Rates and Unit Rates

What makes a rate a rate

A rate is a ratio that compares two quantities measured in different units. The ratio of $2$ cups of flour to $3$ cups of sugar from the last lesson compares two amounts measured in the very same unit: both are cups. Because the unit matches on both sides, you never have to write it, and $2 : 3$ already says everything there is to say. A rate compares two quantities measured in different units instead, so its units cannot be dropped the same way:

Because the units stay, you must say them out loud, and the order still matters exactly as it did for ratios. “Miles per hour” is miles on top and hours on the bottom; “dollars per pound” is dollars on top and pounds on the bottom. The little word per is the signpost: it means “for each” and marks the quantity that goes in the denominator. Read the bar as “per,” so $\frac{120 \text{ miles}}{2 \text{ hours}}$ is ”$120$ miles per $2$ hours.”

The unit rate

A rate written with a denominator of $1$ is called a unit rate: the amount for a single unit of the second quantity. ”$60$ miles per hour” is a unit rate because the hour count is $1$. ”$2$ dollars per pound” is a unit rate because the pound count is $1$. You will see unit rates everywhere, on speedometers, price tags, and nutrition labels. Unit rates appear there because one number that means “per one” is the easiest kind of rate to compare and to scale.

One division turns a rate into its unit rate. The car above traveled $120$ miles in $2$ hours, so

Suppose the car holds a steady speed. Those $2$ hours then share the $120$ miles equally, and each hour’s share is $60$ miles. To turn any rate into a unit rate, divide the first quantity by the second.

Why dividing gives the amount per one

For a steady rate with a whole-number second quantity, dividing does exactly what the words “amount per one” ask: it splits a total into equal shares, one share for each unit. That picture is not the whole reason division works, though, since the second quantity is not always a whole number of shares. The real reason is scaling.

Unit rates can use decimals

The two quantities will not always divide evenly, and that is fine. The quotient is then just a decimal, exactly the kind you practiced in the Decimals chapter. If $3$ pounds of grapes cost $7$ dollars, the price per pound is

The $3$s keep repeating forever, so round to a sensible number of places for the situation. For money, two decimal places (the nearest cent) is standard, so about $2.33$ dollars per pound.

Comparing rates: the better deal

Two deals are hard to compare when they use different amounts. But once you reduce each deal to “per one”, the two deals line up on the same scale and the comparison is immediate. This is exactly how unit pricing works in a grocery store.

To compare two options, find the unit rate for each using the same units in the same order, then read off which is better. For a price, the lower dollars-per-item is the better buy; for a speed, the higher miles-per-hour is faster. The direction depends on what you want more or less of.

Scaling a rate to a new amount

A unit rate also predicts totals. Once you know the amount per one unit, multiply by however many units you want. This is the same scaling you used for ratios: a unit rate is the rate scaled down to $1$. So scaling that unit rate back up by any number gives an equivalent rate for that many units.

Converting between rates

Sometimes a rate is measured over one stretch of time and you want it over a longer or shorter stretch. Turning gallons per second into gallons per minute is one example. You convert by scaling, using the same equivalent-ratio move from the last lesson. Multiply or divide both quantities by the same number so the rate stays equal. Choose the number that resizes the bottom quantity to the stretch you want. That stretch may then have a shorter name: $60$ seconds is $1$ minute.

Scaling can shrink an interval too, using the same move in reverse. If that same tank keeps draining at this steady rate, the $6$ gallons every $60$ seconds can be restated over a $15$-second window. Since $60$ seconds holds $4$ windows of $15$ seconds, divide both quantities by $4$:

So at that steady rate, the tank drains $1.5$ gallons every $15$ seconds, the same rate seen over a shorter stretch. If the drain instead sped up or slowed down along the way, $1.5$ gallons per $15$ seconds would still be the correct average, just not the exact amount in every single window.

Common mistakes