Section 3A

Uses and Abuses of Percentages ... exercises

...

Brief review of fractions and percentages (p.121)

Exercise 29 p.132

Express the reduced fraction \( \frac{7}{5} \) as a decimal

Solution

Simply divide on a calculator:

Answer: \( \frac{7}{5} = 1.4 \)

Brief review of fractions and percentages (p.121)

Exercise 29 p.132

Express the reduced fraction \( \frac{7}{5} \) as a percentage

Solution

We already know that \( \frac{7}{5} = 1.4 \)

In order to get percentage, we multiply it by \(100\):

\(1.4 \times 100 =140\)

Answer: \( \frac{7}{5} = 140\% \)

Brief review of fractions and percentages (p.121)

Exercise 26 p.132

Express the percentage \( 44 \% \) as a decimal

Solution

Divide it by \( 100 \) to get

\( 44\% = \frac{44}{100} = 0.44\)

Answer: \(44\%=0.44 \)

Brief review of fractions and percentages (p.121)

Exercise 26 p.132

Express the percentage \( 44 \% \) as a reduced fraction

Solution

Divide it by \( 100 \) to get

\( 44\% = \frac{44}{100} = \frac{11}{25}\)

Answer: \(44\%= \frac{11}{25}\)

Brief review of fractions and percentages (p.121)

Exercise 39 p.132

\(A= 1.5 \) million is the 2012 population of Philadelphia, and \(B=2.1 \) million is the 2012 population of Houston.

Find the ratio of \(A\) to \(B\).

Solution

The ratio is \( \frac{A}{B}=\frac{1.5}{2.1} \approx 0.71 \)

Brief review of fractions and percentages (p.121)

Exercise 39 p.132

\(A= 1.5 \) million is the 2012 population of Philadelphia, and \(B=2.1 \) million is the 2012 population of Houston.

Find the ratio of \(B\) to \(A\).

Solution

The ratio is \( \frac{B}{A}=\frac{2.1}{1.5} =1.4 \)

Brief review of fractions and percentages (p.121)

Exercise 39 p.132

\(A= 1.5 \) million is the 2012 population of Philadelphia, and \(B=2.1 \) million is the 2012 population of Houston.

Complete the sentence: \(A\) is _______ percent of \( B \).

Solution

Their ratio is \( \frac{A}{B} \times 100 =\frac{1.5}{2.1} \times 100 \approx 71 \)

Answer: \(A\) is approximately \( 71 \% \) of \(B\).

Percenrages and fractions

Exercise 45 p.132

The full-time year-round median salary for U.S. men in 2010 was \(\$ 42,800 \), and the full-time year-round median salary for U.S. women in 2010 was \(\$ 34,700 \).

Express the first number as a percentage of the second number.

Solution

\( \frac{42800}{34700} \times 100 \approx 123 \% \)

Answer: Approximately \(123 \% \)

Percenrage Change

Exercise 53 p.133

The number of daily newspapers in the United States was \( 2226 \) in 1900, and \( 1382 \) in 2011.

Find the absolute change and the percentage change.

Solution

Absolute change is \(1382 - 2226 = -844\). The amount of newspapers decreased by \( 844 \)

Percentage (relative) change is \( \frac{\text{absolute change}}{\text{reference value}} \times 100 = \frac{-844}{2226} \times 100 \approx -38 \% \)

Percenrage Comparisons

Exercise 59 p.133

Complete the following sentence

The number of deaths due to poisoning in the United States in 2009 (\(39,000\)) is _______ percent greater than the number of deaths due to falls ( \(26,100\)).

Solution. The absolute difference is \(39,000 - 26,100 = 12900 \), and

the relative difference is \( \frac{\text{absolute difference}}{\text{reference value}} \times 100 = \frac{12900}{26100} \times 100 \approx 49 \% \)

Answer: approximately \(49 \% \).

\( Of \) versus \(More\) \(than \)

Exercise 63 p.133

The population of Virginia is \(18 \% \) less than the population of Georgia, so Virginia's population is ________% of Georgia's.

Solution. Georgia's population, being the reference value is considered as \( 100 \% \).

Then Virginia's population is \( 100 - 18 = 82 \% \) of Georgia's population.

Answer: The population of Virginia is \(18 \% \) less than the population of Georgia, so Virginia's population is \( 82 \% \) of Georgia's population

Prices and sales

Exercise 67 p.133

The retail cost of a TV is \( 30 \% \) more than its wholesale cost. Therefore, the retail cost is ______ times the wholesale cost.

Solution. Since \( P\% \) more means \( (100+P)% \) of, the retail cost of the TV is \( 130 \% \) of its wholesale cost.

\(130 \% \) of means \( 1.3 \) times more

Answer: the retail cost is \( 1.3 \) times the wholesale cost.

Percentages of percentages

Exercise 71 p.133

The percentage of Americans accessing the Internet increased from \(67 \% \) in 2000 to \(83 \% \) in 2012.

Describe the change as an absolute change in percentage points, and as a relative change in terms of percentage.

Solution. Absolute change is \(83-67=16\) percentage points.

Relative change is \( \frac{\text{absolute change}}{\text{reference value}} \times 100 = \frac{16}{67} \approx 24 \% \)

Solving percentage problems

Exercise 76 p.133

The final cost of your new shoes is \(\$ 107.69 \). The local sales tax rate is \(6.2 \%\). What was the retail (pre-tax) price?

You've paid \(100+6.2 = 106.2 \%\) of the price.

That is the retail price multiplied by \(1.062\)

The retail price thus was \( \frac{107.69}{1.062} \approx \$ 101.40 \)

Solving percentage problems

Exercise 77 p.133

Between 2000 and 2010, the percentage of U.S households with cordless phones increased by \( 13.7 \% \) to \(91 \% \). What percentage of households had cordless phones in 2000?

Solution. \(91 \% \) in 2010 is \( 113.7 \% \) of what it was in 2000. Thus, in 2000 it was

\( \frac{91}{1.137} \approx 80 \% \)

Is it possible?

Exercise 83 p.134

By turning off her lights and closing all windows at night, Maria saved \(120\% \) of her monthly energy bill.

Answer. No, that is impossible. The new bill would be \( 100\% - 120\% = -20\% \),

which is negative \(20 \% \) of her previous bill. That cannot happen.

Shifting reference value

Exercise 79 p.134 True or False?

If the national economy shrank by \(4\%\) per year for three consecutive years, then the economy shrank by \(12\%\) over the three year period.

False! ... although not far away from the correct answer in this case

After the first year, the economy indeed shrank by \(4\%\), and became \(96\%\) of what it was.

However, already after the second year the economy shrank by \(4\%\) of this new value, not the original one!

Shifting reference value

Exercise 79 p.134 ... continued ... how to calculate

If the national economy shrank by \(4\%\) per year for three consecutive years, then the economy shrank by \(12\%\) over the three year period.

The economy shrank by \(4\%\) means it is multiplied by \(0.96 \).

Thus after three years we have the original economy multiplied by

\(0.96 \times 0.96 \times 0.96 \approx 0.885 =88.5 \%\)

Three years later, the economy is \(88.5 \%\) of what it was

It thus shrank by \(100 \% - 88.5 \% = 11.5 \% \)

Further applications

Exercise 107 p.134

A major state university reported increases in in-state tuition of \(9.3\%, 8.8\%, 8.9\%, 9.3\%, 5.0\%, 8.7\% \) in years 2008 -- 2013, respectively. What was the percentage increase in tuition over the five year period?

Solution. The tuition was multiplied every time by the corresponding fraction. All together,

\(\text{new tuition = old tuition} \times 1.093 \times 1.088 \times 1.089 \times 1.093 \ldots \)

\(1.093 \times 1.088 \times 1.089 \times 1.093 \times 1.05 \times 1.087 \approx 1.62 \)

Answer: tuition increase is \(62 \%\).

Further applications

Exercise 107 p.134 ... continued

A major state university reported increases in in-state tuition of \(9.3\%, 8.8\%, 8.9\%, 9.3\%, 5.0\%, 8.7\% \) in years 2008 -- 2013, respectively. What was the percentage increase in tuition over the five year period?

We found that tuition increase is \(62 \%\).

If we simply add these numbers:

\(9.3+8.8+8.9+9.3+5.0+8.7 = 50 \%\), and this answer differs substantially from the correct one!