Rate and Work

A rate compares a quantity to a unit of something else, most often time — pages per minute, miles per hour, or jobs per day. This lesson turns rates into a single reusable equation, amount equals rate times time, and shows how two separate work rates combine into one when people or machines work together.

What you will learn

• Define rate as a quantity per unit of another quantity
• Combine individual work rates into a combined rate
• Solve work problems using rate times time equals amount

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

A rate is an amount per unit

A rate is a ratio where one of the two quantities is a fixed unit, usually a unit of time. "A printer produces 25 pages per minute" is a rate: 25 pages for every 1 minute. Because the denominator is fixed at 1 unit, a rate can be multiplied directly by however many units are actually involved to get a total amount, without first finding a scale factor the way a general ratio problem requires.

Finding a rate from raw data means dividing the amount produced by the time it took: a machine that makes 150 pages in 6 minutes has a rate of $150\div6=25$ pages per minute. This division step is essential whenever a rate is not stated outright but must be computed from a total amount and a total time first.

Combining work rates

When two workers or machines act on the same job at the same time, their rates add — but only when the rates are expressed as "fraction of the job per unit of time," not as "hours to finish the whole job." A painter who finishes a fence alone in 6 hours completes $\frac{1}{6}$ of the fence every hour; a second painter who takes 3 hours alone completes $\frac{1}{3}$ of the fence every hour. Working together, they complete $\frac{1}{6}+\frac{1}{3}=\frac{1}{2}$ of the fence every hour.

This combined rate is itself a rate, so it converts back into a total time the same way any rate does: if $\frac{1}{2}$ of the job is completed every hour, the whole job (1 job) takes $1\div\frac{1}{2}=2$ hours. A common error is to average the two individual times instead of adding the rates — averaging $6$ and $3$ hours gives $4.5$, which is not the correct combined time.

The rate-time-amount equation

Every rate problem, whether it involves distance, printing, or work, follows the same equation: $\text{amount}=\text{rate}\times\text{time}$. Depending on which two quantities a problem provides, this equation is solved for whichever one is missing — dividing amount by rate to find time, or dividing amount by time to find rate.

This equation also handles partial completion: if a hose fills a pool at $\frac{1}{8}$ of the pool per hour and runs for 3 hours, the amount filled is $\frac{1}{8}\times3=\frac{3}{8}$ of the pool, leaving $1-\frac{3}{8}=\frac{5}{8}$ of the pool still empty. Watching for whether a question asks for the amount completed or the amount remaining is essential, since the two are complements of each other.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Averaging two workers' individual completion times instead of adding their rates.
• Forgetting to invert a combined rate back into a time by dividing 1 by the rate.
• Answering with the amount completed when the problem asks for the amount remaining, or the reverse.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A rate is an amount divided by a unit of time, and it converts directly into a total amount by multiplying by any number of units.
• Combine work rates by adding fractions of the job completed per unit of time, never by averaging completion times, then invert the sum for the combined time.
• Use amount equals rate times time to solve for whichever quantity is missing, and watch for whether a question asks for the amount completed or the amount remaining.