Lesson 49: Solving Rate Problems with Proportions and Equations

New Concept

Proportional relationships can be expressed with two key equations, allowing you to solve rate problems using either proportions or a direct formula.

All proportional relationships can be expressed with these two equations:

1. The ratio is constant: $\frac{y}{x} = k$
2. The unit rate is constant: $y = kx$

What’s next

You're now ready to see these methods in action. Next, you'll walk through worked examples applying both proportions and rate equations to solve the same problems.

Property

A unit rate is a rate with a denominator of 1, showing the value 'per one' item. The formula is $k = \frac{y}{x}$.

Examples

$\frac{20.25 \text{ dollars}}{3 \text{ tickets}} = 6.75$ dollars per ticket.
$\frac{15 \text{ pounds}}{6 \text{ books}} = 2.5$ pounds per book.

Explanation

Think of it as the price tag for just one! Knowing the unit rate helps you find the best deal or compare speeds. It’s your secret weapon for becoming a super-shopper and making smart choices!

Property

Proportions show that two ratios are equal. Use this when a relationship is proportional to find a missing value: $\frac{a}{b} = \frac{c}{d}$.

Examples

$\frac{6 \text{ books}}{15 \text{ lbs}} = \frac{20 \text{ books}}{p \text{ lbs}} \implies p=50$ lbs.
$\frac{15 \text{ km}}{27 \text{ min}} = \frac{50 \text{ km}}{m \text{ min}} \implies m=90$ min.

Explanation

Think of this as your ultimate scaling tool! If you know the ratio for one situation, you can scale it to find a missing piece in another, keeping everything fair and balanced like a math seesaw.

Property

Use the equation $y = kx$, where $k$ is the constant unit rate. Multiply the rate ($k$) by the quantity ($x$) to find the total ($y$).

Examples

Unit rate is $2.5$ lbs/book. Weight of 20 books is $w = 2.5 \cdot 20 = 50$ lbs.

Explanation

First, find the 'per one' value, then multiply away! This direct method is super fast once you have the unit rate. No complex fractions, just simple multiplication to get your answer.