This grade 7 unit rates fractions lesson explains unit rates with fractions in one printable page, with definitions, a method and a worked example.
A unit rate with fractions compares two quantities when one or both are fractions, such as \(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour. Divide the first quantity by the second to find the amount for 1 unit. Here \(\frac{1}{2}: \frac{1}{4}=2\), so the unit rate is 2 miles per hour.
Grade 7 Unit Rates Fractions Lesson: key facts
- A unit rate is the amount of one quantity per 1 unit of another quantity.
- A rate with fractions is a complex fraction, and it equals the first fraction divided by the second.
- Dividing by a fraction means multiplying by its reciprocal.
- Convert units first when the two quantities use different units, such as minutes and hours.
- Check by multiplying the unit rate by the second quantity to get the first quantity.
What is a unit rate with fractions?
A rate compares two different quantities, such as miles and hours. When one or both quantities are fractions, the rate is a complex fraction. Maya walks \(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour, so her rate is \(\dfrac{\frac{1}{2}\text{ mi}}{\frac{1}{4}\text{ h}}\).
If \(\frac{a}{b}\) of one quantity goes with \(\frac{c}{d}\) of another, the unit rate is the amount of the first quantity for 1 unit of the second: \(\frac{a}{b}: \frac{c}{d}\).
How do you find a unit rate when the numbers are fractions?
- Write the rate as (first quantity) \(: \) (second quantity).
- Change the division into multiplication by the reciprocal of the second fraction.
- Multiply, then simplify.
- Say the answer with units: miles per hour, cups per batch, dollars per pound.
Example. Sofia reads \(\frac{2}{3}\) of a chapter in \(\frac{1}{6}\) hour. \(\frac{2}{3}: \frac{1}{6}=\frac{2}{3}\times \frac{6}{1}=\frac{12}{3}=4\), so she reads 4 chapters per hour. Check: \(4\times \frac{1}{6}=\frac{2}{3}\).
What if the two quantities use different units?
Convert one of them first so the rate is in the unit you want. Ethan walks \(\frac{1}{2}\) mile in 10 minutes. Since 10 minutes is \(\frac{1}{6}\) of an hour, \(\frac{1}{2}: \frac{1}{6}=\frac{1}{2}\times 6=3\) miles per hour.
Can a unit rate involve area?
Yes. Priya paints \(\frac{3}{4}\) square foot of a poster in \(\frac{1}{8}\) hour. \(\frac{3}{4}: \frac{1}{8}=\frac{3}{4}\times 8=6\) square feet per hour.
What mistakes should you avoid?
- Multiplying the two fractions instead of dividing. A rate of \(\frac{1}{2}\) mile in \(\frac{1}{4}\) hour is not \(\frac{1}{8}\) mile per hour.
- Dividing in the wrong order. The unit named before the word per goes first, so miles per hour is miles \(: \) hours.
- Forgetting to convert. Ten minutes must become \(\frac{1}{6}\) hour before you divide to get miles per hour.
- Leaving off the units. The number 3 means little until you write 3 miles per hour.
How can you use a table of rates?
| Situation | Division | Unit rate |
|---|---|---|
| \(\frac{1}{2}\) mi in \(\frac{1}{4}\) h | \(\frac{1}{2}: \frac{1}{4}\) | 2 mi per h |
| \(\frac{3}{4}\) cup per \(\frac{1}{2}\) batch | \(\frac{3}{4}: \frac{1}{2}\) | \(1\frac{1}{2}\) cups per batch |
| $3 for \(\frac{3}{4}\) lb | \(3: \frac{3}{4}\) | $4 per lb |
Unit Rates with Fractions: frequently asked questions
What is a complex fraction?
A complex fraction has a fraction in its numerator, its denominator, or both. In a rate such as \(\frac{1/2\text{ mi}}{1/4\text{ h}}\), the fraction bar means division, so it equals \(\frac{1}{2}: \frac{1}{4}\).
How do you divide by a fraction?
Multiply by the reciprocal of the fraction you are dividing by. For example, \(\frac{3}{4}: \frac{1}{8}=\frac{3}{4}\times \frac{8}{1}=6\).
Why does dividing by a small fraction give a big rate?
A small fraction of an hour fits many times into 1 hour. If you cover \(\frac{1}{2}\) mile in only \(\frac{1}{4}\) hour, you would cover four times as much in a full hour, which is 2 miles.
Which quantity goes first in the division?
The quantity named before the word per goes first. Miles per hour means miles \(: \) hours, and cups per batch means cups \(: \) batches.
How can I check a unit rate?
Multiply the unit rate by the second quantity. If the unit rate is 3 miles per hour and the time is \(\frac{1}{6}\) hour, then \(3\times \frac{1}{6}=\frac{1}{2}\) mile, which matches the original distance.
How to use this grade 7 unit rates fractions lesson
Use this grade 7 unit rates fractions lesson before the worksheet, then check what you learned with the quiz.
See every chapter on the Grade 7 math page.
