Ratios and Rates

Understanding ratios, rates, and unit rates

What is a Ratio?

A ratio compares two quantities. It can be written three ways:

Example: If there are 15 boys and 10 girls in a class, the ratio of boys to girls is: $\frac{15}{10} = \frac{3}{2} \quad \text{or} \quad 3:2$

A rate is a ratio that compares two quantities with different units.

Example: 60 miles in 2 hours = $\frac{60 \text{ miles}}{2 \text{ hours}} = 30 \text{ mph}$

A unit rate is a rate with a denominator of 1.

Example: If 12 apples cost $6, the unit rate is: $\frac{\$6}{12 \text{ apples}} = \$0.50 \text{ per apple}$

Simplify ratios like fractions by dividing by the GCF.

Example: $12:16 = 3:4$ (divide both by 4)

Simplify the ratio $15:25$

💡 Show Solution

Find the GCF of 15 and 25:

Divide both numbers by 5: $15:25 = \frac{15}{5}:\frac{25}{5} = 3:5$

Answer: $3:5$

A car travels 240 miles in 4 hours. What is the unit rate in miles per hour?

💡 Show Solution

To find the unit rate, divide the distance by the time:

$\text{Unit rate} = \frac{240 \text{ miles}}{4 \text{ hours}}$

$= \frac{240}{4} = 60 \text{ miles per hour}$

Answer: 60 mph

If 8 pencils cost $2.40, what is the cost per pencil?

💡 Show Solution

Divide the total cost by the number of pencils:

$\text{Cost per pencil} = \frac{\$2.40}{8}$

$= \$0.30$

Answer: $0.30 per pencil

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