# Section 3A

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## Brief review of fractions and percentages (p.121)

Exercise 29 p.132

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

Solution

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!