Ratios and Proportions

The ratio of 5 to 2 is $\frac{5}{2}$.

The ratio of 6 to 4 is $\frac{6}{4}$, which simplifies to $\frac{3}{2}$.

The ratio of 15 to 5 is $\frac{15}{5}$, which simplifies to $3$.

A 12-ounce can of Coca-Cola ^® costs 60 cents.

The ratio of 60 cents to 12 ounces is $\frac{60\mathrm{cent}}{12\mathrm{ounce}}$.

$$\frac{60\mathrm{cent}}{12\mathrm{ounce}}$$

$$\frac{60}{12}·\frac{\mathrm{cent}}{\mathrm{ounce}}$$

$$5·\frac{\mathrm{cent}}{\mathrm{ounce}}$$

In ordinary speech, $\frac{\mathrm{cent}}{\mathrm{ounce}}$ is read, " cents per ounce."

The meaning of the ratio of the cost (60 cents) to the amount (12 ounces) turns out to be the unit price, 5 cents per ounce.

Compute the unit prices. Use a calculator. Round, but keep results precise enough that the unit prices are distinguishable.

Size	Price	Unit Price
16 oz	$0.69	$\frac{0.69\mathrm{dollar}}{16\mathrm{ounce}}=0.043125\frac{\mathrm{dollar}}{\mathrm{ounce}}$
24 oz	$0.89	$\frac{0.89\mathrm{dollar}}{24\mathrm{ounce}}\cong 0.037083\frac{\mathrm{dollar}}{\mathrm{ounce}}$
32 oz	$1.19	$\frac{1.19\mathrm{dollar}}{32\mathrm{ounce}}\cong 0.037188\frac{\mathrm{dollar}}{\mathrm{ounce}}$
44 oz	$1.69	$\frac{1.69\mathrm{dollar}}{44\mathrm{ounce}}\cong 0.038409\frac{\mathrm{dollar}}{\mathrm{ounce}}$
64 oz	$2.49	$\frac{2.49\mathrm{dollar}}{64\mathrm{ounce}}\cong 0.038906\frac{\mathrm{dollar}}{\mathrm{ounce}}$

Find the least unit price.

Size	Price	Unit Price
16 oz	$0.69	$\frac{0.69\mathrm{dollar}}{16\mathrm{ounce}}=0.043125\frac{\mathrm{dollar}}{\mathrm{ounce}}$
24 oz	$0.89	$\frac{0.89\mathrm{dollar}}{24\mathrm{ounce}}\cong 0.037083\frac{\mathrm{dollar}}{\mathrm{ounce}}$
32 oz	$1.19	$\frac{1.19\mathrm{dollar}}{32\mathrm{ounce}}\cong 0.037188\frac{\mathrm{dollar}}{\mathrm{ounce}}$
44 oz	$1.69	$\frac{1.69\mathrm{dollar}}{44\mathrm{ounce}}\cong 0.038409\frac{\mathrm{dollar}}{\mathrm{ounce}}$
64 oz	$2.49	$\frac{2.49\mathrm{dollar}}{64\mathrm{ounce}}\cong 0.038906\frac{\mathrm{dollar}}{\mathrm{ounce}}$

The size that has the least unit price is the best deal.

Size	Price	Unit Price
16 oz	$0.69	$\frac{0.69\mathrm{dollar}}{16\mathrm{ounce}}=0.043125\frac{\mathrm{dollar}}{\mathrm{ounce}}$
24 oz	$0.89	$\frac{0.89\mathrm{dollar}}{24\mathrm{ounce}}\cong 0.037083\frac{\mathrm{dollar}}{\mathrm{ounce}}$
32 oz	$1.19	$\frac{1.19\mathrm{dollar}}{32\mathrm{ounce}}\cong 0.037188\frac{\mathrm{dollar}}{\mathrm{ounce}}$
44 oz	$1.69	$\frac{1.69\mathrm{dollar}}{44\mathrm{ounce}}\cong 0.038409\frac{\mathrm{dollar}}{\mathrm{ounce}}$
64 oz	$2.49	$\frac{2.49\mathrm{dollar}}{64\mathrm{ounce}}\cong 0.038906\frac{\mathrm{dollar}}{\mathrm{ounce}}$

The 24-ounce drink is the best deal.

Solve. $$\frac{x}{5}=\frac{x+3}{6}$$

$$\frac{x}{5}=\frac{x+3}{6}$$

Cross-multiply.

$$\left(x\right)\left(6\right)=\left(5\right)\left(x+3\right)$$

Solve.

$$6x=5x+15$$

$$6x-5x=5x+15-5x$$

$$x=15$$

$$\left\{15\right\}$$

Solve. $$\frac{2w+5}{9}=\frac{7w-2}{12}$$

$$\frac{2w+5}{9}=\frac{7w-2}{12}$$

Cross-multiply.

$$\left(2w+5\right)\left(12\right)=\left(9\right)\left(7w-2\right)$$

Solve.

$$24w+60=63w-18$$

$$24w+60-63w=63w-18-63w$$

$$-39w+60=-18$$

$$-39w+60-60=-18-60$$

$$-39w=-78$$

$$\frac{-39w}{-39}=\frac{-78}{-39}$$

$$w=2$$

$$\left\{2\right\}$$

If 5 cups of vinegar cost $1.50, how much do 2 cups of vinegar cost?

Set up a proportion to solve the problem.

• One one side, write a ratio describing the known situation.
• Make both numerators have the same unit or describe the same feature.

Let x be the cost of 2 cups of vinegar.

The left side of the proportion describes the known situation—buying 5 cups of vinegar for $1.50.

The right side has the same unit in its numerator (cup) as the left side has in its numerator (cup). The right side has the same unit in its denominator (dollar) that the left side has in its denominator (dollar).

$$\frac{\stackrel{\stackrel{\text{cup}}{⏞}}{5}}{\underset{\underset{\text{dollar}}{⏟}}{1.50}}=\frac{\stackrel{\stackrel{\text{cup}}{⏞}}{2}}{\underset{\underset{\text{dollar}}{⏟}}{x}}$$

Cross-multiply.

$$5x=\left(1.50\right)\left(2\right)$$

$$5x=3$$

$$\frac{5x}{5}=\frac{3}{5}$$

$$x=0.60$$

The unit for x is the dollar.

It costs $0.60 for two cups of vinegar.

The proportion $\frac{\stackrel{\stackrel{\text{dollar}}{⏞}}{1.50}}{\underset{\underset{\text{cup}}{⏟}}{5}}=\frac{\stackrel{\stackrel{\text{dollar}}{⏞}}{x}}{\underset{\underset{\text{cup}}{⏟}}{2}}$ leads to the same answer.

Let d be the actual distance between Fraction City and Expressionopolis. Write a proportion in which the numerators are actual distances in miles, and the denominators are map distances in inches.

$$\frac{d}{18}=\frac{16}{12}$$

Cross-multiply.

$$d·12=18·16$$

$$12d=288$$

$$\frac{12d}{12}=\frac{288}{12}$$

$$d=24$$

The unit for d is mile.

The actual distance between Fraction City and Expressionopolis is 24 miles.

Ratios of corresponding sides are equal, so the ratios $\frac{x}{15}$, $\frac{y}{13}$, and $\frac{21}{14}$ are equal.

Solve $\frac{y}{13}=\frac{21}{14}$ to find the value of y. It is 19.5.

Solve $\frac{x}{15}=\frac{21}{14}$ to find the value of x. It is 22.5.

The value of x is 22.5, and the value of y is 19.5.