Percents for Firefighter Exam Study Guide (page 3)

What Is a Percent?

A percent is a special kind of fraction or part of something. The bottom number (the denominator) in such a fraction is always 100. For example, 17% is the same as . Literally, the word percent means per 100 parts. The root cent means 100: A century is 100 years, there are 100 cents in a dollar, etc. Thus, 17% means 17 parts out of 100. Because fractions can also be expressed as decimals, 17% is also equivalent to .17, which is 17 hundredths.

You come into contact with percents every day. Sales tax, interest, and discounts are just a few common examples.

If you are shaky on fractions, you may want to review the fraction section before reading further.

Changing a Decimal to a Percent and Vice Versa

To change a decimal to a percent, move the decimal point two places to the right and tack on a percent sign (%) at the end. If the decimal point moves to the very right of the number, you don't have to write the decimal point. If there aren't enough places to move the decimal point, add zeros on the right before moving the decimal point.

To change a percent to a decimal, drop off the percent sign and move the decimal point two places to the left. If there aren't enough places to move the decimal point, add zeros on the left before moving the decimal point.

Try changing the following decimals to percent. The answers can be found at the end of this section on page 174.

55. .45
56. .008
57. 

Now change these percents to decimals:

58. 12%
59. 
60. 250%

Changing a Fraction to a Percent and Vice Versa

To change a fraction to a percent, there are two techniques. Each is illustrated by changing the fraction to a percent:

Technique 1:	Multiply the fraction by 100%.
	Multiply by 100%:
Technique 2:	Divide the fraction's bottom number into the top number; then move the decimal point two places to the right and tack on a percent sign (%).
	Divide 4 into 1 and move the decimal point 2 places to the right:

To change a percent to a fraction, remove the percent sign and write the number over 100. Then reduce if possible.

Example: Change 4% to a fraction.

1. Remove the % and write the fraction 4 over 100:  
2. Reduce:  

Here's a more complicated example: Change % to a fraction.

1. Remove the % and write the fraction over 100:
2. Since a fraction means "top number divided by bottom number," rewrite the fraction as a division problem:   ÷ 100
3. Change the mixed number () to an improper fraction ():       ÷
4. Flip the second fraction () and multiply:    

Try changing these fractions to percents:

Now, change these percents to fractions:

64. 95%
65. 
66. 125%

Sometimes it is more convenient to work with a percent as a fraction or a decimal. Rather than have to calculate the equivalent fraction or decimal, consider memorizing the equivalence table on page 171. Not only will this increase your efficiency on the math test, but it will also be practical for real-life situations.

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?

Although 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:

To solve each of the three varieties, we are going to 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:

67. 1% of 25
68. 18.2% of 50
69. of 100
70. 125% of 60

Find what percent one number is of another number.

71. 10 is what percent of 20?
72. 4 is what percent of 12?
73. 12 is what percent of 4?

Find the whole when the percent of it is given.

74. 15% of what number is 15?
75. of what number is 3?
76. 200% of what number is 20?

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

77. The deputy chief, after reviewing the roll call, finds that 15% of the 115 members of B shift are absent due to illness and approved vacation. How many personnel have to be called in to fill in for the absent personnel?

a. 15       b. 17       c. 115       d. 132

78. Twenty percent of Engine 5's firefighters are women. If there are 30 women on the roster, how many total members are there?

a. 36       b. 70       c. 100       d. 150

79. During June, Firefighter Hanson's company responded to 150 alarms. Of these, 26 were wildland responses. What percent of the total alarms were wildland responses?

a. 12%       b. 17%       c. 22%       d. 26%

80. Chief Smith has approved a purchase of new turnout gloves at $25.50 a pair. If his company received a government discount of 15%, what was the original price per pair?

a. $25.50       b. $27.50       c. $28.45       d. $30.00