Percentages
In This Chapter
- Converting percentages into fractions and vice versa
- Changing percentages into decimal numbers
- Finding the percentage of a given quantity
- Getting approximate percentages
- Increasing or decreasing a quantity by a certain percentage
A percentage is a number or ratio expressed as a fraction of 100. And as you probably know, percentages are denoted by the percent sign: %. Percentages are generally used to express how big or small one quantity is when comparing it with another percentage. You probably deal with them frequently in your daily life, so why not make it easier?
In this chapter, you learn how to work with percentages more efficiently, including converting a percentage into a fraction, finding the percentage of a quantity, estimating the percentage of something, and much more.
Converting Percentages into Fractions
To convert a percentage to a fraction, all you need to do is put the number over 100 and simplify by the greatest common factor, if necessary.
Example 1
Convert 75% into a fraction.
Step 1: Turn the percentage into a fraction out of 100.
Step 2: Simplify the fraction by the greatest common factor. For 
, the greatest common factor is 25. Dividing by 25 gives you the fraction 
.
Solution: The fraction is 
.
QUICK TIP
Don’t remember what greatest common factor means? Let me jog your memory: it’s the highest number that divides exactly into two or more numbers. For example 1, the greatest common factor is 25, because when you divide the numerator and denominator by it, you get a number without a remainder or decimal.
Convert 40% into a fraction.
Step 1: Turn the percentage into a fraction out of 100.
Step 2: Simplify the fraction by the greatest common factor. For 
, the greatest common factor is 20. Dividing by 20 gives you the fraction 
.
Solution: The fraction is 
.
Example 3
Convert 65% into a fraction.
Step 1: Turn the percentage into a fraction out of 100.
Step 2: Simplify the fraction by the greatest common factor. For 
, the greatest common factor is 5. Dividing by 5 gives you the fraction 
.
Solution: The fraction is 
.
Converting Fractions into Percentages
To convert a fraction to a percentage, you do the exact opposite of what you did in the previous section—you multiply by 100.
Example 1
Convert 
into a percentage.
Multiply the fraction by 100. You can reduce by the greatest common factor (in this case, 4) before multiplying to get your answer.
Solution: The percentage is 75%.
Example 2
Convert 
into a percentage.
Multiply the fraction by 100. You can reduce by the greatest common factor (in this case, 5) before multiplying to get your answer.
Solution: The percentage is 40%.
Example 3
Convert 
into a percentage.
Multiply the fraction by 100. You can reduce by the greatest common factor (in this case, 20) before multiplying to get your answer.
Solution: The percentage is 85%.
Converting Percentages into Decimals
To convert percentages into decimals, all you need to do is move the decimal two places to the left.
Example 1
Convert 45.6% into a decimal.
Because 45.6% is technically 45.6 divided by 100, simply move the decimal two places to the left.
Solution: The decimal is 0.456.
Example 2
Convert 8.09% into a decimal.
This is the same as dividing 8.09 by 100, so all you need to do is shift the decimal two places to the left.
Solution: The decimal is 0.0809.
Example 3
Convert 0.674% into a decimal.
Because you’re basically dividing .674 by 100, just shift the decimal point two places to the left.
Solution: The decimal is 0.00674.
Finding the Percentage of a Given Quantity
You can find the percentage of a given quantity by using our old friend, the vertically and crosswise method (see Chapter 2 for the process).
Two-Digit Numbers
If the percentage and quantity are two-digit numbers, you can use the two-digit version of the vertically and crosswise method.
Example 1
Find 45% of 81.
Step 1: Multiply 45 and 81 using the vertically and crosswise method. For two-digit numbers, that means you start by multiplying the right column: 5 × 1 = 5. Put down 5.
Step 2: Multiply crosswise and add: (4 × 1) + (5 × 8) = 4 + 40 = 44. Put down 4 and carry over the 4 to the next step.
Step 3: Multiply the left column: 4 × 8 = 32. Add the carryover: 32 + 4 = 36. Put down 36.
Step 4: Put the decimal in the answer. Because you’re dividing 3,645 by 100, put the decimal two places to the left.
Solution: The answer is 36.45.
Find 73% of 98.
Step 1: Multiply 73 and 98 using the vertically and crosswise method. Begin by multiplying the right column: 3 × 8 = 24. Put down 4 and carry over the 2 to the next step.
Step 2: Multiply crosswise and add: (7 × 8) + (3 × 9) = 56 + 27 = 83. Add the carryover: 83 + 2 = 85. Put down 5 and carry over the 8 to the next step.
Step 3: Multiply the left column: 7 × 9 = 63. Add the carryover: 63 + 8 = 71. Put down 71.
Step 4: Put the decimal in the answer. Because you’re dividing 7,154 by 100, put the decimal two places to the left.
Solution: The answer is 71.54.
Example 3
Find 23% of 67.
Step 1: Multiply 23 and 67 using the vertically and crosswise method. Start with the right column: 3 × 7 = 21. Put down 1 and carry over the 2 to the next step.
Step 2: Multiply crosswise and add: (3 × 6) + (2 × 7) = 18 + 14 = 32. Add the carryover: 32 + 2 = 34. Put down 4 and carry over the 3 to the next step.
Step 3: Multiply the left column: 2 × 6 = 12. Add the carryover: 12 + 3 = 15. Put down 15.
Step 4: Put the decimal in the answer. Because you’re dividing 1,541 by 100, put the decimal two places to the left.
Solution: The answer is 15.41.
Three-Digit Numbers
If the percentage and quantity are three-digit numbers, you can use the three-digit version of the vertically and crosswise method.
QUICK TIP
If you’d like a visual of the multiplication steps for the three-digit version of the vertically and crosswise method, flip back to Chapter 2.
Example 1
Find 14.2% of 682.
Step 1: Ignoring the decimal point, multiply 142 and 682 using the vertically and crosswise method. Begin by multiplying the right column: 2 × 2 = 4. Put down 4.
Step 2: Multiply the right and middle columns crosswise and add: (2 × 4) + (8 × 2) = 8 + 16 = 24. Put down 4 and carry over the 2 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (2 × 1) + (8 × 4) + (6 × 2) = 2 + 32 + 12 = 46. Add the carryover: 46 + 2 = 48. Put down 8 and carry over the 4 to the next step.
Step 4: Multiply the right and middle columns crosswise and add: (6 × 4) + (8 × 1) = 24 + 8 = 32. Add the carryover: 32 + 4 = 36. Put down 6 and carry over the 3 to the next step.
Step 5: Multiply left column: 6 × 1 = 6. Add the carryover: 6 + 3 = 9. Put down 9.
Step 6: Put the decimal in the answer. Because the percentage had a decimal one to the left, place it there so 96,844 becomes 9,684.4. Now you have to divide 9,684.4 by 100, which simply means putting the decimal two places to the left.
Solution: The answer is 96.844.
Find 71.1% of 475.
Step 1: Ignoring the decimal point, multiply 711 and 475 using the vertically and crosswise method. Begin by multiplying the right column: 5 × 1 = 5. Put down 5.
Step 2: Multiply the right and middle columns crosswise and add: (5 × 1) + (7 × 1) = 5 + 7 = 12. Put down 2 and carry over the 1 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (5 × 7) + (7 × 1) + (4 × 1) = 35 + 7 + 4 = 46. Add the carryover: 46 + 1 = 47. Put down 7 and carry over the 4 to the next step.
Step 4: Multiply the right and middle columns crosswise and add: (4 × 1) + (7 × 7) = 4 + 49 = 53. Add the carryover: 53 + 4 = 57. Put down 7 and carry over the 5.
Step 5: Multiply the right column: 4 × 7 = 28. Add the carryover: 28 + 5 = 33. Put down 33.
Step 6: Put the decimal in the answer. Because the percentage had a decimal one to the left, place it there so 337,725 becomes 33,772.5. Now you have to divide 33,772.5 by 100, which simply means putting the decimal two places to the left.
Solution: The answer is 337.725.
Example 3
Find 87.2% of 584.
Step 1: Ignoring the decimal point, multiply 872 and 584 using the vertically and crosswise method. Begin by multiplying the right column: 4 × 2 = 8. Put down 8.
Step 2: Multiply the right and middle columns crosswise and add: (4 × 7) + (8 × 2) = 28 + 16 = 44. Put down 4 and carry over the 4 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (5 × 2) + (8 × 7) + (4 × 8) = 10 + 56 + 32 = 98. Add the carryover: 98 + 4 = 102. Put down 2 and carry over the 10 to the next step.
Step 4: Multiply the right and middle columns crosswise and add: (8 × 8) + (7 × 5) = 64 + 35 = 99. Add the carryover: 99 + 10 = 109. Put down 9 and carry over the 10 to the next step.
Step 5: Multiply the right column: 5 × 8 = 40. Add the carryover: 40 + 10 = 50. Put down 50.
Step 6: Put the decimal in the answer. Because the percentage had a decimal one to the left, place it there so 509,248 becomes 50,924.8. Now you have to divide 50,924.8 by 100, which simply means putting the decimal two places to the left.
Solution: The answer is 509.248.
Expressing One Quantity as a Percentage of Another
For this scenario, all you have to do is divide the numbers and then multiply the answer by 100 to get the percentage.
Example 1
Find 75 as a percentage of 250.
Step 1: Divide 75 by 250.
75 ÷ 250 = .3
Step 2: Multiply the number by 100. If you recall, you simply need to move the decimal two places to the right to get the number.
.3 × 100 = 30
Solution: The answer is 30%.
Example 2
Find 82 as a percentage of 450.
Step 1: Divide 82 by 450.
82 ÷ 450 = .1822
Step 2: Multiply the number by 100. If you recall, you simply need to move the decimal two places to the right to get the number.
.1822 × 100 = 18.22
Solution: The answer is 18.22%.
Example 3
Find 35 as a percentage of 890.
Step 1: Divide 35 by 890.
35 ÷ 890 = .03932
Step 2: Multiply the number by 100. If you recall, you simply need to move the decimal two places to the right to get the number.
.03932 × 100 = 3.932
Solution: The answer is 3.932%.
Approximating Percentages
Sometimes, you may need to get an idea or estimate of how much one number is a percentage of the other. Without the aid of a calculator, you can use the percentages you do know for the larger number and build off that information to get the answer.
Example 1
Estimate 42 as a percentage of 800.
Step 1: Start with the simplest percentage that will reduce the larger number to two digits. For 800, you can use 10%, which gives you 80.
Step 2: Build off that percentage until you get the parts of your answer. Half of 10% is 5%, and half of 80 is 40; that gives you one piece of your answer, because it’s close to 42. To get the percentage for 2, start with 1% of 800, which is 8. The 2 is one fourth of 8, so the percentage has to be one fourth of 1%, or 0.25%.
10% of 800 = 80
5% of 800 = 40
1% of 800 = 8
0.25% of 800 = 2
Step 3: Add the percentages you found for 40 and 2.
5% + 0.25% = 5.25%.
Solution: The answer is 5.25%.
Example 2
Estimate 45 as a percentage of 650.
Step 1: Start with the simplest percentage that will reduce the larger number to two digits. For 650, you can use 10%, which gives you 65.
Step 2: Build off that percentage until you get the parts of your answer. Half of 10% is 5%, and half of 65 is 32.5; that gives you one piece of your answer, because it’s close to 45. To get the percentage for the rest of the answer, which is a little over 12, start with 1% of 650, which is 6.5.
5% of 650 = 32.5
1% of 650 = 6.5
Step 3: Add the percentages you found for 32.5 and 6.5; in case, add the 6.5 twice, because the remainder above was over 12. The number equal to this percentage is 45.5, slightly higher than 45, so the percentage is a little less than what you get when you add.
5% + 1% + 1% = 7%
Solution: The answer is a little less than 7%.
Example 3
Estimate 73 as a percentage of 3,568.
Step 1: Start with the simplest percentage that will reduce the larger number to two digits. For 3,568, you can use 1%, which gives you 35.68.
Step 2: Build off that percentage until you get the parts of your answer. The double of 1% is 2%, and the double of 35.68 is 71.36; this is extremely close, so you can use this percentage as your answer. Because 71.36 is less than 73, that means your percentage is a little higher than 2%.
2% of 3568 = 71.36
Solution: The answer is a little more than 2%.
QUICK TIP
I used percentages like 10% and 1% as starting points in the examples, because those involve simply moving the decimal to find the value. If you still have trouble with knowing where to move the decimal, feel free to revisit Chapter 7.
Percentage Increase or Decrease
To increase or decrease a number by a certain percentage, you utilize the vertically and crosswise method yet again. The following sections break down how to perform the process for each type.
Percentage Increase
To find the solution for a number increased by a percentage, using the vertically and crosswise method, multiply the number by the decimal version of the percentage plus 1. You add the 1 because you’re finding how much more it is than the number.
Example 1
Increase 673 by 23%.
Step 1: The decimal version of 23% is .23; add 1 to that, so it becomes 1.23. Now, ignoring the decimal for the moment, multiply 673 and 1.23. Start with the right column: 3 × 3 = 9. Put down 9.
Step 2: Multiply the right and middle columns crosswise and add: (3 × 7) + (3 × 2) = 21 + 6 = 27. Put down 7 and carry over the 2 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (3 × 6) + (7 × 2) + (3 × 1) = 18 + 14 + 3 = 35. Add the carryover: 35 + 2 = 37. Put down 7 and carry over the 3 to the next step.
Step 4: Multiply the left and middle columns crosswise and add: (1 × 7) + (2 × 6) = 7 + 12 = 19. Add the carryover: 19 + 3 = 22. Put down 2 and carry over the 2 to the next step.
Step 5: Multiply the left column: 1 × 6 = 6. Add the carryover: 6 + 2 = 8. Put down 8. Now add the decimal point. The decimal in 1.23 is two places to the left, so for 82,779, put the decimal two places to the left.
Solution: The answer is 827.79.
Example 2
Increase 450 by 34%.
Step 1: The decimal version of 34% is .34; add 1 to that, so it becomes 1.34. Now, ignoring the decimal for the moment, multiply 450 and 1.34. Start with the right column: 4 × 0 = 0. Put down 0.
Step 2: Multiply the right and middle columns crosswise and add: (4 × 5) + (3 × 0) = 20 + 0 = 20. Put down 0 and carry over the 2 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (4 × 4) + (1 × 0) + (3 × 5) = 16 + 0 + 15 = 31. Add the carryover: 31 + 2 = 33. Put down 3 and carry over the 3 to the next step.
Step 4: Multiply the left and middle columns crosswise and add: (3 × 4) + (1 × 5) = 12 + 5 = 17. Add the carryover: 17 + 3 = 20. Put down 0 and carry over the 2 to the next step.
Step 5: Multiply the right column: 1 × 4 = 4. Add the carryover: 4 + 2 = 6. Now add the decimal point. The decimal in 1.34 is two places to the left, so for 60,300, put the decimal two places to the left.
Solution: The answer is 603.00.
Percentage Decrease
Like you did for percentage increase, you use the vertically and crosswise method to get your answer. However, this time, you multiply the number by the decimal value of the percentage minus 1, because you’re finding how much less it is than the number.
Example 1
Decrease 500 by 25%.
Step 1: The decimal version of 25% is .25; subtract 1 from that, so it becomes .75. Now, ignoring the decimal for the moment, multiply 500 and 0.75. Start with the right column: 5 × 0 = 0. Put down 0.
Step 2: Multiply the right and middle columns crosswise and add: (5 × 0) + (7 × 0) = 0 + 0 = 0. Put down 0.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (5 × 5) + (0 × 0) + (7 × 0) = 25 + 0 + 0 = 25. Put down 5 and carry over the 2 to the next step.
Step 4: Multiply the left and middle columns crosswise and add: (7 × 5) + (0 × 0) = 35 + 0 = 35. Add the carryover: 35 + 2 = 37. Put down 7 and carry over the 3 to the next step.
Step 5: Multiply the left column: 0 × 5 = 0. Add the carryover: 0 + 3 = 3. Put down 3. Now add the decimal point. The decimal in .75 is two places to the left, so for 37,500, put the decimal two places to the left.
Solution: The answer is 375.00.
QUICK TIP
In the first step of the process, rather than subtracting the decimal version of the percentage from 1, you can just figure out what percentage added to the one in the problem equals 100%. For example 1, you probably know 75% added to 25% equals 100%, so feel free to use that shortcut to speed up your calculations.
Decrease 878 by 62%.
Step 1: The decimal version of 62% is .62; subtract 1 from that, so it becomes .38. Now, ignoring the decimal for the moment, multiply 878 and 0.38. Start with the right column: 8 × 8 = 64. Put down 4 and carry over the 6 to the next step.
Step 2: Multiply the right and middle columns crosswise and add: (8 × 7) + (3 × 8) = 56 + 24 = 80. Add the carryover: 80 + 6 = 86. Put down 6 and carry over the 8 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (8 × 8) + (0 × 8) + (3 × 7) = 64 + 0 + 21 = 85. Add the carryover: 85 + 8 = 93. Put down 3 and carry over the 9 to the next step.
Step 4: Multiply the left and middle columns crosswise and add: (3 × 8) + (0 × 7) = 24 + 0 = 24. Add the carryover: 24 + 9 = 33. Put down 3 and carry over the 3 to the next step.
Step 5: Multiply the left column: 0 × 8 = 0. Add the carryover: 0 + 3 = 3. Put down 3. Now add the decimal point. The decimal in .38 is two places to the left, so for 33,364, put the decimal two places to the left.
Solution: The answer is 333.64.
Example 3
Decrease 345 by 18%.
Step 1: The decimal version of 18% is .18; subtract 1 from that, so it becomes .82. Now, ignoring the decimal for the moment, multiply 878 and 0.82. Start with the right column: 2 × 5 = 10. Put down 0 and carry over the 1 to the next step.
Step 2: Multiply the right and middle columns crosswise and add: (2 × 4) + (8 × 5) = 8 + 40 = 48. Add the carryover: 48 + 1 = 49. Put down 9 and carry over the 4 to the next step.
Step 3: Multiply in the star pattern—the bottom-right and top-left, the top and bottom middle, and the bottom-left and top-right numbers—and add: (2 × 3) + (0 × 5) + (8 × 4) = 6 + 0 + 32 = 38. Add the carryover: 38 + 4 = 42. Put down 2 and carry over the 4 to the next step.
Step 4: Multiply the left and middle columns crosswise and add: (8 × 3) + (0 × 4) = 24 + 0 = 24. Add the carryover: 24 + 4 = 28. Put down 8 and carry over the 2 to the next step.
Step 5: Multiply the left column: 0 × 3 = 0. Add the carryover: 0 + 2 = 2. Put down 2. Now add the decimal point. The decimal in .82 is two places to the left, so for 28,290, put the decimal two places to the left.
Solution: The answer is 282.90.
- Put the percentage over 100 and simplify by the greatest common factor in order to convert a fraction to a percentage.
- When converting fractions to percentages, multiply by 100 and simplify as necessary.
- You only have to move the decimal two places to the left to change a percentage into a decimal number.
- To find the percentage of a certain quantity, multiply the quantity and the percentage using the vertically and crosswise method.
- You can easily find the approximate percentage for a number by breaking down the number into the most recognizable percentages.
- To find the percentage increase or decrease of a number, multiply the number by the decimal version of the percentage plus or minus 1.