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Core Mathematics

Katherine Yoshiwara

Section 2.2 More Fractions and Percents

  • Percents
  • Calculating Percents
  • What Percent Is It?
  • Eighths
  • Thirds
  • Decimal Form of One-Third

Subsection 2.2.1 Percents

A percent is a fraction. For example,
\begin{equation*} 75\% = 0.75 = \dfrac{75}{100} \end{equation*}
"Percent" means "for each 100" or "out of 100." So a percent is just a fraction whose denominator is 100. If you understand hundredths, you also understand percents!

Example 2.2.1.

Jaycee is an interior designer and takes a 10% commission on the cost of a remodel. A recent remodel cost $12,500. How much is Jaycee’s commission?

Solution.

10% means "ten out of 100" or \(\dfrac{10}{100}\text{,}\) and \(\dfrac{10}{100}\) reduces to \(\dfrac{1}{10}\text{.}\) To find \(\dfrac{1}{10}\) of 12,500, we divide by 10:
\begin{equation*} \dfrac{1}{10}~ \text{of}~ 12,500 = 12,500 \div 10 = 1250 \end{equation*}
Jaycee’s commission is $1250.
Because 10% is equal to \(\dfrac{1}{10},~\) it is easy to find 10% of a number: we just divide the number by 10. Once we know 10% of a number, we can find other percents. For example, to find 20% of a number we can compute 10% of the number and then double the result.

Checkpoint 2.2.2.

Viktor used a coupon for 10% off the price of hiking boots and saved $12.00. Find the following percents of the price of the boots.
  1. 5%
  2. 20%
  3. 25%
  4. 1%
  5. 3%
  6. 100%
Answer.
  1. $6
  2. $24
  3. $30
  4. $1.20
  5. $3.60
  6. $120

Activity 2.2.1. Percents.

Follow the steps to find each percent. Try to do the calculations in your head!
  1. The Kwans are buying a house for $340,000, and they want to make a down payment of 20% of the price.
    1. What is 10% of the price of the house?
    2. What is 20% of the price of the house?
  2. The restaurant bill came to $26, and Tyrell wants to leave a 15% tip.
    1. What is 10% of the bill?
    2. What is 5% of the bill?
    3. What is 15% of the bill?
  3. The sales tax on a $25 rose bush is 8%.
    1. What is 10% of the price of the rose bush?
    2. What is 1% of the price of the rose bush?
    3. What is 8% of the price of the rose bush?
  4. In 2020 there were approximately 72 million children under the age of 18 in the United States. About 16% of them lived in poverty.
    1. What was 10% of the US population of children?
    2. What was 5% of the US population of children?
    3. What was 1% of the US population of children?
    4. What was 16% of the US population of children?

Subsection 2.2.2 Calculating Percents

Because "percent" means "per 100," we can change a percent to a decimal fraction by dividing it by 100. You can use your calculator, or, even better, you can remember that to divide by 100 we move the decimal point 2 places to the left.

Example 2.2.3.

Change 12% to a decimal fraction.

Solution.

12% means twelve hundredths, so we divide 12 by 100. We move the decimal point 2 places to the left.
We see that 12% is equal to 0.12.

To change a percent to a decimal fraction:.

Divide the percent by 100,
\begin{equation*} \text{or} \end{equation*}
Move the decimal point two places to the left.
Even Easier! You can also think of the % sign as "hundredths," so that we read 12% as "twelve hundredths" and write the fraction 0.12.

Checkpoint 2.2.4.

Change each percent to a decimal fraction.
  1. 50%
  2. 5% (Be careful!)
  3. 23%
  4. 99%
  5. 12.5%
  6. 200%
Answer.
  1. 0.5
  2. 0.05
  3. 0.23
  4. 0.99
  5. 0.125
  6. 2.00

Note 2.2.5.

In Checkpoint 2.2.4 above, do you see the difference between the decimal forms of 50% and 5%? Try one yourself. Write the decimal form of each percent:
\begin{equation*} 8\% = \hphantom{0000000000000000} 80\% = \hphantom{0000000000000000} \end{equation*}
Now we can find any percent of a number.

Example 2.2.6.

The sales tax in Florida is 6%. What is the sales tax on a TV that costs $458?

Solution.

We’d like to find 6% of $458. We must first change 6% to a decimal fraction by moving the decimal point 2 places to the left.
\begin{equation*} 6\% = 0.06 \end{equation*}
(Note that we inserted a zero in front of 6 so that we could move the decimal point 2 places to the left!)
Next, we multiply 0.06 times 458.
\begin{equation*} 0.06 \times 458 = 27.48 \end{equation*}
The sales tax is $27.48.

Note 2.2.7.

In Example 2.2.6, why does multiplying by 0.06 give us 6 hundredths of 458? Here’s why:
To find one hundredth of 458 we divide by 100: \(~~~458 \div 100 = 4.58\)
To find six hundredths of 458 we multiply 4.58 by 6: \(~~~4.58 \times 6 = 27.48\)
You can check with your calculator that multiplying 458 by 0.06 is the same as multiplying by 6 and dividing by 100!

To calculate a percent of a number.

  1. Change the percent to a decimal fraction.
  2. Multiply the number by the decimal fraction.

Checkpoint 2.2.8.

Calculate the sales tax.
State Percent
Sales Tax		 Decimal Form Amount Tax
Indiana 7% \(\hphantom{000000}\) $15 \(\hphantom{0000000000}\)
Oklahoma 11% \(\hphantom{000000}\) $120 \(\hphantom{0000000000}\)
Washington 9.6% \(\hphantom{000000}\) $80 \(\hphantom{0000000000}\)
Texas 8.25% \(\hphantom{000000}\) $300 \(\hphantom{0000000000}\)
Answer.
State Percent
Sales Tax		 Decimal Form Amount Tax
Indiana 7% 0.07 $15 $1.05
Oklahoma 11% 0.11 $120 $13.20
Washington 9.6% 0.096 $80 $7.68
Texas 8.25% 0.0825 $300 $24.75

Subsection 2.2.3 What Percent Is It?

If you follow a diet of 2000 calories per day, your minimum daily requirement (MDR) of potassium is 3500 milligrams.

Example 2.2.9.

A bowl of Cheerios provides 140 milligrams of potassium. What percent of your MDR is that?

Solution.

We’d like to know what percent is 140 out of 3500. Remember that a percent is a fraction, so we make a fraction with the part (140) out of the whole (3500).
\begin{equation*} \dfrac{\text{part}}{\text{whole}} = \dfrac{140}{3500} = 0.04 ~~~~~~~~\blert{\text{Divide 140 by 3500.}} \end{equation*}
As a decimal fraction, Cheerios provides 0.04, or 4 hundredths of the MDR for potassium. Four hundredths is the same as 4%. (Or you can multiply 0.04 by 100 to write it as a percent: \(0.04 \times 100 = 4%\))

To find what percent a part is.

  1. Divide the part by the whole.
  2. Multiply the decimal fraction by 100.

Checkpoint 2.2.10.

Calculate the percent of each MDR provided by Cheerios.
Cheerios MDR Percent
Calories 110 2000 \(\hphantom{0000000000}\)
Sodium 120 mg 2400 mg \(\hphantom{0000000000}\)
Carbohydrate 24 g 300 g \(\hphantom{0000000000}\)
Answer.
Cheerios MDR Percent
Calories 110 2000 5.5%
Sodium 120 mg 2400 mg 5%
Carbohydrate 24 g 300 g 8%

Activity 2.2.2. Using Percents.

  1. The Gonzalez family’s monthly income is $5200. They budget their money according to the circle graph below. Complete the table showing the amount they spend in each category.
    Category Percent Amount Spent
    Food 33% \(\hphantom{0000000000}\)
    Rent 25% \(\hphantom{0000000000}\)
    Clothes 15% \(\hphantom{0000000000}\)
    Miscellaneous 21% \(\hphantom{0000000000}\)
    Savings \(\hphantom{0000000000}\)
  2. At Marilyn’s office, they held a "Losing It" competition: whoever loses the largest percent of their body weight in six months wins a gift certificate. Who won?
    Person Original
    Weight (lbs)    		 New Weight Pounds Lost Percent of
    Original Weight
    Jackie 156 140.4 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    Trinh 128 120 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    Jorge 210 182.7 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    Adam 185 162.8 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    Marilyn 166 157.7 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
  3. Estimate a 15% tip on each restaurant bill by rounding the amount to the nearest dollar.
    Bill Rounded 10% of Bill 5% of Bill 15% of Bill
    $29.78 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    $45.16 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    $11.89 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
    $63.58 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)

Subsection 2.2.4 Eighths

We have studied the benchmark fractions \(\dfrac{1}{4},~~\dfrac{1}{2},~~\) and \(~\dfrac{3}{4},~~\) as well as tenths and hundredths. Along the way we also encountered fifths. Here is another familiar and useful fraction.

Example 2.2.11.

Nelson bought a quarter-pound of sliced roast beef and ate half of it for lunch. How many ounces of roast beef did he eat? What fraction of a pound is that?

Solution.

There are 16 ounces in a pound, so a quarter-pound is
\begin{equation*} \dfrac{1}{4}~ \text{of}~ 16 = 16 \div 4 = 4 ~~\text{ounces} \end{equation*}
Half of 4 ounces is 2 ounces, so Nelson ate 2 ounces of roast beef. As a fraction of a pound, 2 ounces is
\begin{equation*} \dfrac{\text{part}}{\text{whole}} = \dfrac{2}{16} = \dfrac{1}{8}~~~~~~\blert{\text{Reduce: divide top and bottom by 2.}} \end{equation*}
Two ounces is \(\dfrac{1}{8}\) lb, or one-eighth of a pound.

Note 2.2.12.

Recall that \(\dfrac{1}{4}\) is one-half of \(\dfrac{1}{2}\text{.}\) Now we see that \(\dfrac{1}{8}\) is one-half of \(\dfrac{1}{4}\text{.}\) You know that the decimal form of \(\dfrac{1}{4}\) is 0.25 (or 0.250). What is the decimal form of \(\dfrac{1}{8}\text{?}\)

Checkpoint 2.2.13.

Find the decimal and percent equivalents for each eighth. Then shade the corresponding fraction of the figure.
Fraction Decimal Percent
\(\dfrac{1}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{2}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{3}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{4}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{5}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{6}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{7}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{8}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Answer.
Fraction Decimal Percent
\(\dfrac{1}{8}\) 0.125 12.5%
8 by 1 grid with 1 shaded box
\(\dfrac{2}{8}\) 0.25 25%
8 by 1 grid with 2 shaded box
\(\dfrac{3}{8}\) 0.375 37.5%
8 by 1 grid with 3 shaded box
\(\dfrac{4}{8}\) 0.5 50%
8 by 1 grid with 4 shaded box
\(\dfrac{5}{8}\) 0.625 62.5%
8 by 1 grid with 5 shaded box
\(\dfrac{6}{8}\) 0.75 75%
8 by 1 grid with 1 shaded box
\(\dfrac{7}{8}\) 0.875 87.5%
8 by 1 grid with 7 shaded box
\(\dfrac{8}{8}\) 1 100%
8 by 1 grid with 8 shaded box
Converting between fractions and percents can simplify many calculations.

Example 2.2.14.

Devon and Darla Grant together make $3200 per month in take-home pay. They are looking for an apartment, and don’t want to spend more than 37.5% of their income on housing. What is the maximum rent they can afford?

Solution.

We’ll solve this problem in two different ways. First, we’ll compute 37.5% of $3200. We write the percent as a decimal fraction and use a calculator to multiply:
\begin{equation*} 0.375 \times 3200 = 1200 \end{equation*}
The Grants can afford $1200 monthly rent.
We can compute the fraction without a calculator if we remember that 37.5% is the same as \(\dfrac{3}{8}\text{.}\) We first find \(\dfrac{1}{8}\) of $3200 by dividing by 8:
\begin{equation*} 3200 \div 8 = 400 \end{equation*}
Then we multiply by 3 to find \(\dfrac{3}{8}\) of 3200:
\begin{equation*} 3 \times 400 = 1200 \end{equation*}
We get the same answer, and we can do the calculations mentally!

Checkpoint 2.2.15.

Devon and Darla from Example 2.2.14 have budgeted their $3200 monthly income as follows.
Category Percent Fraction Amount
Housing 37.5% \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Food and Clothing 25% \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Auto and Insurance 12.5% \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Entertainment 15% \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Savings 10% \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Total \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
  1. Compute the amount they can spend in each category. Try to do each calculation in your head, by using fractions whenever possible!
  2. Check your calculations by computing the total in each column.
Answer.
Category Percent Fraction Amount
Housing 37.5% \(\dfrac{3}{8}\) $1200
Food and Clothing 25% \(\dfrac{1}{4}\) $800
Auto and Insurance 12.5% \(\dfrac{1}{8}\) $400
Entertainment 15% \(\dfrac{3}{20}\) $480
Savings 10% \(\dfrac{1}{10}\) $320
Total 100% 1 $3200

Subsection 2.2.5 Thirds

Thirds are also useful fractions.
Many fields have a "one-third" rule as a guideline.

Example 2.2.16.

In photography, the one-third rule suggests that an image should be divided into thirds, both horizontally and vertically, and important elements of the picture, including the horizon, should be placed on the division lines or their intersections.
Zack plans to enlarge the farming scene below into a 24-inch by 15-inch photograph. If he crops the photo according to the one-third rule, how far will the base of the tree be from the right-side edge of the picture?

Solution.

One-third of 24 is 8, so the base of the tree should be 8 inches from the edge of the picture.

Checkpoint 2.2.17.

Apply the one-third rule to each situation.
  1. When a home-buyer applies for a mortgage, the monthly payments on the mortgage amount should not exceed one third of the buyer’s net monthly salary. Brandon’s net annual salary is $48,600. What size mortgage payment can he afford?
  2. A manager should allow herself up to one third of the available time for a task to organize the task and delegate responsibility, thus allowing her team the remaining two thirds of the time. Emily has two hours to write her weekly report. How long should her team work on the report?
  3. In scuba diving, one third of the gas supply is used for the outward journey, one third for the return journey, and one third is left as a safety reserve. Farah’s cylinder holds 10 liters. How many liters are left when she starts her return journey?
  4. When mowing grass, you should never remove more than one third of the grass blade. If the mowing height on your lawn mower is 3 inches, how tall should the grass be when you mow, so that you remove one third of the grass blade?
Answer.
  1. $1700
  2. \(\dfrac{4}{3}\) hour, or 1 hour and 20 minutes
  3. \(\dfrac{20}{3}\) liters, or \(6\dfrac{2}{3}\) liters
  4. \(4\dfrac{1}{2}\) inches

Subsection 2.2.6 Decimal Form of One-Third

What about the decimal form of \(\dfrac{1}{3}\text{?}\) Remember that some fractions do not "come out evenly" in their decimal form. One-third is one of those fractions. Use your calculator to verify that
where the … symbol indicates that the 3’s go on forever. (Recall that we encountered repeating decimals in Subsection 2.1.4. We can represent \(\dfrac{1}{3}\) with a repeater bar as \(0.\overline{3}.~\)) This is the exact value of \(\dfrac{1}{3}\) as a decimal fraction. We often round the decimal form to hundredths and say that
Similarly, you can verify that the exact value of \(\dfrac{2}{3}\) is
or \(0.\overline{6},~~\) and by rounding to hundredths we see that
From here we can say that
Can we write exact percent values for \(\dfrac{1}{3}\) and \(\dfrac{2}{3}\text{?}\) Try the following calculation:
To write \(\dfrac{1}{3}\) as a percent: divide 100 by 3 by hand (that is, without a calculator!). You should find that
\begin{equation*} 100 \div 3 = 33\dfrac{1}{3} \end{equation*}
If we double this number we find the percent form of \(\dfrac{1}{3}\text{:}\)
\begin{equation*} 2 \times 33\dfrac{1}{3} = 66\dfrac{2}{3} \end{equation*}
We have found exact percent forms for \(\dfrac{1}{3}\) and \(\dfrac{2}{3}\text{.}\)

Thirds as Percents.

\begin{equation*} \dfrac{1}{3} = 33\dfrac{1}{3} \% \hphantom{0000000000} \text{and} \hphantom{0000000000} \dfrac{2}{3} = 66\dfrac{2}{3} \% \end{equation*}

Note 2.2.18.

You should become familiar these percent values, so that you will recognize \(\dfrac{1}{3}\) and \(\dfrac{2}{3}\) when you see them!

Activity 2.2.3. Percents as Fractions.

Complete the table by entering the (reduced) common fraction equivalent to each percent, and then computing that fraction of 120.
Try to do all of your work without a calculator. Remember that once you know the benchmark fractions of an amount, you can use those to find other fractions of the same amount.
Percent, % Common
Fraction		 Fraction
of 120
1 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
5 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
10 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
12.5 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
20 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
25 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
30 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
33\(\frac{1}{3}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
37.5 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
40 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
50 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Percent, % Common
Fraction		 Fraction
of 120
60 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
62.5 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
66\(\frac{2}{3}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
70 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
75 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
80 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
87.5 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
90 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
95 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
99 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
100 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)

Subsection 2.2.7 Vocabulary

  • percent
  • circle graph

Exercises 2.2.8 Practice 2.2

1.

List several different ways to find 10% of a number.

2.

If you know 10% of a number, how can you find the number?

3.

  1. If you know 10% of a number, how can you find 5% of the number?
  2. How can you find 25% of the number?
  3. Leslie gave $76 to a local charity. That was 10% of her paycheck. Find 25% of her paycheck, using the method you described in parts (a) and (b).

4.

  1. If you know 10% of a number, how can you find 1% of the number?
  2. How can you find 99% of the number?
  3. The Wellness Clinic gave flu shots to 170 seniors last week. That was 10% of the seniors living in Stratton County. Find 99% of the senior population, using the method you described in parts (a) and (b).

5.

Delbert says that to find 4% of a number, we multiply the number by 0.4. Is he correct? Why or why not?

6.

Francine says that we can omit the zero after the 8 in the fraction 0.80, but we cannot omit the zero before the 8 in 0.08. Is she correct? Why or why not?

7.

Moving the decimal point 2 places to the left is a short-cut for what operation?

8.

Which of the following operations does \(\dfrac{3}{8}\) represent?
  1. 3 divided into 8
  2. 3 divided by 8
  3. \(\displaystyle 3 \div 8\)
  4. \(\displaystyle 8 \div 3\)

9.

Describe a way to find 0.625 of $6400 without using a calculator.

10.

One of the numbers below is not equal to any of the others. Which is it? Sort the rest of the numbers into three groups of equal numbers.
\begin{equation*} 0.3 \hphantom{00} 3\% \hphantom{00} 3.0 \hphantom{00} \dfrac{3}{100} \hphantom{00} 0.03 \hphantom{00} \dfrac{30}{100} \hphantom{00} 30\% \hphantom{00} \dfrac{1}{3} \hphantom{00} 0.30 \hphantom{00} \dfrac{30}{10} \hphantom{00} 300\% \hphantom{00} \dfrac{3}{10} \hphantom{00} 3.00 \end{equation*}

11.

  1. Which is a larger fraction, \(\dfrac{3}{8}\) or \(\dfrac{1}{3}\text{?}\)
  2. Is \(\dfrac{1}{9}\) larger or smaller than 10%?

12.

Is \(\dfrac{1}{3}\) equal to 0.3? Try these calculations.
  1. Find \(\dfrac{1}{3}\) of $120 by dividing by 3.
  2. Find \(\dfrac{3}{10}\) of $120 by dividing by 10, then multiplying the result by 3.
  3. Use your calculator to find 0.3 of $120.
  4. Use your calculator to find 0.33 of $120.
  5. Which is a larger fraction, \(\dfrac{1}{3}\) or 310?

13.

Find the percent form of each fraction by dividing the denominator into 10.
  1. \(\displaystyle \dfrac{1}{5}\)
  2. \(\displaystyle \dfrac{1}{25}\)
  3. \(\displaystyle \dfrac{1}{8}\)
  4. \(\displaystyle \dfrac{1}{16}\)

14.

Write exact decimal and percent forms for \(\dfrac{1}{6}\) and \(\dfrac{1}{12}\) as follows. (Hint: Do you remember the decimal forms for \(\dfrac{1}{3}\) and \(\dfrac{2}{3}\text{?}\))
  1. Start with the decimal form for \(\dfrac{1}{3}\) and divide by 2.
  2. Divide each denominator into 100.
  3. Do you get the same results by each method?

15.

On Lincoln’s birthday, Abe’s Furniture Store offered 12% off some basic items. Complete the table showing Abe’s discounted prices.
Item Original Price Discount New Price
Sofa $480 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Dining Table $625 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Dresser $390 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
TV Stand $238 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)

16.

Calculate the percent change in price for each item over the past decade.
Item Price in 2013 Price in 2023 Change in Price Percent of 2013 Price
Dozen eggs $1.31 $1.95 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Pound of Bananas $0.50 $0.61 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Loaf of Bread $0.98 $1.40 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
Quart of milk $2.04 $3.70 \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)

17.

The Bradleys bought a new home for $760,000 and made a down payment of 15%. What is the amount of their mortgage loan?

18.

Simon made a 20% down payment of $8400 on a new hybrid car. What was the cost of the car?

19.

Cliff answered 24 questions correctly on his quiz and got a score of 80%. How many questions were on the quiz?

20.

Cadbury recently reduced the weight of its most popular chocolate bar from 250 grams to 200 grams. What percent of its original weight was that reduction?

21.

In the first half of 2013, there were 258,915 guns sold in Virginia. In the first half of 2014, there were 192,585 guns sold.
  1. What was the decrease in the number of guns sold?
  2. What percent decrease did guns sales experience over 2013?

22.

At the census of 2010, the population of the U.S. was 308,745,000. In 2020, the population was 331,449,000.
  1. What was the increase in population over that decade?
  2. By what percent of the 2010 population did the population increase from 2010 to 2020?
  3. If the population continues to increase by the same percent, what will the U.S. population be in 2030?

Exercise Group.

For Problems 23-26, decide which discount results in a lower price.
23.
A dress: 40% off $80, or \(\dfrac{1}{4}\) off $60
24.
Hiking shoes: \(\dfrac{1}{5}\) off $110, or 25% off $120
25.
Oatmeal soap: \(\dfrac{1}{3}\) off $6, or 30% off $6
26.
Bathing suit: 60% off $50, or \(\dfrac{2}{3}\) off $60

27.

Which of the following are correct ways to find 5% of an amount?
  1. Divide by 10, then divide by 2.
  2. Multiply by 0.5
  3. Move the decimal two places to the left, then multiply by 5
  4. Divide by 20
  5. Multiply by 100, then divide by 5

28.

Which of these are the same?
  1. 1% of $10,000
  2. 10% of $1000
  3. 0.01 of $1000
  4. \(\dfrac{1}{100}\) of $10,000
  5. 10% of $10,000

29.

Allison is planning a birthday party for her daughter. She has a budget of $68 to spend according to the circle graph shown at left below. How much can she spend in each category?

30.

Robert divides his school days as shown in the circle graph shown above at right. How many hours does he spend on each activity?

Exercise Group.

For Problems 31-36, what fraction of the rectangle is shaded? Then reduce the fraction.
31.
32.
33.
34.
35.
36.

Exercise Group.

For Problems 37-46, convert the percent to a fraction, and reduce.
37.
80%
38.
60%
39.
45%
40.
85%
41.
24%
42.
72%
43.
38%
44.
74%
45.
96%
46.
8%

Exercise Group.

For Problems 47-52, find three fractions equivalent to the given fraction.
47.
\(\dfrac{2}{3}\)
48.
\(\dfrac{3}{8}\)
49.
\(\dfrac{1}{6}\)
50.
\(\dfrac{5}{6}\)
51.
\(\dfrac{5}{16}\)
52.
\(\dfrac{1}{8}\)

53.

Fabric is usually sold by the yard, and a yard is equal to 36 inches. The number line below represents one yard.
  1. On the number line, label each benchmark fraction of one yard with its measure in inches.
  2. Here is another copy of the one-yard number line. Mark the eighths of a yard on the number line, and label each with its measure in inches.
  3. Use your answers to parts (a) and (b) to complete the table.
Fraction of One Yard Length in Inches Percent of One Yard
\(\dfrac{1}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{2}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{3}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{4}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{5}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{6}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{7}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
\(\dfrac{8}{8}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)

54.

Each of the lengths in the table below is greater than one yard. Write each length as one yard plus a fraction of a yard.
Length in Inches Length in Yards
54 \(\hphantom{0000000000}\)
45 \(\hphantom{0000000000}\)
40.5 \(\hphantom{0000000000}\)
67.5 \(\hphantom{0000000000}\)
63 \(\hphantom{0000000000}\)

55.

Here is a block from a quilt pattern called Irish Chain.
  1. What fraction of the block is blue?
  2. What fraction of the block is pink?
  3. Use your answers to parts(a) and (b) to find the fraction of the block in the background color.

56.

Here is a block from a quilt pattern called Bow Tie.
  1. What fraction of the block is blue?
  2. What fraction of the block is pink?
  3. Use your answers to parts(a) and (b) to find the fraction of the block in the background color.

57.

In New York, a three-fifths majority is required to pass most tax increases. The Assembly has 150 members. How many Assembly members must vote in favor to pass a tax increase?

58.

The U.S. Constitution gives the Senate sole authority to ratify treaties of the United States but makes this contingent upon a two thirds majority. There are 100 U.S. Senators. How many must vote in favor in order to ratify a treaty?

Exercise Group.

For Problems 59-60, use this picture of a ruler marked in inches.
59.
  1. What is the smallest fraction of an inch marked on the ruler?
  2. If you start at 2 on the ruler and move 20 sixteenths of an inch to the right, where will you end up?
  3. Reduce 20 sixteenths.
60.
  1. From 3 on the ruler, move eleven eighths inches to the right. Where are you now?
  2. From 3 on the ruler, move eleven eighths inches to the left. Where are you now?
  3. What is the distance between your ending points in parts (a) and (b)?

Exercise Group.

For Problems 61-64, remember that rotating through one whole circle sweeps out an angle of 360°. A sector of a circle is a wedge with its point at the center of the circle.
61.
  1. What is the degree measure of each sector of the circle shown at right?
  2. Shade a sector that shows \(\dfrac{2}{3}\) of the circle. What is the degree measure of that sector?
62.
  1. What is the degree measure of each sector of the circle shown at right?
  2. Shade a sector that shows \(\dfrac{5}{8}\) of the circle. What is the degree measure of that sector?
63.
  1. What is the degree measure of each sector of the circle shown at right?
  2. Shade a sector that shows \(\dfrac{5}{6}\) of the circle. What is the degree measure of that sector?
64.
  1. What is the degree measure of each sector of the circle shown at right?
  2. Shade a sector that shows \(\dfrac{3}{8}\) of the circle. What is the degree measure of that sector?

65.

Until recently, a wife was typically entitled to one-third of the common family income when a couple divorced. The amount of the wife’s alimony was calculated as one-third of the sum of both partners’ incomes, minus her own income. If the husband earned $50,000 and the wife earned $10,000, what was the wife’s alimony?

66.

Demographers argue that when a subgroup becomes one-third of a population, it becomes a political force. In 2012 the U.S. population was 314 million, and the populations of some of the U.S. minority groups are given in the table.
Group Population (millions) % of US Population
Hispanic 53 \(\hphantom{0000000000}\)
African-American 44.5 \(\hphantom{0000000000}\)
Asian 18.9 \(\hphantom{0000000000}\)
  1. Compute the percent of the U.S. population made up by each minority group.
  2. Do any of these groups, alone or in combination, make up one-third of the U.S. population?

67.

To prevent injury, fitness experts recommend that athletes increase their training activity no more than 10 percent per week. Use this 10 Percent Rule to complete the workout schedule.
Activity This Week Next Week
Running 20 miles \(\hphantom{0000000000}\)
Weights 45 pounds \(\hphantom{0000000000}\)
Swimming 400 yards \(\hphantom{0000000000}\)

68.

In 1942 the ecologist Raymond Lindemann put forth the ten percent law for the transfer of energy in the food chain. His law suggests that 90% of the energy is discarded or lost in the transfer at each level of the food chain, and only 10% is passed on as useable biological energy.
Use the ten percent law to complete the table showing the amount of energy, in kilocalories, available at each level of the food chain.
Source Sun Plants Cows Humans
Energy Available 40,000 kcl \(\hphantom{0000000000}\) \(\hphantom{0000000000}\) \(\hphantom{0000000000}\)
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