equation:

986.    23

Let x equal the number. Then, set up and solve the following equation:

987.    3

Let x equal the number. Then, set up and solve the following equation:

988.    7.25

Let x equal the number. Then, set up and solve the following equation:

989.    –11

Let x equal the number. Then, set up and solve the following equation:

990.    3.6

Let x equal the number. Then, set up and solve the following equation:

991.    4

Let x equal the number. Then, set up and solve the following equation:

992.    16

Let x equal the number. Then, set up and solve the following equation:

993.    $11

Let p = the number of dollars that Peter has. Then Lucy has p + 5 dollars. Together, they have $27, so

p + p + 5 = 27

Solve for p.

994.    $170

Let m = the cost of the MP3 player in dollars. Then 2m is the cost of the cellphone and 4m is the cost of the laptop computer. So, you can set up the following equation:

Simplify and solve for m.

Therefore, the MP3 player cost $170.

995.    5 years old

Let j be Jane’s age. Then, Cody’s age is j + 8 and Brent’s age is 2j. Cody is 3 years older than Brent, so you can set up the following equation:

Brent + 3 = Cody

Simplify and solve for j.

Therefore, Jane is 5 years old.

996.    2 hours and 20 minutes

Let x be the number of minutes that the class takes. So, the teacher spends minutes going over homework problems and minutes reviewing for a test. Thus, you can set up the following equation:

Raise the terms of every term in this equation to a denominator of 10, then drop the denominators.

Simplify and solve for x.

Therefore, the class is 140 minutes long, which equals 2 hours and 20 minutes.

997.    35

Let x be the first number. Then the other four numbers are x + 1, x + 2, x + 3, and x + 4. Thus, you can set up the following equation:

Simplify and solve for x.

Thus, the five numbers are 31, 32, 33, 34, and 35. So the greatest is 35.

998.    78

Let y be the number of yellow marbles in the jar. Then, the jar contains 3y orange marbles, y + 6 blue marbles, and 2(y + 6) red marbles. So you can set up the following equation:

Simplify and solve for y.

Therefore, the jar contains 22 yellow marbles, so it contains 28 blue marbles and 56 red marbles. Therefore, it contains 22 + 56 = 78 yellow and red marbles.

999.    50 mph

Let s be the speed of the southbound train. Then, 2s is the speed of the northbound train and 2s – 10 is the speed of the eastbound train.

Therefore, the southbound train is traveling at 50 mph.

1000.    $300

Let k equal the number of dollars that Ken has. Then Walter has k – 100 dollars. So, you can set up the following equation:

Simplify on the right and increase the terms of each term to a denominator of 2; then drop the denominators.

Simplify and solve for k.

Therefore, Ken has $300.

1001.    12

Let d be Damar’s age now. So Jessica’s age now is 2d. Three years ago, Damar’s age was d – 3 and Jessica’s age was 2d – 3. And at that time, Jessica was 3 times as old as Damar, so

2d – 3 = 3(d – 3)

Solve for d.

Thus, Damar is 6 years old right now. Jessica is twice his age, so she is now 12 years old.