Lesson 7: Solve Unit Rate Problems

Property

A ratio table organizes equivalent rates.
To find a unit rate, divide both quantities by the value of the quantity you want to be 1.
To find an equivalent rate, multiply both quantities in the unit rate row by the new desired quantity.

Examples

Property

Proportional relationships can be represented by an equation of the form $y = kx$ or $y = rx$. In this equation:

• $x$ is the independent variable (input).
• $y$ is the dependent variable (output).
• $r$ (or $k$) is the constant of proportionality (the unit rate).

This equation shows that the output is always a constant multiple of the input.

Examples

• A machine prints 80 pages in 5 minutes. The unit rate is $r = \frac{80}{5} = 16$ pages per minute. The equation is $p = 16m$, where $p$ is pages and $m$ is minutes.
• The cost for apples is 2.50 dollars per pound. If $C$ is the total cost and $p$ is the number of pounds, the equation is $C = 2.5p$.
• A graph of a proportional relationship passes through $(4, 32)$. The unit rate is $\frac{32}{4}=8$. The equation representing this graph is $y = 8x$.

Explanation

An equation is like a powerful calculator for a proportional relationship. Once you find the constant rate ($r$), you can plug in any amount for $x$ to instantly find its corresponding amount $y$, without having to fill out a huge table.

Property

For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula:

where $d$ = distance, $r$ = rate, and $t$ = time. To solve problems with this formula, first identify the knowns and unknowns, substitute the values into the formula, and then solve for the missing variable.

Examples

• Lindsay drove for 4.5 hours at 60 miles per hour. How much distance did she travel? The distance is $d = rt = 60 \cdot 4.5 = 270$ miles.
• Lee wants to drive 600 miles. If he drives at a steady rate of 75 miles per hour, how many hours will the trip take? The time is $t = \frac{d}{r} = \frac{600}{75} = 8$ hours.
• A train trip from Boston to Washington D.C. is 440 miles and takes 5 hours. What was the average speed of the train? The rate is $r = \frac{d}{t} = \frac{440}{5} = 88$ miles per hour.

Explanation

This formula connects speed, time, and distance. Think of it this way: the distance you cover is a product of how fast you travel (rate) and for how long (time). It is a fundamental principle used to describe motion.