Percents and Averages for Armed Services Vocational Aptitude Battery (ASVAB) Study Guide (page 2)

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