5.6 Ratios and Rate

Ratios and Rates

A ratio compares two quantities using division. A rate is a specific type of ratio that compares two quantities with different units. Together, they show up constantly in everyday life: comparing prices at the store, figuring out how fast you're driving, or scaling a recipe up or down.

Ratios and Rates as Fractions

A ratio can be written three ways, and they all mean the same thing:

• $a:b$
• $a$ to $b$
• $\frac{a}{b}$

For example, if a class has 3 boys for every 2 girls, you can write that as $3:2$, $3$ to $2$, or $\frac{3}{2}$. Notice that order matters: $3:2$ (boys to girls) is not the same as $2:3$ (girls to boys).

A rate works the same way but involves different units. If a car travels 120 miles in 3 hours, the rate is:

$\frac{120 \text{ miles}}{3 \text{ hours}}$

You can simplify that to $\frac{40 \text{ miles}}{1 \text{ hour}}$, which is 40 miles per hour.

Calculation of Unit Rates

A unit rate is a rate where the second quantity equals 1. To find it, divide the first quantity by the second.

Example 1: 5 apples cost $2.00.

$\frac{2.00 \text{ dollars}}{5 \text{ apples}} = \frac{0.40 \text{ dollars}}{1 \text{ apple}}$

Each apple costs $0.40.

Example 2: 3 pounds of bananas cost $4.50.

$\frac{4.50 \text{ dollars}}{3 \text{ pounds}} = \frac{1.50 \text{ dollars}}{1 \text{ pound}}$

Each pound costs $1.50.

Unit rates are especially useful for comparing deals. If Brand A sells 12 oz of cereal for $3.60 and Brand B sells 16 oz for $4.00, find each unit price:

• Brand A: $\frac{3.60}{12} = 0.30$ dollars per oz
• Brand B: $\frac{4.00}{16} = 0.25$ dollars per oz

Brand B is the better deal.

Verbal to Fractional Conversions

When a problem gives you a ratio or rate in words, convert it to a fraction by placing the first quantity mentioned as the numerator and the second as the denominator.

1. Identify the two quantities being compared.
2. Put the first quantity on top and the second on the bottom.

Example: "A car uses 2 gallons of gas for every 50 miles."

$\frac{2 \text{ gallons}}{50 \text{ miles}}$

Watch the wording carefully. "For every 2 boys, there are 3 girls" compares boys to girls, so the ratio of boys to girls is $\frac{2 \text{ boys}}{3 \text{ girls}}$. If a question asks for the ratio of girls to boys, you'd flip it: $\frac{3 \text{ girls}}{2 \text{ boys}}$. Always check what the question is asking for.

Ratios vs. Rates in Context

The key difference is units:

• Ratios compare quantities with the same units (or no units at all). A recipe that uses 2 cups of flour for every 3 cups of sugar gives a ratio of $\frac{2}{3}$ since both are measured in cups.
• Rates compare quantities with different units. A car traveling 60 miles in 2 hours gives a rate of $\frac{60 \text{ miles}}{2 \text{ hours}}$, since miles and hours are different units.

If you're unsure which one you're dealing with, check whether the units match.

Applications of Ratio Concepts

Ratios become really useful when you need to find an unknown quantity. The process uses proportions, which are two equal ratios set side by side.

Steps to solve a proportion problem:

1. Identify the known ratio or rate.
2. Set up a proportion with the unknown value as $x$.
3. Cross multiply and solve for $x$.

Example: A recipe calls for 2 cups of flour for every 3 cups of sugar. How much flour do you need for 9 cups of sugar?

1. Known ratio: $\frac{2 \text{ cups flour}}{3 \text{ cups sugar}}$
2. Set up the proportion: $\frac{2}{3} = \frac{x}{9}$
3. Cross multiply: $2 \times 9 = 3 \times x$, so $18 = 3x$
4. Divide both sides by 3: $x = 6$

You need 6 cups of flour.

Advanced Ratio and Rate Concepts

These ideas build on what you've already learned and connect to topics you'll see later in math:

• Scaling means increasing or decreasing quantities while keeping the ratio the same. Doubling a recipe (multiplying every ingredient by 2) is scaling.
• Dimensional analysis uses conversion factors written as fractions to switch between units. For instance, converting 3 hours to minutes: $3 \text{ hours} \times \frac{60 \text{ minutes}}{1 \text{ hour}} = 180 \text{ minutes}$.
• Direct variation describes a relationship where one quantity changes proportionally with another, written as $y = kx$. The value $k$ is called the constant of variation. If you earn $12 per hour, your pay ($y$) and hours worked ($x$) are in direct variation with $k = 12$.
• Percent is a ratio out of 100. Scoring 45 out of 50 on a test gives $\frac{45}{50} = \frac{90}{100} = 90\%$.