Proportions and Cross-Multiplication

Ratios, Proportions, and Percent

By the end of this lesson, you will be able to set up and solve proportions, convert between fractions, decimals, and percents, and solve percent problems including percentage change.

Ratios

Ratio: a comparison of two quantities. Can be written as:

• $3:4$ (ratio notation)
• $\dfrac{3}{4}$ (fraction)
• $3$ to $4$ (word form)

Ratio vs. fraction: a ratio compares two quantities; these quantities don't have to be part-to-whole (a ratio of boys to girls is part-to-part).

Equivalent ratios: $3:4 = 6:8 = 9:12$ (multiply or divide both parts by the same number).

Unit rate: a ratio with a denominator of 1.

$\dfrac{60 \text{ miles}}{1 \text{ hour}} = 60 \text{ mph}$

Example: A car travels 150 miles in 3 hours. Find the unit rate.

$\dfrac{150}{3} = 50$ miles per hour

Proportions

Proportion: a statement that two ratios are equal.

$$\frac{a}{b} = \frac{c}{d}$$

Cross-multiplication: if $\dfrac{a}{b} = \dfrac{c}{d}$, then $ad = bc$.

Solve: $\dfrac{3}{5} = \dfrac{x}{20}$

$3 \times 20 = 5x$

$60 = 5x$

$x = 12$

Word problem: If 4 gallons of paint cover 800 square feet, how many gallons are needed to cover 1,400 square feet?

$$\frac{4 \text{ gal}}{800 \text{ sq ft}} = \frac{x \text{ gal}}{1400 \text{ sq ft}}$$

$4 \times 1400 = 800x$

$5600 = 800x$

$x = 7$ gallons

Percents

Percent ("per hundred"): a ratio with a denominator of 100.

$45% = \dfrac{45}{100} = 0.45$

Converting:

• Percent → Decimal: divide by 100 (move decimal 2 places left): $37% = 0.37$
• Decimal → Percent: multiply by 100 (move decimal 2 places right): $0.85 = 85%$
• Percent → Fraction: write over 100, simplify: $60% = \dfrac{60}{100} = \dfrac{3}{5}$
• Fraction → Percent: convert to decimal, then multiply by 100: $\dfrac{3}{8} = 0.375 = 37.5%$