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0.6 Percentages

Karen Perilloux; Prakash Ghimire; and Lauren Johnson

Learning Objectives

By the end of this section, you will be able to:

  • Use the definition of percent
  • Convert percents to fractions and decimals
  • Convert decimals and fractions to percents
  • Translate and solve basic percent equations
  • Solve applications of percent
  • Find percent increase and percent decrease

Use the Definition of Percent

How many cents are in one dollar? There are 100 cents in a dollar.

How many years are in a century? There are 100 years in a century.

Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is 100.

We use the percent symbol, %, to show percent.

Percent

A percent is a ratio whose denominator is 100.

According to data from the American Association of Community Colleges (2015), about 57% of community college students are female. This means 57 out of every 100 community college students are female, as Figure 1 shows. Out of the 100 squares on the grid, 57 are shaded, which we write as the ratio [latex]frac{57}{100}[/latex].
The figure shows a hundred flat with 57 units shaded.
Figure 1
Among every 100 community college students, 57 are female.

Similarly, [latex]25%[/latex] means a ratio of [latex]frac{25}{100}[/latex], [latex]3%[/latex] means a ratio of [latex]frac{3}{100}[/latex], and [latex]100%[/latex] means a ratio of [latex]frac{100}{100}[/latex].

In words, “one hundred percent” means the total [latex]100%[/latex] is [latex]frac{100}{100}[/latex], and since [latex]frac{100}{100}=1[/latex], we see that [latex]100%[/latex] means 1 whole.

Example

According to the Public Policy Institute of California (2010), 44% of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.

 Solution

Try It

Write the percent as a ratio.

According to a survey, 89% of college students have a smartphone.

Show Solution

[latex]\frac{89}{100}[/latex]

Try It

Write the percent as a ratio.

A study found that 72% of U.S. teens send text messages regularly.

Show Solution

[latex]\frac{72}{100}[/latex]

 

Example

In 2007, according to a U.S. Department of Education report, 21 out of every 100 first-time freshmen college students at 4-year public institutions took at least one remedial course. Write this as a ratio and then as a percent.

 Solution

 

Try It

Write as a ratio and then as a percent: The American Association of Community Colleges reported that 62 out of 100 full-time community college students balance their studies with full-time or part time employment.

Show Solution

[latex]\frac{62}{100}\text{ or }62\%[/latex]

Try It

Write as a ratio and then as a percent: In response to a student survey, 41 out of 100 Santa Ana College students expressed a goal of earning an Associate’s degree or transferring to a four-year college.

Show Solution

[latex]\frac{41}{100}\text{ or }41\%[/latex]

 

Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per 100, so the denominator of the fraction is 100.

How To

Convert a percent to a fraction.

  1. Write the percent as a ratio with the denominator 100.
  2. Simplify the fraction if possible.

Example

Convert each percent to a fraction:

a) [latex]36\%[/latex]

b)  [latex]125\%[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each percent to a fraction:

a) [latex]48\%[/latex]

b) [latex]110\%[/latex]

Show Solution
a) [latex]\frac{12}{25}[/latex]
b) [latex]\frac{11}{10}[/latex]

 

Try It

Convert each percent to a fraction:

a) [latex]64\%[/latex]

b) [latex]150\%[/latex]

Show Solution
a) [latex]\frac{16}{25}[/latex]
b) [latex]\frac{3}{2}[/latex]

The previous example shows that a percent can be greater than 1.  We saw that [latex]125\%[/latex] means [latex]\frac{125}{100}[/latex], or [latex]\frac{5}{4}[/latex].  These are improper fractions, and their values are greater than one.

Example

Convert each percent to a fraction:

a) [latex]24.5\%[/latex]

b) [latex]33\frac{1}{3}\%[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each percent to a fraction:

a) [latex]64.4\%[/latex]

b) [latex]66\frac{2}{3}\%[/latex]

Show Solution
a) [latex]\frac{161}{250}[/latex]
b) [latex]\frac{2}{3}[/latex]

Try It

Convert each percent to a fraction:

a) [latex]42.5\%[/latex]

b) [latex]8\frac{3}{4}\%[/latex]

Show Solution
a) [latex]\frac{17}{40}[/latex]
b) [latex]\frac{35}{400}[/latex]
To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

How To

Convert a percent to a decimal.

  1. Write the percent as a ratio with the denominator 100.
  2. Convert the fraction to a decimal by dividing the numerator by the denominator.

Example

Convert each percent to a decimal:

a) [latex]6\%[/latex]

b) [latex]78\%[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each percent to a decimal:

a)   [latex]9\%[/latex]

b) [latex]87\%[/latex]

Show Solution
a) [latex]0.09[/latex]
b) [latex]0.87[/latex]

Try It

Convert each percent to a decimal:

a) [latex]3\%[/latex]

b) [latex]91\%[/latex]

Show Solution
a) [latex]0.03[/latex]
b) [latex]0.91[/latex]

Example

Convert each percent to a decimal:

a) [latex]135\%[/latex]

b) [latex]12.5\%[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each percent to a decimal:

a) [latex]115\%[/latex]

b) [latex]23.5\%[/latex]

Show Solution
a) [latex]1.15[/latex]
b) [latex]0.235[/latex]

Try It

Convert each percent to a decimal:

a) [latex]123\%[/latex]

b) [latex]16.8\%[/latex]

Show Solution
a) [latex]1.23[/latex]
b) [latex]0.168[/latex]
Let’s summarize the results from the previous examples in Table 1, and look for a pattern we could use to quickly convert a percent number to a decimal number.
Table 1
Percent Decimal
6%6% 0.060.06
78%78% 0.780.78
135%135% 1.351.35
12.5%12.5% 0.1250.125
Do you see the pattern?

To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the % sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer 6 as 6.0, we can think of 6% as 6.0%.)  Notice that we may need to add zeros in front of the number when moving the decimal to the left.

Figure 2 uses the percents in Table 1 and shows visually how to convert them to decimals by moving the decimal point two places to the left.


The figures shows two columns and five rows . The first row is a header row and it labels each column “Percent” and “Decimal”. Under the “Percent” column are the values: 6%, 78%, 135%, 12.5%. Under the “Decimal” column are the values: 0.06, 0.78, 1.35, 0.125. There are two jumps for each percent to show how to convert it to a decimal.
Figure 2

Example

Among a group of business leaders, 77% believe that poor math and science education in the U.S. will lead to higher unemployment rates.

Convert the percent to: (a) a fraction (b) a decimal.

 Solution to part (a)
 Solution to part (b)

Try It

Convert the percent to: (a) a fraction and (b) a decimal.

Twitter’s share of web traffic jumped 24% when one celebrity tweeted live on air.

Show Solution
a) [latex]\frac{6}{25}[/latex]
b) [latex]0.24[/latex]

Try It

Convert the percent to:  (a) a fraction and (b) a decimal.

The U.S. Census estimated that in 2013, 44% of the population of Boston age 25 or older have a bachelor’s or higher degrees.

Show Solution
a) [latex]\frac{11}{25}[/latex]
b) [latex]0.44[/latex]

Example

There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is 25%.

Convert the percent to:  (a) a fraction and (b) a decimal.


The figure shows someone holding a deck of cards.
Figure 3 (credit: Riles32807, Wikimedia Commons)
 Solution to part (a)
 Solution to part (b)

Try It

Convert the percent to:  (a) a fraction, and (b) a decimal.

The probability that it will rain Monday is 30%.

Show Solution
a) [latex]\frac{3}{10}[/latex]
b) [latex]0.3[/latex]

Try It

Convert the percent to:  (a) a fraction, and (b) a decimal.

The probability of getting heads three times when tossing a coin three times is 12.5%.

Show Solution
a) [latex]\frac{1}{8}[/latex]
b) [latex]0.125[/latex]

Convert Decimals and Fractions to Percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.

How To

Convert a decimal to a percent.

  1. Write the decimal as a fraction.
  2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
  3. Write this ratio as a percent.

Example

Convert each decimal to a percent:

a) [latex]0.05[/latex]

b) [latex]0.83[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each decimal to a percent:

a) [latex]0.01[/latex]

b) [latex]0.17[/latex]

Show Solution
a) [latex]1\%[/latex]
b) [latex]17\%[/latex]

Try It

Convert each decimal to a percent:

a) [latex]0.04[/latex]

b) [latex]0.41[/latex]

Show Solution
a) [latex]4\%[/latex]
b) [latex]41\%[/latex]

To convert a mixed number to a percent, we first write it as an improper fraction.

Example

Convert each decimal to a percent:

a)  [latex]1.05[/latex]

b) [latex]0.075[/latex]

 Solution to part (a)
 Solution to part (b)

Try It

Convert each decimal to a percent:

a) [latex]1.75[/latex]

b) [latex]0.0825[/latex]

Show Solution
a) [latex]175\%[/latex]
b) [latex]8.25\%[/latex]

Try It

Convert each decimal to a percent:

a) [latex]2.25[/latex]

b) [latex]0.0925[/latex]

Show Solution
a) [latex]225\%[/latex]
b) [latex]9.25\%[/latex]

Let’s summarize the results from the previous examples in Table 2 so we can look for a pattern.

Decimal Percent
0.050.05 5%5%
0.830.83 83%83%
1.051.05 105%105%
0.0750.075 7.5%7.5%
Table 2

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

Figure 4 uses the decimal numbers in Table 2 and shows visually to convert them to percents by moving the decimal point two places to the right and then writing the % sign.


The figure shows two columns and five rows. The first row is a header row and it labels each column “Decimal” and “Percent”. Under the “Decimal” column are the values: 0.05, 0.83, 1.05, 0.075, 0.3. Under the “Percent” column are the values: 5%, 83%, 105%, 7.5%, 30%. There are two jumps for each decimal to show how to convert it to a percent.
Figure 4

We have learned how to convert fractions to decimals. Now we also know how to change decimals to percents. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

How To

Convert a fraction to a percent.

  1. Convert the fraction to a decimal.
  2. Convert the decimal to a percent.

Example

Convert each fraction or mixed number to a percent:

a) [latex]\frac{3}{4}[/latex]

b) [latex]\frac{11}{8}[/latex]

c) [latex]2\frac{1}{5}[/latex]

 Solution to part (a)
 Solution to part (b)
 Solution to part (c)

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.

Try It

Convert each fraction or mixed number to a percent:

a) [latex]\frac{5}{8}[/latex]

b) [latex]\frac{11}{4}[/latex]

c) [latex]3\frac{2}{5}[/latex]

Show Solution
a) [latex]62.5\%[/latex]
b) [latex]275\%[/latex]
c) [latex]340\%[/latex]

Try It

Convert each fraction or mixed number to a percent:

a) [latex]\frac{7}{8}[/latex]

b) [latex]\frac{9}{4}[/latex]

c) [latex]1\frac{3}{5}[/latex]

Show Solution
a) [latex]87.5\%[/latex]
b) [latex]225\%[/latex]
c) [latex]160\%[/latex]

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we will round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases in this book we will round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

Example

Convert [latex]\frac{5}{7}[/latex] to a percent.

 Solution

Try It

Convert the fraction to a percent: [latex]\frac{3}{7}[/latex].

Show Solution

[latex]42.9\%[/latex]

Try It

Convert the fraction to a percent: [latex]\frac{4}{7}[/latex].

Show Solution

[latex]57.1\%[/latex]

When we first looked at fractions and decimals, we saw that some fractions converted to a repeating decimal. For example, when we converted the fraction [latex]\frac{4}{3}[/latex] to a decimal, we wrote the answer as [latex]1.\bar{3}[/latex].

We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.

Example

An article in a medical journal claimed that approximately [latex]\frac{1}{3}[/latex] of American adults are obese. Convert the fraction [latex]\frac{1}{3}[/latex] to a percent.

 Solution

Try It

Convert the fraction to a percent:

According to the U.S. Census Bureau, about [latex]\frac{1}{9}[/latex] of United States housing units have just 1 bedroom.

Show Solution

[latex]11.\bar{1}\%[/latex]

Try It

Convert the fraction to a percent:

According to the U.S. Census Bureau, about [latex]\frac{1}{6}[/latex] of Colorado residents speak a language other than English at home.

Show Solution

[latex]16.\bar{6}\%[/latex]

Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.

We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Aolani and her friends ate dinner at a restaurant, the bill came to $80. They wanted to leave a 20% tip. What amount would the tip be?

To solve this, we want to find what amount is 20% of $80. The $80 is called the base. The amount of the tip would be 0.20(80), or $16. See Figure 6.6. To find the amount of the tip, we multiplied the percent by the base.
The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.
Figure 6.6: 20% of an $80 restaurant bill comes out to $16.

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

Example 

What number is 35% of 90?

Solution

Try It

What number is 45% of 80?

Show Solution

36

Try It

What number is 55% of 60?

Show Solution

33

Example

125% of 28 is what number?

 Solution

Try It

150% of 78 is what number?

Show Solution

117

Try It

175% of 72 is what number?

Show Solution

126

Example

Translate and solve: 36 is 75% of what number?

 Solution

 

Try It

17 is 25% of what number?

Show Solution

68

Try It

40 is 62.5% of what number?

Show Solution

64

Example

6.5% of what number is $1.17?

Try It

7.5% of what number is $1.95?

Show Solution

$26

Try It

8.5% of what number is $3.06?

$3.06?

Show Solution

$36

In the next examples, we will solve for the percent.

 

Example 

What percent of 36 is 9?

 Solution

 

Try It

What percent of 76 is 57?

Show Solution

75%

 

Try It

What percent of 120 is 96?

96?

Show Solution

80%

Example 

144144 is what percent of 96?

96?

 Solution

 

Try It

110 is what percent of 88?

Show Solution
125%

Try It

126 is what percent of 72?

Show Solution
175%

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

How to

Solve an application

Step 1. Identify what you are asked to find and choose a variable to represent it.

Step 2. Write a sentence that gives the information to find it.

Step 3. Translate the sentence into an equation.

Step 4. Solve the equation using good algebra techniques.

Step 5. Check the answer in the problem and make sure it makes sense.

Step 6. Write a complete sentence that answers the question.

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we’ll solve involve everyday situations, you can rely on your own experience.

Example 

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $68.50. They want to leave an 18% tip. If the tip will be 18% of the total bill, how much should the tip be?

 Solution

Try It

Cierra and her sister enjoyed a special dinner in a restaurant, and the bill was $81.50. If she wants to leave 18% of the total bill as her tip, how much should she leave?

Show Solution

$14.67

Try It

Kimngoc had lunch at her favorite restaurant. She wants to leave 15% of the total bill as her tip. If her bill was $14.40, how much will she leave for the tip?

Show Solution

$2.16

Example

The label on Masao’s breakfast cereal said that one serving of cereal provides 85 milligrams (mg) of potassium, which is 2% of the recommended daily amount. What is the total recommended daily amount of potassium?

The figures shows the nutrition facts for cereal.

 

 Solution

Try It

One serving of wheat square cereal has 7 grams of fiber, which is 29% of the recommended daily amount. What is the total recommended daily amount of fiber?

Show Solution

24.1 grams

Try It

One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

Show Solution

2,375 mg

Example

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240
calories of fat. What percent of the total calories in each brownie comes from fat?

 Solution

Try It

Veronica is planning to make muffins from a mix. The package says each muffin will be 230 calories and 60 calories will be from fat. What percent of the total calories is from fat? (Round to the nearest whole percent.)

Show Solution

26%

Try It

The brownie mix Ricardo plans to use says that each brownie will be 190 calories, and 70 calories are from fat. What percent of the total calories are from fat?

Show Solution

37%

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

 

How To

Find Percent Increase

Step 1. Find the amount of increase.

  • increase=new amount−original amountincrease=new amount−original amount

Step 2. Find the percent increase as a percent of the original amount.

Example 

In 2011, the California governor proposed raising community college fees from $26 per unit to $36 per unit. Find the percent increase. (Round to the nearest tenth of a percent.)

 Solution

 

Try it

In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents. Find the percent increase. (Round to the nearest tenth of a percent.)

Show Solution

8.8%

Try it

In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was $2.25. Find the percent increase. (Round to the nearest tenth of a percent.)

Show Solution

50%

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

How to

Find Percent Decrease

Step 1. Find the amount of decrease.

  • decrease=original amount−new amountdecrease=original amount−new amount

Step 2. Find the percent decrease as a percent of the original amount.

    Example

    The average price of a gallon of gas in one city in June 2014 was $3.71. The average price in that city in July was $3.64. Find the percent decrease.

     Solution

    Try it

    The population of one city was about 672,000 in 2010. The population of the city is projected to be about 630,000 in 2020. Find the percent decrease. (Round to the nearest tenth of a percent.)

    Show Solution

    6.3%

    Try it

    Last year Sheila’s salary was $42,000. Because of furlough days, this year her salary was $37,800. Find the percent decrease. (Round to the nearest tenth of a percent.)

    Show Solution

    10%

     

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    Key Concepts

     

    Convert a percent to a fraction.

    • Step 1. Write the percent as a ratio with the denominator 100.
    • Step 2. Simplify the fraction if possible.

    Convert a percent to a decimal.

    • Step 1. Write the percent as a ratio with the denominator 100.
    • Step 2. Convert the fraction to a decimal by dividing the numerator by the denominator.

    Convert a decimal to a percent.

    • Step 1. Write the decimal as a fraction.
    • Step 2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100.
    • Step 3. Write this ratio as a percent.

    Convert a fraction to a percent.

    • Step 1. Convert the fraction to a decimal.
    • Step 2. Convert the decimal to a percent.

    Solve an application.

    Step 1. Identify what you are asked to find and choose a variable to represent it.
    Step 2. Write a sentence that gives the information to find it.
    Step 3. Translate the sentence into an equation.
    Step 4. Solve the equation using good algebra techniques.
    Step 5. Write a complete sentence that answers the question.
    Step 6. Check the answer in the problem and make sure it makes sense.

    Find percent decrease.

    Step 1. Find the amount of decrease.

      • decrease = original amount – new amount

    Step 2 .Find the percent increase as a percent of the original amount.

     

    Section Exercises

    Use the Definition of Percents

    In the following exercises, write each percent as a ratio.

    1. In 2014, the unemployment rate for those with only a high school degree was 6.0%.

     Solution

    2. In 2015, among the unemployed, 29% were long-term unemployed.

    3. The unemployment rate for those with Bachelor’s degrees was 3.2% in 2014.

     Solution

    4. The unemployment rate in Michigan in 2014 was 7.3%.

     

    In the following exercises, write as (a) a ratio and (b) a percent.

    5. 57 out of 100 nursing candidates received their degree at a community college.

     Solution

    6. 80 out of 100 firefighters and law enforcement officers were educated at a community college.

    7. 42 out of 100 first-time freshmen students attend a community college.

     Solution

    8. 71 out of 100 full-time community college faculty have a master’s degree.

     

    Convert Percents to Fractions and Decimals

    In the following exercises, convert each percent to a fraction and simplify all fractions.

    9. [latex]4\%[/latex]

     Solution

    10. [latex]8\%[/latex]

    11. [latex]17\%[/latex]

     Solution

    12. [latex]19\%[/latex]

    13. [latex]52\%[/latex]

     Solution

    14. [latex]78\%[/latex]

    15. [latex]125\%[/latex]

     Solution

    16. [latex]135\%[/latex]

    17. [latex]37.5\%[/latex]

     Solution

    18. [latex]42.5\%[/latex]

    19. [latex]18.4\%[/latex]

     Solution

    20. [latex]46.4\%[/latex]

    21. [latex]9\frac{1}{2}\%[/latex]

     Solution

    22. [latex]8\frac{1}{2}\%[/latex]

    23. [latex]5\frac{1}{3}\%[/latex]

     Solution

    24. [latex]6\frac{2}{3}\%[/latex]

     

    In the following exercises, convert each percent to a decimal.

    25. [latex]5\%[/latex]

     Solution

    26. [latex]9\%[/latex]

    27. [latex]1\%[/latex]

     Solution

    28. [latex]2\%[/latex]

    29. [latex]63\%[/latex]

     Solution

    30. [latex]71\%[/latex]

    31. [latex]40\%[/latex]

     Solution

    32. [latex]50\%[/latex]

    33. [latex]115\%[/latex]

     Solution

    34. [latex]125\%[/latex]

    35. [latex]150\%[/latex]

     Solution

    36. [latex]250\%[/latex]

    37. [latex]21.4\%[/latex]

     Solution

    38. [latex]39.3\%[/latex]

    39. [latex]7.8\%[/latex]

     Solution

    40. [latex]6.4\%[/latex]

     

    In the following exercises, convert each percent to (a) a simplified fraction and (b) a decimal.

    41. In 2010, 1.5% of home sales had owner financing. (Source: Bloomberg Businessweek, 5/23–29/2011)

     Solution

    42. In 2000, 4.2% of the United States population was of Asian descent. (Source: www.census.gov)

    43. According to government data, in 2013 the number of cell phones in India was 70.23% of the population.

     Solution

    44. According to the U.S. Census Bureau, among Americans age 25 or older who had doctorate degrees in 2014, 37.1% are women.

    45. A couple plans to have two children. The probability they will have two girls is 25%.

     Solution

    46. Javier will choose one digit at random from 0 through 9. The probability he will choose 3 is 10%.

    47. According to the local weather report, the probability of thunderstorms in New York City on July 15 is 60%.

     Solution

    48. A club sells 50 tickets to a raffle. Osbaldo bought one ticket. The probability he will win the raffle is 2%.

     

    Convert Decimals and Fractions to Percents

    In the following exercises, convert each decimal to a percent.

    49. [latex]0.01[/latex]

     Solution

    50. [latex]0.03[/latex]

    51. [latex]0.18[/latex]

     Solution

    52. [latex]0.15[/latex]

    53. [latex]1.35[/latex]

     Solution

    54. [latex]1.56[/latex]

    55. [latex]3[/latex]

     Solution

    56. [latex]4[/latex]

    57. [latex]0.009[/latex]

     Solution

    58. [latex]0.008[/latex]

    59. [latex]0.0875[/latex]

     Solution

    60. [latex]0.0625[/latex]

    61. [latex]1.5[/latex]

     Solution

    62. [latex]2.2[/latex]

    63. [latex]2.254[/latex]

     Solution

    64. [latex]2.317[/latex]

     

    In the following exercises, convert each fraction to a percent.

    65. [latex]\frac{1}{4}[/latex]

     Solution

    66. [latex]\frac{1}{5}[/latex]

    67. [latex]\frac{3}{8}[/latex]

     Solution

    68. [latex]\frac{5}{8}[/latex]

    69. [latex]\frac{7}{4}[/latex]

     Solution

    70. [latex]\frac{9}{8}[/latex]

    71. [latex]6\frac{4}{5}[/latex]

     Solution

    72. [latex]5\frac{1}{4}[/latex]

    73. [latex]\frac{5}{12}[/latex]

     Solution

    74. [latex]\frac{11}{12}[/latex]

    75. [latex]2\frac{2}{3}[/latex]

     Solution

    76. [latex]1\frac{2}{3}[/latex]

    77. [latex]\frac{3}{7}[/latex]

     Solution

    78. [latex]\frac{6}{7}[/latex]

    79. [latex]\frac{5}{9}[/latex]

     Solution

    80. [latex]\frac{4}{9}[/latex]

     

    In the following exercises, convert each fraction to a percent.

    81. [latex]\frac{1}{4}[/latex] of washing machines needed repair.

     Solution

    82. [latex]\frac{1}{5}[/latex]of dishwashers needed repair.

    83. According to the National Center for Health Statistics, in 2012, [latex]\frac{7}{20}[/latex] of American adults were obese.

     Solution

    84. The U.S. Census Bureau estimated that in 2013, 85% of Americans lived in the same house as they did 1 year before.

     

    In the following exercises, complete the table.

    85.

    Fraction Decimal Percent
    1212
    0.450.45
    18%18%
    1313
    0.0080.008
    22
     Solution
    86.
    Fraction Decimal Percent
    1414
    0.650.65
    22%22%
    2323
    0.0040.004
    33

    Everyday Math

    87. Sales tax Felipa says she has an easy way to estimate the sales tax when she makes a purchase. The sales tax in her city is 9.05%. She knows this is a little less than 10%.

    a) Convert 10% to a fraction.

    b) Use your answer from (a) to estimate the sales tax Felipa would pay on a $95 dress.

     Solution

    88. Savings Ryan has 25% of each paycheck automatically deposited in his savings account.

    a) Write 25% as a fraction.

    b) Use your answer from (a) to find the amount that goes to savings from Ryan’s $2,400 paycheck.

     

    Amelio is shopping for textbooks online. He found three sellers that are offering a book he needs for the same price, including shipping. To decide which seller to buy from he is comparing their customer satisfaction ratings. The ratings are given in the chart.

    Use the chart to answer the following questions

    Seller Rating
    AA 4/54/5
    BB 3.5/43.5/4
    CC 85%85%

    89. Write seller C’s rating as (a) a fraction and (b) a decimal.

     Solution

    90. Write seller B’s rating as (a) a percent and (b) a decimal.

    91. Write seller A’s rating as (a) a percent and (b) a decimal.

     Solution

    92. Which seller should Amelio buy from and why?

    Writing Exercises

    93. Convert 25%, 50%, 75%, and 100% to fractions. Do you notice a pattern? Explain what the pattern is.

    94. Convert [latex]\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{4}{10}, \frac{5}{10}, \frac{6}{10}, \frac{7}{10, \frac{8}{10}[/latex], and [latex]\frac{9}{10}[/latex] to percents. Do you notice a pattern? Explain what the pattern is.

    95. When the Szetos sold their home, the selling price was 500% of what they had paid for the house 30 years ago. Explain what 500% means in this context.

    96. According to cnn.com, cell phone use in 2008 was 600% of what it had been in 2001. Explain what 600% means in this context.

    Practice Makes Perfect

    Translate and Solve Basic Percent Equations

    In the following exercises, translate and solve.

    1. What number is 45% of 120?

     Solution
    54

    2. What number is 65% of 100?

    3. What number is 24% of 112?

     Solution
    26.9

    4. What number is 36 of 124?

    5. 250% of 65 is what number?

     Solution
    162.5

    6. 150% of 90 is what number?  

    7. 800% of 2,250 is what number?

     Solution
    18,000

    8. 600% of 1,740 is what number? 

    9. 28 is 25% of what number? 

     Solution
    112

    10. 36 is 25% of what number?

    11. 81 is 75% of what number?

     Solution
    108

    12. 93 is 75% of what number?

    13. 8.2% of what number is $2.87?

     Solution
    $35

    14. 6.4% of what number is $2.88?

    15. 11.5% of what number is $108.10?

     Solution
    $940

    16. 12.3% of what number is $92.25?

    17. What percent of 260 is 78?

     Solution
    30%

    18. What percent of 215 is 86?

    19. What percent of 1,500 is 540?

     Solution
    36%

    20. What percent of 1,800 is 846?

    21. 30 is what percent of 20?

     Solution
    150%

    22. 50 is what percent of 40?

    23. 840 is what percent of 480?

     Solution
    175%

    24. 790 is what percent of 395?

     

    Solve Applications of Percents

    In the following exercises, solve the applications of percents.

    25. Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. She wants to leave 16% of the total bill as a tip. How much should the tip be?

     Solution
    $11.88

    26. When Hiro and his co-workers had lunch at a restaurant the bill was $90.50. The want to leave 18% of the total bill as a tip. How much should the tip be?

    27. Trong has 12% of each paycheck automatically deposited to his savings account. His last paycheck was $2,165. How much money was deposited to Trong’s savings account?

     Solution
    $259.80

    28. Cherise deposits 8% of each paycheck into her retirement account. Her last paycheck was $1,485. How much did Cherise deposit into her retirement account?

    29. One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

     Solution
    24.2 grams

    30. One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates? 

    31. A bacon cheeseburger at a popular fast food restaurant contains 2,070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

     Solution
    2,407 mg

    32. A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium? 

    33. The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

     Solution
    45%

    34. The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat? 

    35. Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

     Solution
    25%

    36. Dimple gets paid $3,200 per month. She pays $960 a month for rent. What percent of her monthly pay goes to rent? 

     

    Find Percent Increase and Percent Decrease

    In the following exercises, find the percent increase or percent decrease.

    37. Tamanika got a raise in her hourly pay, from $15.50 to $17.55. Find the percent increase.

     Solution
    13.2%

    38. Ayodele got a raise in her hourly pay, from $24.50 to $25.48. Find the percent increase. 

    39. Annual student fees at the University of California rose from about $4,000 in 2000 to about $9,000 in 2014. Find the percent increase.

     Solution
    125%

    40. The price of a share of one stock rose from $12.50 to $50. Find the percent increase. 

    41. According to Time magazine (7/19/2011) annual global seafood consumption rose from 22 pounds per person in 1960 to 38 pounds per person today. Find the percent increase. (Round to the nearest tenth of a percent.)

     Solution
    72.7%

    42. In one month, the median home price in the Northeast rose from $225,400 to $241,500. Find the percent increase. (Round to the nearest tenth of a percent.)

    43. A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent decrease.

     Solution
    2.5%

    44. The price of a share of one stock fell from $8.75 to $8.54. Find the percent decrease. 

    45. Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent decrease.

     Solution
    11%

    46. From 2000 to 2010, the population of Detroit fell from about 951,000 to about 714,000. Find the percent decrease. (Round to the nearest tenth of a percent.)

    47. In one month, the median home price in the West fell from $203,400 to $192,300. Find the percent decrease. (Round to the nearest tenth of a percent.)

     Solution
    5.5%

    48. Sales of video games and consoles fell from $1,150 million to $1,030 million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)

    Everyday Math

    49. Tipping At the campus coffee cart, a medium coffee costs $1.65. MaryAnne brings $2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave? 

     Solution
    21.2%

    50. Late Fees Alison was late paying her credit card bill of $249. She was charged a 5% late fee. What was the amount of the late fee?

    Writing Exercises

    51. Without solving the problem “44 is 80% of what number”, think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

     Solution
    Greater than 44

    52. Without solving the problem “What is 20% of 300?” think about what the solution might be. Should it be a number that is greater than 300 or less than 300? Explain your reasoning. 

    53. After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

     Solution
    He should have packed half as many shorts and twice as many shirts. 

    Glossary

    percent
    A percent is a ratio whose denominator is 100.
    discount
    An amount of discount is a percent off the original price, determined by the discount rate.
    mark-up
    The mark-up is the amount added to the wholesale price, determined by the mark-up rate.

    percent decrease

    The percent decrease is the percent the amount of decrease is of the original amount.

    percent increase
    The percent increase is the percent the amount of increase is of the original amount.

    License

    Icon for the Creative Commons Attribution 4.0 International License

    0.6 Percentages Copyright © by Karen Perilloux; Prakash Ghimire; and Lauren Johnson is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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