Lesson 31: Using Rates, Ratios, and Proportions

New Concept

Algebra is built on relationships between quantities. A core example is the proportion, an equation that shows two ratios are equal.

Proportion:
The equation $\frac{a}{b} = \frac{c}{d}$ is a proportion.

Cross Products Property:
If $\frac{a}{b} = \frac{c}{d}$ and $b \ne 0$ and $d \ne 0$, then $ad = bc$.

What’s next

Our journey begins with this lesson on rates, ratios, and proportions. Soon, you'll use these fundamental tools to practice setting up and solving your first algebraic equations.

Property

A unit rate is a rate whose denominator is 1. A unit price is the cost per unit.

Examples

Which is cheaper: 4 candy bars for 3.20 dollars or 5 for 3.50 dollars? $\frac{3.20 \text{ dollars}}{4} = 0.80$ dollars per bar vs $\frac{3.50 \text{ dollars}}{5} = 0.70$ dollars per bar. The 5-pack is the better buy!
A car travels 120 miles on 4 gallons of gas. The unit rate is $\frac{120 \text{ miles}}{4 \text{ gallons}} = 30$ miles per gallon.
You earn 45 dollars for 5 hours of work. Your unit rate is $\frac{45 \text{ dollars}}{5 \text{ hours}} = 9$ dollars per hour.

Explanation

Ever wondered which deal is truly better? Unit rates are your secret weapon! By finding the cost for just one item, like a can of soda, you can easily compare prices. This makes you a super-savvy shopper because you always know the cost 'per one,' which helps you spot the best value every single time.

Property

Set up the conversion factor so the units of measure cancel.

Examples

Convert 45 miles per hour to miles per minute: $\frac{45 \text{ miles}}{1 \text{ hour}} \cdot \frac{1 \text{ hour}}{60 \text{ minutes}} = \frac{45 \text{ miles}}{60 \text{ minutes}} = 0.75$ miles per minute.
Convert 2 gallons per minute to quarts per minute: $\frac{2 \text{ gallons}}{1 \text{ minute}} \cdot \frac{4 \text{ quarts}}{1 \text{ gallon}} = \frac{8 \text{ quarts}}{1 \text{ minute}}$.

Explanation

Need to switch from miles per hour to feet per second? That's converting rates! The trick is multiplying by a special fraction, called a conversion factor, that equals one, like $\frac{60 \text{ seconds}}{1 \text{ minute}}$. You cleverly arrange it so the old units on top and bottom cancel out, leaving you with the brand new units you want.

Property

Cross Products Property: If $\frac{a}{b} = \frac{c}{d}$ and $b \ne 0$ and $d \ne 0$, then $ad = bc$.

Examples

Solve $\frac{x}{10} = \frac{4}{5}$. Cross multiply to get $5 \cdot x = 10 \cdot 4$, so $5x = 40$, which means $x=8$.
Solve $\frac{y+2}{8} = \frac{5}{4}$. Cross multiply to get $4(y+2) = 8 \cdot 5$. This simplifies to $4y + 8 = 40$, so $4y=32$ and $y=8$.

Explanation

Think of this as the ultimate shortcut for solving proportions! Instead of guessing the missing number, you just multiply the numbers that are diagonal from each other across the equals sign. This 'cross-multiplication' trick magically turns a tricky fraction problem into a simple equation you already know how to solve. It is a powerful tool!