Percents and Averages for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide

Percent Word Problems

Word problems involving percents come in three main varieties:

• Find a percent of a whole.
  Example: What is 30% of 40?
• Find what percent one number is of another number.
  Example: 12 is what percent of 40?
• Find the whole when the percent of it is given.
  Example: 12 is 30% of what number?

While each variety has its own approach, there is a single shortcut formula you can use to solve each of these:

The is is the number that usually follows or is just before the word is in the question.

The of is the number that usually follows the word of in the question.

The % is the number that is in front of the % or percent in the question.

Or you may think of the shortcut formula as:

part × 100 = whole × %

To solve each of the three varieties, let's use the fact that the cross-products are equal. The cross-products are the products of the numbers diagonally across from each other. Remembering that product means multiply, here's how to create the cross-products for the percent shortcut:

part × 100 = whole × %

Here's how to use the shortcut with cross-products:

• Find a percent of a whole.
  What is 30% of 40?
  30 is the % and 40 is the of number:        
  Cross-multiply and solve for is:                 is × 100 = 40 × 30
                                                                  is × 100 = 1,200
                                                                  12 × 100 = 1,200
  Thus, 12 is 30% of 40.
• Find what percent one number is of another number.
  12 is what percent of 40?
  12 is the is number and 40 is the of number:        
  Cross-multiply and solve for %:                           12 × 100 = 40 × %
                                                                            1,200 = 40 × %
                                                                            1,200 = 40 × 30
  Thus, 12 is 30% of 40.
• Find the whole when the percent of it is given.
  12 is 30% of what number?
  12 is the is number and 30 is the %:                    
  Cross-multiply and solve for the of number:         12 × 100 = of × 30
                                                                            1,200 = of × 30
                                                                            1,200 = 40 × 30
  Thus, 12 is 30% of 40.

You can use the same technique to find the percent increase or decrease. The is number is the actual increase or decrease, and the of number is the original amount.

Example: If a merchant puts his $20 hats on sale for $15, by what percent does he decrease the selling price?

1. Calculate the decrease, the is number:                         $20 – $15 = $5
2. The of number is the original amount, $20
3. Set up the equation and solve for of by cross-multiplying:    
                                                                                   5 × 100 = 20 × %
                                                                                   500 = 20 × %
                                                                                   500 = 20 × 25
4. Thus, the selling price is decreased by 25%.
   If the merchant later raises the price of the hats from $15 back to $20,            
   don't be fooled into thinking that the percent increase is also 25%! It's             5 × 100 = 15 × %
   actually more, because the increase amount of $5 is now based on a lower     500 = 15 × %
   original price of only $15:                                                                              
   Thus, the selling price is increased by 33%.

Find a percent of a whole:

13. 1% of 25
14. 18.2% of 50
15. of 100
16. 125% of 60

Find what percent one number is of another number.

17. 30 is what % of 60?
18. 4 is what % of 12?
19. 12 is what % of 4?

Find the whole when the percent of it is given.

20. 15% of what number is 15?
21. of what number is 3?
22. 200% of what number is 20?

Now try your percent skills on some real-life problems.

23. Last Monday, 20% of 140 staff members was absent. How many employees were absent that day?
    1. 14
    2. 20
    3. 28
    4. 112
    5. 126
24. 30% of Vero's postal service employees are women. If there are 90 women in Vero's postal service, how many men are employed there?
    1. 150
    2. 180
    3. 210
    4. 260
    5. 300
25. Of the 840 shirts sold at a retail store last month, 42 had short sleeves. What percent of the shirts were short sleeved?
    1. .5%
    2. 2%
    3. 5%
    4. 20%
    5. 50%
26. Sam's Shoe Store put all of its merchandise on sale for 20% off. If Jason saved $10 by purchasing one pair of shoes during the sale, what was the original price of the shoes before the sale?
    1. $12
    2. $20
    3. $40
    4. $50
    5. $70

Answers to Percent Questions

1. 67%
2. 0.8%
3. 
4. 0.12
5. 0.7825
6. 2.5
7. 
8. 64%
9. 
10. 
11. 
12. 
13. 
14. 9.1
15. 
16. 75
17. 50%
18. 
19. 300%
20. 100
21. 8
22. 10
23. c.
24. c.
25. c.
26. d.

Averages

An average, also called an arithmetic mean, is a number that typifies a group of numbers, a measure of central tendency. You come into contact with averages on a regular basis: your bowling average, the average grade on a test, the average number of hours you work per week.

To calculate an average, add up the number of items being averaged and divide by the number of items.

Example: What is the average of 6, 10, and 20?
Solution: Add the three numbers together and divide by 3: = 12

Shortcut

Here's a shortcut for some average problems.

• Look at the numbers being averaged. If they are equally spaced, like 5, 10, 15, 20, and 25, then the average is the number in the middle, or 15 in this case.
• If there is an even number of such numbers, say 10, 20, 30, and 40, then there is no middle number. In this case, the average is half-way between the two middle numbers. In this case, the average is half-way between 20 and 30, or 25.
• If the numbers are almost evenly spaced, you can probably estimate the average without going to the trouble of actually computing it. For example, the average of 10, 20, and 32 is just a little more than 20, the middle number.

Try these average questions.

1. Bob's bowling scores for the last five games were 180, 182, 184, 186, and 188.What was his average bowling score?
   1. 182
   2. 183
   3. 184
   4. 185
   5. 186
2. Jackson averaged 45 miles an hour for the three hours he drove in town and 60 miles an hour for the three hours he drove on the highway. What was his average speed in miles per hour?
   1. 26
   2. 
   3. 48
   4. 
   5. 105
3. There are 10 females and 20 males in a history class. If the females achieved an average score of 85 and the males achieved an average score of 95, what was the class average? (Hint: don't fall for the trap of taking the average of 85 and 95; there are more 95s being averaged than 85s, so the average is closer to 95.)
   1. 
   2. 
   3. 92
   4. 
   5. 95

Answers to Averages Questions

1. c.
2. d.
3. b.