Unit rates when the amounts are fractions

When you reach for it

A unit rate says how much of one quantity goes with exactly one of another: miles in one hour, cups of honey in one batch, how much of a lawn is mowed in one hour. When both amounts are whole numbers, finding it is one division: 6 miles in 2 hours is 6 \div 2 = 3 miles per hour.

Real measurements are rarely that tidy. A walk of \displaystyle \frac{2}{3} of a mile takes a quarter of an hour. A painter covers \displaystyle \frac{3}{8} of a fence in \displaystyle \frac{3}{4} of an hour. A three-quarter batch of granola uses \displaystyle \frac{1}{2} cup of honey. Neither amount is “one” of anything, and you cannot count up in steps of 1 to reach a whole hour. Reach for this skill whenever both sides of a rate are fractions or mixed numbers and you want the amount that goes with one.

The answer is still a division, and the rate can be written as a single fraction whose top and bottom are themselves fractions, such as

This lesson shows why that division gives the amount per one, how to carry it out, and how to use the result. It is taught in two parts. Part 1 builds the idea with fractions of a mile and of an hour, and treats fractions built out of fractions as numbers in their own right. Part 2 brings in mixed numbers, jobs and areas, ingredients per batch, and comparing two rates.

The procedure

Part 1: from a fraction of an hour to a whole hour

A unit fraction of an hour. Femi walks \displaystyle \frac{2}{3} of a mile in \displaystyle \frac{1}{4} of an hour, at a steady pace. Lay the hours and the miles side by side on a double number line, where points that sit one above the other go together.

An hour is four quarter-hours. At a steady pace every quarter-hour adds the same distance, \displaystyle \frac{2}{3} of a mile, so one hour covers four times as much:

Femi walks at \displaystyle 2\frac{2}{3} miles per hour. Multiplying by 4 was right because 4 is the number of quarter-hours in an hour, and that is the same thing as dividing by \displaystyle \frac{1}{4}, since \displaystyle \frac{1}{4} fits into 1 exactly four times:

Any fraction of an hour. Hana swims \displaystyle \frac{9}{10} of a mile in \displaystyle \frac{3}{5} of an hour. Now the time is three fifths of an hour, not one fifth, so reach one hour in two moves.

Down to one part. \displaystyle \frac{3}{5} of an hour is three fifths. At a steady pace each fifth carries the same distance, so one fifth of an hour carries a third of the distance:

Up to a whole. An hour is five fifths, so it carries five times that:

Hana swims at \displaystyle 1\frac{1}{2} miles per hour. The two moves were “divide by 3, then multiply by 5”, which together is “multiply by \displaystyle \frac{5}{3}”, and \displaystyle \frac{5}{3} is the reciprocal of the time, \displaystyle \frac{3}{5}. So the two moves are exactly dividing by multiplying by the reciprocal:

The same two moves work for any fraction of an hour \displaystyle \frac{c}{d}. Divide by c to get the amount for \displaystyle \frac{1}{d} of an hour, then multiply by d to get the amount for \displaystyle \frac{d}{d} = 1 hour. Together that is multiplying by \displaystyle \frac{d}{c}, which is dividing by \displaystyle \frac{c}{d}. So in every case

where the second amount is the one you want exactly one of.

Fractions built out of fractions. A rate is often written with a fraction bar: 6 miles in 2 hours is \displaystyle \frac{6}{2} miles per hour. Written the same way, Hana’s rate is

A fraction whose top or bottom, or both, contains a fraction is called a complex fraction. “Complex” here means built from parts, not difficult. Its long bar means division, exactly as in any fraction, so a complex fraction is simplified by carrying out that division. A second way is often quicker: multiply the top and the bottom by one number chosen to clear both small denominators. Here 10 works, since it is a multiple of both 10 and 5:

This is allowed because multiplying top and bottom by 10 multiplies the whole fraction by \displaystyle \frac{10}{10}, which is 1, and multiplying by 1 changes nothing. It also makes sense as a rate: 9 miles in 6 hours is ten times as far in ten times as long, the same pace as \displaystyle \frac{9}{10} of a mile in \displaystyle \frac{3}{5} of an hour.

Two unit rates from one pair. The pair “\displaystyle \frac{9}{10} of a mile with \displaystyle \frac{3}{5} of an hour” answers two questions. Miles for one hour is miles divided by hours, which gave \displaystyle \frac{3}{2}. Hours for one mile is hours divided by miles:

Hana takes \displaystyle \frac{2}{3} of an hour, which is 40 minutes, for each mile. The two unit rates are reciprocals of each other, \displaystyle \frac{3}{2} \times \frac{2}{3} = 1, because each is the other’s division done the other way round. The question decides which one you want: “how far in an hour?” asks for miles per hour, and “how long for a mile?” asks for hours per mile.

Part 2: mixed numbers, jobs, batches and comparisons

Mixed numbers become single fractions first. \displaystyle 2\frac{1}{2} means \displaystyle 2 + \frac{1}{2}. Division acts on the whole amount, so first rewrite it as an improper fraction, \displaystyle \frac{5}{2}, and then divide exactly as in Part 1.

A whole job as the unit. When someone paints part of a fence or mows part of a lawn, the natural “one” is the whole fence or the whole lawn. Dividing the fraction of the job done by the time taken gives the fraction of the job done per hour, and dividing the other way gives the hours for one whole job. The job can also be measured as an area, in square meters or acres, and then the rate is square meters or acres per hour. Example 5 works one through with a picture.

Per batch. An ingredient works the same way. Cups divided by batches gives cups per batch. Batches divided by cups gives batches per cup, which says how many batches fit into the amount you have.

Comparing two rates. Two walkers who cover different distances in different times cannot be compared by either number alone: the one who went further may also have taken longer. Turn each pair into a unit rate with the same “per”, then compare those. If you compare hours per mile, or hours per job, the smaller number belongs to the faster one, since it needs less time for the same amount.

Keep the fractions exact. Fractions such as \displaystyle \frac{1}{3} and \displaystyle \frac{1}{6} have decimals that never end. Rounding them before dividing gives a unit rate that is slightly off, and the error grows when the rate is used to predict a larger amount (Example 8). Divide the fractions exactly and round only at the end, if at all.

Worked examples

Examples 1 to 3 belong to Part 1. Examples 4 to 8 belong to Part 2.

Boundaries

The second amount cannot be zero. A rate per one hour needs some time to have passed. Dividing by 0 has no value, so a complex fraction with 0 as its whole bottom is not a number. The first amount may be 0: no distance in \displaystyle \frac{1}{4} of an hour is 0 miles per hour.

The pace has to be steady. Every step above assumed each quarter-hour, or each fifth of an hour, carries the same distance. If the pace changes, the same division gives the average rate over the stretch that was measured, and using it to predict another amount assumes that average holds.

The order of division decides which unit rate you get. Miles divided by hours is miles per hour; hours divided by miles is hours per mile. When neither amount is 0, the two are reciprocals, and they are equal only when both are 1. When the first amount is 0, only one of them exists: 0 miles per hour, but no hours per mile.

Units travel with the numbers. Miles divided by hours is in miles per hour, and fraction of a fence divided by hours is in fences per hour. Two amounts in the same unit, such as cups of oats divided by cups of honey, give a plain number: how many times as much of one there is.

A mixed number is one number. \displaystyle 2\frac{1}{2} is \displaystyle 2 + \frac{1}{2}, not \displaystyle 2 \times \frac{1}{2}, and it has to be rewritten as \displaystyle \frac{5}{2} before dividing.

Where this goes next

The unit rate is the number that turns one quantity into the other. When two quantities keep the same rate all the way along, as the miles and hours of a steady swim do, that number is the constant of proportionality, and the pair is one of the proportional relationships: every row of Hana’s table has \displaystyle \text{miles} = \frac{3}{2} \times \text{hours}. The fractions met here are what make those constants come out as fractions rather than whole numbers.

Speed is the most familiar rate of all, and converting a rate to other units, from miles per hour to feet per minute, multiplies by fractions of exactly the kind used here. Comparing two rates by putting them on the same “per” is also how unit prices decide which package is the better buy.

Complex fractions come back in algebra with letters in them, in the lesson on adding and subtracting rational expressions, which also covers complex fractions. A fraction whose top and bottom are themselves such expressions is simplified by the same two moves: divide, or multiply top and bottom by one expression that clears every small denominator. And division by a fraction extends to negative fractions in multiplying and dividing rational numbers, with the sign worked out separately.

Exercises

Part 1: fractions built out of fractions

Simplify.

1. \displaystyle \dfrac{\frac{5}{6}}{\frac{1}{3}}
2. \displaystyle \dfrac{\frac{3}{8}}{\frac{9}{10}}
3. \displaystyle \dfrac{4}{\frac{2}{7}}
4. \displaystyle \dfrac{\frac{9}{16}}{3}
5. \displaystyle \dfrac{\frac{7}{12}}{\frac{7}{8}}, by multiplying top and bottom by 24.

Part 1: walking and swimming

Each person moves at a steady pace.

6. A walker covers \displaystyle \frac{3}{4} of a mile in \displaystyle \frac{1}{5} of an hour. Find the speed in miles per hour.
7. A swimmer covers \displaystyle \frac{2}{5} of a mile in \displaystyle \frac{1}{4} of an hour. Find the miles per hour and the hours per mile.
8. A walker covers \displaystyle \frac{7}{10} of a mile in 10 minutes. Find the speed in miles per hour.
9. A swimmer covers \displaystyle \frac{3}{8} of a mile in \displaystyle \frac{3}{10} of an hour. Find the speed in miles per hour, and how many minutes one mile takes.
10. A walker covers \displaystyle \frac{4}{5} of a mile in \displaystyle \frac{4}{15} of an hour. Find the speed in miles per hour, and how far the walker goes in \displaystyle \frac{1}{2} of an hour.

Part 2: mixed numbers

11. A walker covers \displaystyle 3\frac{1}{3} miles in \displaystyle 1\frac{1}{6} hours. Find the speed in miles per hour.
12. A swimmer covers \displaystyle 1\frac{1}{8} miles in \displaystyle \frac{5}{6} of an hour. Find the speed in miles per hour.

Part 2: painting and mowing

Each person works at a steady pace.

13. Ade paints \displaystyle \frac{2}{5} of a wall in \displaystyle \frac{1}{3} of an hour. How many walls does he paint per hour, and how many minutes does one whole wall take?
14. Ruth mows \displaystyle \frac{5}{8} of a lawn in \displaystyle \frac{5}{6} of an hour. What fraction of the lawn does she mow per hour, and how long does the whole lawn take?
15. A fence has 30 square meters to paint. Ilse paints \displaystyle 13\frac{1}{2} square meters of it in \displaystyle \frac{3}{4} of an hour. Find her rate in square meters per hour, and how many minutes the rest of the fence takes.
16. Bram mows \displaystyle \frac{2}{3} of a field in \displaystyle 1\frac{1}{4} hours. What fraction of the field does he mow per hour, and how long does the rest take?

Part 2: per batch

17. A third of a batch of pancakes uses \displaystyle \frac{1}{2} cup of milk. How much milk does one batch use, and how many batches does one cup of milk make?
18. Two thirds of a batch of muffins uses \displaystyle 1\frac{1}{4} cups of flour. How much flour does one batch use, and how many batches can be made with 5 cups of flour?

Part 2: comparing rates

19. Yusuf swims \displaystyle \frac{5}{8} of a mile in \displaystyle \frac{1}{3} of an hour. Mirela swims \displaystyle \frac{3}{4} of a mile in \displaystyle \frac{2}{5} of an hour. Who swims faster?
20. Sven paints \displaystyle \frac{3}{10} of a room in \displaystyle \frac{1}{2} of an hour. Kemi paints \displaystyle \frac{1}{4} of a room of the same size in \displaystyle \frac{2}{5} of an hour. Who paints faster, and by how much of a room per hour?

Reasoning

21. A walker covers some distance in \displaystyle \frac{3}{4} of an hour. Explain, without the rule for dividing fractions, why the distance per hour is the distance multiplied by \displaystyle \frac{4}{3}.
22. A walk takes less than one hour. Explain why the number of miles per hour is greater than the number of miles walked. What happens if the walk takes more than one hour?
23. Explain why, for any pair of amounts, the miles per hour and the hours per mile multiply to 1.
24. Someone rounds \displaystyle \frac{2}{3} of a mile to 0.67 and 10 minutes to 0.17 hour, and finds a speed of about 3.94 miles per hour. Find the exact speed, and explain the difference.
25. Explain why multiplying the top and the bottom of a complex fraction by the same non-zero number does not change its value, both as arithmetic and in terms of a rate.