Why do I flip the second fraction instead of the first?

Because dividing by a number and multiplying by its reciprocal give the same result, and only the divisor — the number you are dividing by — gets replaced. Dividing 3/4 by 1/2 asks how many halves fit in 3/4, and the answer, 3/4 × 2 = 3/2, comes from flipping the 1/2. Flipping the first fraction changes the quantity you started with, which changes the problem entirely.

How do I know which number goes on the bottom?

Look for the word "per," "each," or "for one." Whatever comes right after it becomes 1 and goes in the divisor spot. "Cups per serving" puts servings on the bottom. "Cost for each pound" puts pounds on the bottom. If the question just says "find the unit rate," the standard order is the first quantity mentioned divided by the second.

Can a unit rate be a fraction or a mixed number?

Yes. Unit rates are often not whole numbers. A pace of 2/27 hour per mile or 15/16 cubic yards per hour is perfectly valid. The word "unit" refers to the second quantity being 1, not to the answer being a whole number.

Should I turn the fractions into decimals first?

You can when the fractions convert cleanly, like 1/4 = 0.25, and a decimal answer is often easier to compare with prices. But fractions like 2/3 become repeating decimals, and rounding early makes the final answer inexact. Divide with fractions to stay exact, then convert to a decimal at the end if the situation calls for it.

M7MATH-3.2

Unit Rates with Fractions

Learn to find unit rates from ratios of fractions — divide by multiplying by the reciprocal, handle like and different units, and avoid the flip-the-wrong-fraction trap.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Unit Rates with Fractions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Unit Rates with Fractions free

What this lesson covers

You already know how to find a unit rate when the numbers are whole: 12 dollars for 4 pounds is 3 dollars per pound. But real situations rarely stay tidy. A runner covers

\frac{3}{4}

of a mile in

\frac{1}{6}

of an hour. A painter uses

\frac{2}{3}

of a quart to cover

\frac{5}{8}

of a wall. The question is exactly the same as before — how much per one? — but the arithmetic now involves dividing a fraction by a fraction.

In this lesson you will learn to write a ratio of fractions as a division problem, carry out that division by multiplying by the reciprocal, and attach the correct units to your answer. You will also learn to decide which quantity belongs on the bottom, because "miles per hour" and "hours per mile" are both true statements about the same trip but they are very different numbers.

What a Unit Rate Means When the Numbers Are Fractions

A rate compares two quantities with different units, like miles and hours. A unit rate is the rate rewritten so the second quantity is exactly 1. Nothing about that definition changes when fractions show up.

Suppose a snail crawls

\frac{1}{2}

foot in

\frac{1}{4}

hour. The rate is

\frac{1}{2}

foot per

\frac{1}{4}

hour. To reach a unit rate you need feet per one hour. Since

\frac{1}{4}

hour is one quarter of an hour, and there are 4 quarter-hours in an hour, the snail travels 4 times as far in a full hour:

4 \times \frac{1}{2} = 2

feet per hour.

That reasoning works, but it only works easily when one number is a unit fraction. The general tool is division. Every unit rate is a quotient:

\text{unit rate} = \frac{\text{first quantity}}{\text{second quantity}}

So the snail's rate is

\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2

feet per hour. Same answer, and this method never breaks down.

A ratio of fractions written as a single fraction, like

\frac{\frac{1}{2}}{\frac{1}{4}}

, is called a complex fraction. It looks intimidating, but the main fraction bar simply means "divide." Reading it out loud as "one half divided by one fourth" turns a scary-looking expression into a problem you already know how to do.

One useful check: if the second quantity is less than 1, the unit rate will be larger than the first quantity, because you are scaling up to a full unit.

Dividing Fractions to Get the Rate

The reliable procedure has four steps. Write the two quantities as a division problem with the quantity you want "per one" in the divisor position. Change any mixed numbers to improper fractions. Multiply by the reciprocal of the divisor. Simplify and attach the unit label.

Try it: a recipe uses

2\frac{1}{4}

cups of flour for

1\frac{1}{2}

loaves of bread. How many cups per loaf?

2\frac{1}{4} \div 1\frac{1}{2} = \frac{9}{4} \div \frac{3}{2} = \frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = \frac{3}{2}

So the rate is

1\frac{1}{2}

cups per loaf. Notice the mixed numbers had to become

\frac{9}{4}

and

\frac{3}{2}

first — you cannot flip a mixed number.

Decimals are welcome too, but fractions usually stay exact. If you converted

\frac{2}{3}

0.67

early on, your answer drifts off. Keep fractions as fractions until the end.

Step	Example: $\frac{5}{6}$ mile in $\frac{2}{3}$ hour
Set up division	$\frac{5}{6} \div \frac{2}{3}$
Multiply by reciprocal	$\frac{5}{6} \times \frac{3}{2}$
Multiply across	$\frac{15}{12}$
Simplify and label	$\frac{5}{4} = 1\frac{1}{4}$ miles per hour

A second route to the same answer is scaling the ratio. Multiply both quantities by the same number until the divisor becomes 1:

\frac{5}{6} : \frac{2}{3}

, multiplied by

\frac{3}{2}

on both sides, gives

\frac{15}{12} : 1

. Both methods rest on the same idea — a rate stays the same rate when you scale both quantities equally.

Choosing Which Quantity Goes on the Bottom

Unit rates come in pairs, and the words tell you which one is wanted. The phrase "per" points at the quantity that becomes 1. In "miles per hour," hours goes on the bottom. In "hours per mile," miles goes on the bottom. These two rates are reciprocals of each other, so mixing them up produces an answer that is not just off by a little — it is upside down.

A cyclist rides

\frac{9}{10}

mile in

\frac{1}{15}

hour.

Question	Setup	Answer
Miles per hour	$\frac{9}{10} \div \frac{1}{15}$	$\frac{9}{10} \times 15 = \frac{27}{2} = 13\frac{1}{2}$ mi/h
Hours per mile	$\frac{1}{15} \div \frac{9}{10}$	$\frac{1}{15} \times \frac{10}{9} = \frac{2}{27}$ h/mi

This lesson also covers ratios of like units, where the labels cancel. If one garden plot has area

\frac{3}{8}

square meter and another has area

\frac{1}{4}

square meter, the ratio

\frac{3}{8} \div \frac{1}{4} = \frac{3}{2}

means the first plot is

1.5

times as large. There is no "per" unit left — the answer is a plain number, and that is fine. Comparing two lengths works the same way: a

\frac{2}{3}

-foot ribbon is

\frac{2}{3} \div \frac{5}{6} = \frac{4}{5}

as long as a

\frac{5}{6}

-foot ribbon.

When units differ, always write the unit label on your final answer. "

13\frac{1}{2}

" alone does not communicate anything; "

13\frac{1}{2}

miles per hour" does. Getting in the habit of labeling also catches setup mistakes, because a nonsense label like "hours per hour" signals that something got divided the wrong way.

Where Students Go Wrong, and How to Check

Four mistakes account for most wrong answers on this topic.

Flipping the wrong fraction. In

\frac{3}{4} \div \frac{2}{5}

, only the divisor

\frac{2}{5}

becomes

\frac{5}{2}

. Flipping the first fraction instead gives a completely different value. Say the rule out loud: "keep the first, flip the second, multiply."

Dividing in the wrong order. Division is not commutative, so

\frac{3}{4} \div \frac{2}{5}

and

\frac{2}{5} \div \frac{3}{4}

are different. Reread the question for the word "per" or the phrase "for each" before you set up.

Forgetting to convert mixed numbers. You cannot take the reciprocal of

1\frac{1}{2}

by flipping it. Change it to

\frac{3}{2}

first, whose reciprocal is

\frac{2}{3}

.

Losing the units. A number without a label is an incomplete answer, and the label is also your best error detector.

To check reasonableness, use estimation. In

\frac{7}{8}

mile in

\frac{1}{4}

hour, a quarter hour is a small slice of an hour, so the miles-per-hour answer must be several times

\frac{7}{8}

— around 3 or 4. Computing

\frac{7}{8} \times 4 = \frac{7}{2} = 3\frac{1}{2}

fits.

If the divisor is...	The unit rate is...
less than 1	greater than the first quantity
equal to 1	equal to the first quantity
greater than 1	less than the first quantity

This table is worth memorizing, since it catches upside-down answers instantly. In the next lessons, these unit rates become the constant of proportionality, so a clean setup habit now pays off immediately.

Key terms

Rate.: A comparison of two quantities measured in different units, such as $\frac{5}{6}$ mile per $\frac{2}{3}$ hour.
Unit rate.: A rate written so the second quantity equals 1, such as $1\frac{1}{4}$ miles per hour. It is found by dividing the first quantity by the second.
Complex fraction.: A fraction whose numerator, denominator, or both are themselves fractions, such as $\frac{\frac{3}{4}}{\frac{2}{5}}$ . The main bar means divide.
Reciprocal.: The multiplicative inverse of a number; flipping a fraction gives its reciprocal, so the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ and their product is 1.
Divisor.: The number you are dividing by — the quantity that becomes 1 in a unit rate. It is the fraction you flip.
Like units.: Two quantities measured in the same unit, such as two areas in square meters. Their ratio simplifies to a plain number with no unit label.
Scaling a ratio.: Multiplying both quantities in a ratio by the same nonzero number, which keeps the rate unchanged and can be used to make the second quantity 1.

Worked example

A landscaper spreads

\frac{7}{8}

of a cubic yard of mulch over

\frac{2}{3}

of an hour, working at a steady pace. How many cubic yards does she spread per hour? At that pace, how long does one cubic yard take?

Start by naming what "per" is asking for. "Cubic yards per hour" means hours becomes 1, so hours is the divisor.

Set up the division:

\frac{7}{8} \div \frac{2}{3}

.

Keep the first fraction, flip the divisor, and multiply:

\frac{7}{8} \times \frac{3}{2} = \frac{21}{16}

.

Simplify and label:

\frac{21}{16} = 1\frac{5}{16}

cubic yards per hour.

Check for reasonableness. The divisor

\frac{2}{3}

is less than 1, so the unit rate should be bigger than

\frac{7}{8}

. Since

1\frac{5}{16}

is bigger than

\frac{7}{8}

, the direction is right.

Now the second question. "Hours per cubic yard" flips which quantity becomes 1, so cubic yards is now the divisor:

\frac{2}{3} \div \frac{7}{8} = \frac{2}{3} \times \frac{8}{7} = \frac{16}{21}

hour per cubic yard.

Notice that

\frac{16}{21}

is the reciprocal of

\frac{21}{16}

, which is exactly what should happen — the two unit rates describe the same steady pace from opposite directions. Since

\frac{16}{21}

is a little less than 1 hour, and she spreads a little more than 1 cubic yard in an hour, the two answers agree with each other.

Practice questions

A hiker walks

\frac{5}{8}

mile in

\frac{1}{6}

hour. What is his speed in miles per hour?

$\frac{5}{48}$ mile per hour
$\frac{15}{4}$ miles per hour
$\frac{4}{15}$ mile per hour
$\frac{3}{4}$ mile per hour

Answer: $\frac{15}{4}$ miles per hour

Miles per hour means hours is the divisor:

\frac{5}{8} \div \frac{1}{6} = \frac{5}{8} \times \frac{6}{1} = \frac{30}{8} = \frac{15}{4}

, or

3\frac{3}{4}

miles per hour. The choice

\frac{5}{48}

comes from multiplying instead of dividing, and

\frac{4}{15}

is the upside-down rate (hours per mile). Since the divisor

\frac{1}{6}

is less than 1, the answer had to be larger than

\frac{5}{8}

, which rules out two choices immediately.

Rectangle A has an area of

\frac{3}{10}

square foot and Rectangle B has an area of

\frac{4}{5}

square foot. Write the ratio of Rectangle A's area to Rectangle B's area as a unit rate, and explain in words what that number means.

Answer: $\frac{3}{8}$ ; Rectangle A's area is $\frac{3}{8}$ of Rectangle B's area.

Set up the division in the order given:

\frac{3}{10} \div \frac{4}{5} = \frac{3}{10} \times \frac{5}{4} = \frac{15}{40} = \frac{3}{8}

. Because both quantities are areas in square feet, the units cancel and the answer is a plain number rather than a rate with a label. The number

\frac{3}{8}

is less than 1, which makes sense since Rectangle A is the smaller one. A good explanation says A is

\frac{3}{8}

times as large as B, not that A is

\frac{3}{8}

square feet.

Two students each buy fabric. Maya pays 9 dollars for

\frac{3}{4}

yard. Devon pays 14 dollars for

1\frac{1}{4}

yards. Who is paying less per yard, and by how much?

Answer: Devon pays less, by 0.80 dollars per yard (Maya: 12 dollars per yard; Devon: 11.20 dollars per yard).

Find each unit rate with yards as the divisor. Maya:

9 \div \frac{3}{4} = 9 \times \frac{4}{3} = 12

dollars per yard. Devon: first convert the mixed number,

1\frac{1}{4} = \frac{5}{4}

, then

14 \div \frac{5}{4} = 14 \times \frac{4}{5} = \frac{56}{5} = 11.2

dollars per yard. Comparing, 11.20 is less than 12, so Devon gets the better deal by 0.80 dollars per yard. Comparing the total prices (9 versus 14) would be misleading because they bought different amounts — that is why unit rates exist.

FAQ

Why do I flip the second fraction instead of the first?: Because dividing by a number and multiplying by its reciprocal give the same result, and only the divisor — the number you are dividing by — gets replaced. Dividing $\frac{3}{4}$ by $\frac{1}{2}$ asks how many halves fit in $\frac{3}{4}$ , and the answer, $\frac{3}{4} \times 2 = \frac{3}{2}$ , comes from flipping the $\frac{1}{2}$ . Flipping the first fraction changes the quantity you started with, which changes the problem entirely.
How do I know which number goes on the bottom?: Look for the word "per," "each," or "for one." Whatever comes right after it becomes 1 and goes in the divisor spot. "Cups per serving" puts servings on the bottom. "Cost for each pound" puts pounds on the bottom. If the question just says "find the unit rate," the standard order is the first quantity mentioned divided by the second.
Can a unit rate be a fraction or a mixed number?: Yes. Unit rates are often not whole numbers. A pace of $\frac{2}{27}$ hour per mile or $1\frac{5}{16}$ cubic yards per hour is perfectly valid. The word "unit" refers to the second quantity being 1, not to the answer being a whole number.
Should I turn the fractions into decimals first?: You can when the fractions convert cleanly, like $\frac{1}{4} = 0.25$ , and a decimal answer is often easier to compare with prices. But fractions like $\frac{2}{3}$ become repeating decimals, and rounding early makes the final answer inexact. Divide with fractions to stay exact, then convert to a decimal at the end if the situation calls for it.

Learn this with a teacher, not a page

The Crimsora tutor teaches Unit Rates with Fractions live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.

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