Chapter 7
Arithmetic Reasoning: Math Word Problems
IN THIS CHAPTER
Solving life’s little (math) problems
Multiplying your chances for a better score
How many miles per gallon does your brand-new SUV get? How long does it take to go over the river and through the woods to Grandmother’s house? How much wood would a woodchuck chuck? These are examples of everyday questions that can be answered by arithmetic reasoning. (Okay, maybe the woodchuck situation doesn’t happen every day.)
The rest of the world calls this type of question math word problems. The ASVAB calls them Arithmetic Reasoning. No matter what they’re called, these problems help you apply mathematical principles to the real world (at least the real world according to the people who think up word problems). Your job is to read a word problem, determine what the question asks, and select the correct answer.
Arithmetic Reasoning is an important part of the Armed Forces Qualification Test (AFQT) score, which is used to determine your general qualification for enlistment in all the service branches (see Chapter 1 for more information). Also, certain military jobs require that you score well on this subtest (see Appendix A).
The test administrator will supply you with scratch paper and a trusty number two pencil, but one thing he or she won’t give you (or even let you bring) is a calculator. You can use your paper and lead to clarify the data, write formulas, and mathematically solve the problem. You can even use them to draw pretty pictures to help you understand the problem. Don’t get too artistic, though — you have only 36 minutes to answer 30 questions if you’re taking the paper version and 39 minutes to answer 16 questions if you’re taking the CAT-ASVAB.
To do well on the Arithmetic Reasoning subtest, you have to remember that there are two parts: arithmetic and reasoning. You usually have to use both of these skills for each problem. The arithmetic part comes in when you have to perform mathematical operations such as addition, subtraction, multiplication, and division. The reasoning comes in when you figure out which numbers to use in your calculations. In other words, Arithmetic Reasoning tests how you apply your ability to perform calculations to real-life problems. If you slept through high school math, don’t worry. This chapter helps you decipher these math problems, focusing on the reasoning part. For additional info on the arithmetic, flip to Chapter 6.
Tackling the Real World of Word Problems
Test-takers often waste a lot of time reading and rereading word problems as if the answer might reveal itself to them by some miracle; however, correctly solving math word problems requires you to perform a series of organized steps:
- Read the problem completely.
- Figure out what the question is asking.
- Dig out the relevant facts.
- Set up one or more equations to arrive at a solution and then solve the problem.
- Review your answer.
I cover these steps in detail throughout this section.
Reading the entire problem
The first step in solving a word problem is reading the entire problem to discover what it’s all about. Try forming a picture about the problem in your mind or — better yet — draw a sketch of the problem on your scratch paper. Ask yourself whether you’ve ever seen a problem like this before. If so, what’s similar about it, and what did you do to solve it in the past?
As plain as the nose on a fly: Figuring out what the question is asking
The second and most important step in solving a word problem is to determine exactly what the question is asking. Sometimes the question is asked directly. At other times, identifying the actual question may be a little more difficult. Suppose you’re asked the following question:
What’s the volume of a cardboard box measuring 12 inches long by 14 inches wide by 10 inches tall?
(A) 52 cubic inches
(B) 88 cubic inches
(C) 120 cubic inches
(D) 1,680 cubic inches
The problem directly asks you to determine the volume of a cardboard box. Recall from your high school algebra and geometry classes that the volume of a rectangular container is 
, or 
. So 
. The correct answer is Choice (D).
Now take a look at the next example:
How many cubic inches of sand can a cardboard box measuring 12 inches long by 14 inches wide by 10 inches tall contain?
(A) 52 cubic inches
(B) 88 cubic inches
(C) 120 cubic inches
(D) 1,680 cubic inches
This is the same problem, but the question you need to answer isn’t as directly stated. Therefore, you have to use clues embedded in the problem to figure out what the actual question is. Would figuring out the perimeter of the box help you with this question? Nope. Would figuring out the area of one side of the box help you? Nope — you’re not painting the box; you’re filling it. The question wants you to determine the volume of the container.
Clue words can be a big help when trying to figure out which question is being asked. Look for the following clue words:
- Addition: Sum, total, in all, perimeter, increased by, combined, added
- Division: Share, distribute, ratio, quotient, average, per, out of, percent
- Equals: Is, was, are, were, amounts to
- Multiplication: Product, total, area, cubic, times, multiplied by, of
- Subtraction: Difference, how much more, exceed, less than, fewer than, decreased
Digging for the facts
After you figure out which question you’re answering, the next step is to figure out which data is necessary to solve the problem and which data is extra. Start by identifying all the information and variables in the problem and listing them on your scratch paper. Make sure you attach units of measurement contained in the problem (miles, feet, inches, gallons, quarts, and so on). After you’ve made a list of the facts, try to eliminate those facts that aren’t relevant to the question. Look at the following example:
To raise money for the school yearbook project, Tom sold 15 candy bars, Becky sold 12 candy bars, Debbie sold 17 candy bars, and Jane sold the most at 50. How many candy bars were sold by the girls?
The list of facts may look something like this:
Because the question is the total number of candy bars sold by the girls, the number of bars sold by Tom isn’t relevant to the problem and can be scratched off the list. Just add the remaining bars from your list. The answer is 79.
Setting up the problem and working your way to the answer
You need to decide how the problem can be solved and then use your math skills to arrive at a solution. For instance, a question may ask the following:
Joan just turned 37. For 12 years, she’s dreamed of traveling to Key West to become a beach bum. To finance this dream, she needs to save a total of $15,000. How much does Joan need to save each year if she wants to become a beach bum by her 40th birthday?
Write down, in mathematical terms, what the question is asking you to determine. Because the question is asking how much money Joan needs to save per year to reach $15,000, you can say y (years Joan has to save) 
(money she needs to save each year) = $15,000. Or to put it more mathematically,
You don’t know the value of m (yet) — that’s the unknown you’re asked to find. But you can find out the value of y — the number of years Joan has to save. If she’s 37 and wants to be a beach bum by the time she’s 40, she has 3 years to save. So now the formula looks like this:
To isolate the unknown on one side of the equation, you simply divide each side by 3, so 
. (If you don’t remember how to isolate unknowns, flip to Chapter 6.) Therefore, your answer is
Joan needs to save $5,000 each year for 3 years to reach her goal of $15,000 by the time she’s 40. You may be tempted to include the 12 years Joan has been dreaming of this trip in your formula. This number was put into the problem as a distraction. It has no bearing on solving the problem.
Reviewing your answer
Before marking your answer sheet or punching in that choice on the computer, review your answer to make sure it makes sense. Review by asking yourself the following questions:
- Does your solution seem probable? Use your common sense. If you determine that a 12-x-16-foot roof is covered in only 12 shingles, you’ve probably made a mistake in your calculations.
- Does it answer the question asked? Reread the problem. For example, if a question asks you to calculate the number of trees remaining after 10% of the total was cut down, the correct answer wouldn’t be 10% of the trees but rather the 90% still standing.
- Are you sure? Double-check your answer. Those tricky test-makers often supply false answers that are very, very close to the correct answer.
- Is your answer expressed using the same units of measurement as used in the problem? A question may ask how many cubic feet of concrete are required to cover a driveway. Your answer in cubic yards would have to be converted to cubic feet so you can select the correct answer choice.
Although you may have been taught in school to round 5 or above up and below 5 down, rounding real-world problems requires a different mindset. For example, if someone needs 2.2 cans of paint for a particular job, she really needs 3 cans of paint to make sure she has enough, even though you’d generally round down. And if someone gets a 15-minute break for every 4 hours of work but works only 7 hours, he’d get only one break, even though 7 divided by 4 equals 1.75, which is generally rounded up to 2.
You may find that the solution you arrived at doesn’t fit the facts presented in the problem. If this is the case, back up and go through the steps again until you arrive at an answer that seems probable.
The Guessing Game: Putting Reason in Your Guessing Strategy
Guessing wrong on any of the ASVAB subtests doesn’t count against you (unless you guess incorrectly on a bunch of questions in a row at the end of the subtest when taking the CAT-ASVAB). If you don’t guess, your chances of getting that answer right are zero, but if you take a shot at it, your chances increase to 25%, or 1 in 4. Eliminate two wrong answers, and you have a 50-50 shot.
If you’re taking the paper version of the ASVAB, you can always skip the tough questions and come back to them after you’ve finished the easier ones. If you’re taking the computerized version of the ASVAB, the software won’t let you skip questions, so you need to make your guess right then and there.
If you’re taking the paper version of the test and elect to skip questions until later, make sure you mark the next answer in the correct space on the answer sheet. Otherwise, you may wind up wearing out the eraser on your pencil when you discover your error at the end of the test. Or even worse, you may not notice the error and wind up getting several answers wrong because you mismarked your answer sheet.
Using the process of elimination
Guessing doesn’t always mean “pick an answer, any answer.” You can increase your chances of picking the right answer by eliminating answers that can’t be right. To eliminate some obvious wrong answers, you can do the following:
- Make sure the answer is realistic in relation to the question asked. For example, if a question asks you how much water would be required to fill a child’s wading pool, 17,000 gallons isn’t a realistic answer. You can save time by eliminating this potential answer choice immediately.
- Pay attention to units of measurement. If a question asks how many feet of rope you’ll need, answer choices listed in inches or cubic feet are probably incorrect.
- Consider easier answer choices first. Remember, you’re not allowed to use a calculator on the ASVAB, so math answers that you’d arrive at by using complicated formulas are probably not correct.
Solving what you can and guessing the rest
Sometimes you may know how to solve part of a problem but not all of it. If you don’t know how to do all the calculations — or don’t have time for them — don’t give up. You can still narrow down your choices by doing what you can. Here’s how partially solving problems can help:
- When adding mixed numbers (a whole number and a fraction), add the whole-number parts first; then immediately eliminate answer choices that are too low. Or when adding lengths, add full feet first and cross off choices that are too small, even before considering the inches.
- Multiply just the last digits and cross off all answers that don’t end in the right numbers (assuming the answers aren’t rounded).
Making use of the answer choices
If you’re stuck on a particular problem, sometimes plugging possible answers into an equation can help you find the right answer. Here’s how using the answer choices can improve your guessing:
- Plug in each remaining answer choice until you get the right answer. Plugging in all the answer choices is time-consuming, so make sure you eliminate obviously wrong choices first.
- Estimate and plug in numbers that involve easy mental calculations. For instance, if Choice (A) is 9 and Choice (B) is 12, plug in 10 and solve the equation in your head. Think about whether the right answer has to be higher or lower than 10, and choose from there.
- Using a little logic, do calculations with an obviously wrong answer choice. Sometimes a wrong answer choice — especially one that differs drastically from the other answers — represents an intermediate step in the calculations, so you can use it to solve the problem. For instance, take this example:
A security guard walks the equivalent of six city blocks when he makes a circuit around the building. If he walks at a pace of eight city blocks every 30 minutes, how long will it take him to complete a circuit around the building, assuming he doesn’t run into any thieves?
(A) 20.00 minutes
(B) 3.75 minutes
(C) 22.50 minutes
(D) 24.00 minutes
Choice (B) is obviously way too low to be the right answer, but it would be a logical guess for the security guard’s rate for a single lap. Multiply 3.75 minutes/block by 6 blocks, and you probably have a good candidate for the right answer — 22.50 minutes, Choice (C).
Arithmetic Reasoning (Math Word Problems) Practice Questions
Arithmetic Reasoning questions are math problems expressed in a story format. Your goal is to determine what the question is asking by picking out the relevant factors needed to solve the problem, set up mathematical equations as needed, and arrive at the correct solution. Sounds easy, right? Try your hand at the following questions.
1. If apples are on sale at 15 for $3, what’s the cost of each apple?
(A) 50 cents
(B) 25 cents
(C) 20 cents
(D) 30 cents
2. A noncommissioned officer challenged her platoon of 11 enlisted women to beat her record of performing a 26-mile training run in 4 hours. If all the enlisted women match her record, how many miles will they have run?
(A) 71.5 miles
(B) 6.5 miles
(C) 286 miles
(D) 312 miles
3. Diane gets her hair cut and colored at an expensive salon in town. She’s expected to leave a 15% tip for services. If a haircut is $45 and a color treatment is $150, how much of a tip should Diane leave?
(A) $22.50
(B) $29.25
(C) $20.00
(D) $224.25
4. A bag of sand holds 1 cubic foot of sand. How many bags of sand are needed to fill a square sandbox measuring 5 feet long and 1 foot high?
(A) 25 bags
(B) 5 bags
(C) 10 bags
(D) 15 bags
5. The day Samantha arrived at boot camp, the temperature reached a high of 90 degrees in the shade and a low of –20 degrees at night in the barracks. What was the average temperature for the day?
(A) 35 degrees
(B) 45 degrees
(C) 55 degrees
(D) 62 degrees
6. Farmer Beth has received an offer to sell her 320-acre farm for $3,000 per acre. She agrees to give the buyer $96,000 worth of land. What fraction of Farmer Beth’s land is the buyer getting?
(A) 
(B) 
(C) 
(D) 
7. A large wall map is drawn so that 1 inch equals 3 miles. On the map, the distance from Kansas City to Denver is 
. How far is the round trip from Kansas City to Denver in miles?
(A) 
(B) 
(C) 385 miles
(D) 1,155 miles
8. Margaret and Julie can sell their tattoo parlor for $150,000. They plan to divide the proceeds according to the ratio of the money they each invested in the business. Margaret put in the most money at a 3:2 ratio to Julie’s contribution. How much money should Julie get from the sale?
(A) $50,000
(B) $30,000
(C) $60,000
(D) $90,000
9. Mr. Cameron purchased a shirt for $20. He sold it for $26. By what percentage did he increase the price?
(A) 5
(B) 20
(C) 30
(D) 25
10. In the military, 
of an enlisted person’s time is spent sleeping and eating, 
is spent standing at attention, 
is spent staying fit, and 
is spent working. The rest of the time is spent at the enlisted person’s own discretion. How many hours per day does this discretionary time amount to?
(A) 6.0 hours
(B) 1.6 hours
(C) 2.4 hours
(D) 3.2 hours
11. A designer sells a square yard of carpet for $15.00. The same carpet can be purchased at the carpet outlet store for $12.50. As a percentage, how much more expensive is the designer’s carpet?
(A) The designer’s carpet costs about 17% more than the outlet-store carpet.
(B) The designer’s carpet costs about 20% more than the outlet-store carpet.
(C) The designer’s carpet costs about 25% more than the outlet-store carpet.
(D) The designer’s carpet costs about 12% more than the outlet-store carpet.
12. Terry got a haircut for $32.50, a hair color for $112.20, and a manicure for $17.25. How much total money did she spend at the salon?
(A) $167.45
(B) $144.70
(C) $161.95
(D) $156.95
13. Mailing the first ounce of a letter costs $0.49, and it costs $0.21 to mail each additional ounce. How much does it cost to mail a 5-ounce letter?
(A) $1.85
(B) $1.05
(C) $1.54
(D) $1.33
14. Larry travels 60 miles per hour going to a friend’s house and 50 miles per hour coming back, using the same road. He drove a total of 5 hours. What is the distance from Larry’s house to his friend’s house, rounded to the nearest mile?
(A) 110
(B) 126
(C) 136
(D) 154
15. Joe ran around a pentagon-shaped track with sides each measuring 1,760 feet. If he made three complete trips around the track, how far did he run?
(A) 37,500 feet
(B) 15,300 feet
(C) 20,150 feet
(D) 26,400 feet
16. It takes Steve 56 hours to paint his fence. If his 3 children each work 7 hours per day with him, how many days will it take the family to paint the fence, assuming the children keep up with their dad’s pace?
(A) 2
(B) 4
(C) 2.5
(D) 1
17. To buy a new car priced at $32,000, Martha takes out a five-year loan with an interest rate of 6.5%. By the time she owns the car, how much will she have paid including principal and interest?
(A) $45,000
(B) $41,500
(C) $40,000
(D) $42,400
18. What is the width of a rectangular vegetable garden whose perimeter is 150 feet and length is 50 feet?
(A) 100 feet
(B) 25 feet
(C) 200 feet
(D) 50 feet
19. Mike took Jen bowling for the first time. He bowled two games with scores of 157 and 175. Jen had never bowled before and scored 78 and 98. What was Mike’s average score?
(A) 88
(B) 127
(C) 156
(D) 166
20. The cost of 4 shirts, 4 pairs of dress pants, and 2 ties is $560. The cost of 9 shirts, 9 pairs of dress pants, and 6 ties is $1,290. What is the total cost of 1 shirt, 1 pair of dress pants, and 1 tie?
(A) $150
(B) $230
(C) $175
(D) $195
21. A can of pork and beans has a radius of 3 inches and a height of 7 inches. What is the volume of the can?
(A) 198 cubic inches
(B) 156 cubic inches
(C) 21 cubic inches
(D) 42 cubic inches
22. Edward’s electric bill for the month of July was $90.12. The electric company charges a flat monthly fee of $20.00 for service plus $0.14 per kilowatt-hour of electricity used. Approximately how many kilowatt-hours of electricity did Edward use in July?
(A) 361.11
(B) 424.12
(C) 500.86
(D) 567.17
23. Billy left the house without his wallet. When he went to purchase his lunch, he had to dig into his change stash to buy it. How much did he have left if he had 15 quarters, 15 dimes, 22 nickels, and 12 pennies and the lunch cost $5.52?
(A) $0.45
(B) $1.15
(C) $0.95
(D) $1.03
24. What will it cost to carpet a room 10 feet wide and 12 feet long if carpet costs $12.51 per square yard?
(A) $166.80
(B) $213.50
(C) $186.23
(D) $165.12
25. Jack eats three hot dogs per minute, while Jeff eats two hot dogs per minute. How many total hot dogs do they eat in 12 minutes?
(A) 35
(B) 40
(C) 60
(D) 65
26. The interest on Jerry’s fixed sum of money depends on the length of time the money is invested. If it draws $60 in 4 months, how much will it draw in 1.5 years?
(A) $320
(B) $240
(C) $270
(D) $200
27. A rancher is driving along the edge of a round sinkhole on his property. The sinkhole’s diameter is 14 kilometers. If he walked around the sinkhole, how far would he walk?
(A) 34 kilometers
(B) 54 kilometers
(C) 44 kilometers
(D) 35 kilometers
28. What is the price of a $200 item after successive discounts of 10% and 15%?
(A) $75
(B) $175
(C) $153
(D) $150
Answers and Explanations
Use this answer key to score the Arithmetic Reasoning practice questions.
- C. Divide $3 by 15. The answer is $0.20, so the correct answer is Choice (C).
- C. Multiply
. The other information in the question is irrelevant — it’s there to throw you off. The correct answer is Choice (C). You can immediately eliminate Choice (B) because it isn’t a reasonable answer. Identifying unreasonable answers (through the process of elimination) can help you choose the correct answer choice faster. - B. Add 45 and 150 to get the cost of the services
; then multiply the answer by 0.15 (15%) to find the tip. The question asks for the amount of the tip, so the correct answer is Choice (B), $29.25. You can immediately eliminate Choice (D), because the amount is far too high to make sense. - A. To find the volume of the sandbox, you take length times width times height
. Don’t forget that the measurements are for a square sandbox, so you can assume that if the box is 5 feet long, then it’s also 5 feet wide. So 
is 25 cubic feet. Each bag holds 1 cubic foot of sand, and 
. Choice (A) is the correct answer. If you were thinking answer Choice (B) sounded good, remember that the answer should make sense. Five cubic feet of sand would not fill a very large sandbox, would it? - A. Add the two temperatures:
. Divide 70 by 2 to reach the average temperature, 35. Choice (A) is the correct answer. - B. The buyer’s price, $96,000, divided by $3,000 (price per acre) equals 32 acres. Thirty-two acres divided by 320 acres (total of the farm) equals 10%, or
, of the land. The correct answer is Choice (B). - D. Multiply
to get the distance in miles, and then double the answer to account for both legs of the trip. Choice (D) is the correct answer. Note: A quick approach here involves rounding. The distance is about 200 inches, or 400 inches round trip. Multiply that by 3, and you get 1,200 miles. The only choice that comes close is Choice (D). - C. According to the ratio, Margaret should get
of the money and Julie should get 
of the money. You calculate the fractions by adding both sides of the ratio together 
to determine the denominator — Margaret gets 3 parts of the total, and Julie gets 2 parts, so there are 5 total parts. Each side of the ratio then becomes a numerator. Multiply the total amount of money by the fraction representing Julie’s share: Multiply $150,000 by 2, and then divide the answer by 5 to determine Julie’s share of the money. The correct answer is Choice (C). - C. Let x = the percentage of profit. Set up the following proportion and solve for x by cross-multiplying:
Mr. Cameron increased the price by 30% on the shirt he sold.
- C. Calculate this answer by first assigning a common denominator of 60 to all the fractions and adjusting the numerators accordingly:
, 
, 
, and 
. Add the fractions to find out how much time is allotted to all of these tasks. The total time is 
, which leaves 
or 
of the day to the enlisted person’s discretion. One-tenth of 24 hours is 2.4 hours. Therefore, Choice (C) is the correct answer. - B. You want the cost of the designer’s carpet in terms of the outlet-store carpet, so divide the difference in costs by the lower price:
, and 
. The correct answer is Choice (B) — the designer’s carpet is 20% more expensive. - C. Simply add the amounts together:
. Choice (C) is the correct answer. - D. The first ounce costs $0.49. The next 4 ounces cost $0.21 each. Multiply
and then add $0.49 to determine how much mailing a 5-ounce letter costs: 
, and 
, the cost of mailing a 5-ounce letter. Choice (D) is the correct answer. - C. Let x be the distance traveled to Larry’s friend’s house. The time it takes to drive to the house looks like this:
The time it takes to return looks like this:
The total time to travel and return is 5 hours. Therefore,
Next, find the common denominator in order to add the fractions and solve for x:
The answer is 136 miles (rounded to the nearest mile), Choice (C).
- D. A pentagon has 5 sides. If each side measures 1,760 feet, the formula for finding the total feet Joe ran looks like this:
- A. Four people are working on the fence for 7 hours a day; therefore, the family is spending
person-hours on the job each day. Divide the total number of hours needed to complete the job by the total number of person-hours to find the total number of days: 
. It will take 2 days for the team to finish painting the fence. - D. The formula for interest is
. Simply substitute what you know to solve this problem:
Now add the interest to the principal:
Martha is paying $42,400 for her new car.
- B. Use the formula for perimeter to find the answer; w = width and l = length:
- D. Don’t let Jen’s scores throw you off. They’re completely irrelevant. To solve this problem, first add up Mike’s scores and then divide by 2:
-
A. Let x = the price of one shirt.
Let y = the price of one pair of dress pants.
Let z = the price of one tie.
Here’s what you know:
Now subtract the smaller equation from the larger one:
Use that simplified equation to find the value of z:
Finally, add the three prices together:
- A. To solve this problem, you have to know the formula for volume of a cylinder:
where h is the height and r is the radius. Simply plug in the numbers from the question to solve:
Round to the nearest whole number to get Choice (A).
- C. First, subtract the monthly fee from the bill to find the total amount actually spent on electricity:
Now divide your answer by the charge per kilowatt-hour:
Choice (C) is the winner here.
- C. Add the change to see how much Billy has before he buys lunch:
Subtract the cost of the lunch:
- A. The room measures
square feet. Because there are 3 feet in a yard, 1 square yard = 9 square feet. Use this conversion factor to find out how many square yards you need to carpet:
Now just multiply to find the total price of the carpet:
- C. If Jack eats 3 hot dogs in one minute, you know he eats
in 12 minutes. If Jeff eats 2 per minute, he eats 
in 12 minutes. Add 
and you have your answer. That’s a lot of hot dogs. - C. Let i = the interest earned in 1.5 years and convert 1.5 years to months
. Then set up a proportion that represents the ratios of interest earned to time invested:
Cross-multiply to solve:
Jerry will make $270, Choice (C).
- C. Use the formula
, where d is the diameter, to solve this problem:
Rounded to the nearest whole number, the sinkhole’s circumference is 44 kilometers, Choice (C).
- C. The item originally sold for $200. The first reduction was 10%, taking the item down to $180:
The second reduction was 15%, taking the item down to $153:
Choice (C) is your answer.