Unit Rate as Slope

Three ways to write a ratio

The same comparison can be written with the word 'to' ($12$ to $18$), with a colon ($12:18$), or as a fraction ($\dfrac{12}{18}$). They all mean the same thing. The fraction form is the most useful one, because everything you know about fractions carries over — especially simplifying.

To simplify a ratio, divide both parts by their greatest common factor, exactly like reducing a fraction. $12:18$ simplifies to $2:3$ because both parts divide by $6$. A simplified ratio still says the same thing: for every $2$ girls there are $3$ boys.

From rate to unit rate

To turn any rate into a unit rate, divide the first quantity by the second. If a car goes $180$ miles in $4$ hours, the unit rate is $\dfrac{180}{4} = 45$ miles per hour. The word 'per' is your cue — it literally means 'divided by,' and the number after 'per' should end up as $1$.

Unit rates are how you compare two options fairly. Two printers with different page counts and different times can't be compared directly — but pages per one minute can. Whoever has the bigger 'per one' number is faster, period.

Keeping units straight

Always write the units next to the numbers: $\dfrac{180 \text{ miles}}{4 \text{ hours}}$, not just $\dfrac{180}{4}$. The units tell you which number goes on top. 'Miles per hour' means miles on top, hours on the bottom. If your answer's units read backwards — hours per mile when you wanted miles per hour — you divided in the wrong order.

Worked examples

Try one yourself

Common questions