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Divide a fraction by a fraction, understand why the reciprocal works, and check by multiplying.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you divide a fraction by a fraction, and whole numbers and mixed numbers by fractions. You read a division as 'how many fit?' or 'how much for one whole?', see why multiplying by the reciprocal gives the same answer as counting pieces, predict whether the answer will be bigger or smaller, and check every quotient by multiplying it back.
You can multiply two fractions by multiplying the tops together and the bottoms together: $\tfrac{2}{3} \times \tfrac{4}{5} = \tfrac{8}{15}$. In grade 5 you divided a whole number by a unit fraction, such as $3 \div \tfrac{1}{4} = 12$, because four quarters fit in each of the 3 wholes, and a unit fraction by a whole number, such as $\tfrac{1}{3} \div 2 = \tfrac{1}{6}$, because half of a third is a sixth. You know that multiplication and division undo each other: $20 \div 4 = 5$ because $5 \times 4 = 20$. And you can change a mixed number such as $2\tfrac{3}{4}$ into the fraction $\tfrac{11}{4}$. This lesson joins these ideas into one method that divides any fraction by any fraction.
| Term | What it means |
|---|---|
| Dividend | The number being divided: the 6 in $6 \div \tfrac{3}{4}$. |
| Divisor | The number you divide by: the $\tfrac{3}{4}$ in $6 \div \tfrac{3}{4}$. |
| Quotient | The answer to a division: $6 \div \tfrac{3}{4} = 8$, so 8. |
| Reciprocal | A fraction turned upside down. The reciprocal of $\tfrac{3}{4}$ is $\tfrac{4}{3}$, and a number times its reciprocal is always 1. |
| Unit fraction | A fraction with a top of 1, such as $\tfrac{1}{5}$: one piece of a whole cut into equal parts. |
| Mixed number | A whole number and a fraction written together, such as $3\tfrac{1}{2}$, which is $\tfrac{7}{2}$. |
| Lowest terms | A fraction whose top and bottom share no factor except 1, such as $\tfrac{5}{8}$. |
Every division asks a question. $12 \div 3$ asks how many 3s fit into 12, and the answer is 4. Fractions ask the same question. $\tfrac{3}{4} \div \tfrac{1}{8}$ asks how many eighths fit into three quarters. Cut each quarter into two eighths and three quarters become six eighths, so the answer is 6.
Counting pieces like that always works, but it is slow when the pieces do not line up. So mathematicians found a shortcut: to divide by a fraction, multiply by its reciprocal. The reciprocal is the fraction turned upside down. For example,
$$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6,$$
the same 6 we counted. People remember the method as keep, flip, multiply: keep the first fraction, flip the second, and multiply.
One result surprises almost everyone the first time. Dividing by a fraction smaller than 1 gives an answer bigger than the number you started with. That is not a mistake. When the pieces are small, many of them fit. Six eighths fit into three quarters, and 6 is much more than $\tfrac{3}{4}$. Division makes a number smaller only when the divisor is bigger than 1.
Every division can be checked by multiplying. If $\tfrac{3}{4} \div \tfrac{1}{8} = 6$, then $6 \times \tfrac{1}{8}$ must be $\tfrac{3}{4}$, and it is: six eighths.
Another way: picture
Draw a strip and shade three quarters of it. Now draw lines that cut the whole strip into eighths. Count the eighths inside the shading: there are 6. The picture shows the division as a count of small pieces inside a bigger amount, and it shows why the count is larger than the amount.
Another way: story
You have 2 cups of rice and a scoop that holds $\tfrac{1}{3}$ cup. Each cup takes 3 scoops, so 2 cups take 6 scoops: $2 \div \tfrac{1}{3} = 6$. Notice that dividing by $\tfrac{1}{3}$ gave the same answer as multiplying by 3, and 3 is $\tfrac{1}{3}$ flipped.
A division like $a \div b$ can be read in two ways, and both matter for fractions.
How many groups? You know the size of each group and want to know how many groups there are. 'A ribbon is $4\tfrac{1}{2}$ feet long. How many pieces $\tfrac{3}{4}$ foot long can be cut from it?' Here $4\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{9}{2} \times \tfrac{4}{3} = \tfrac{36}{6} = 6$ pieces.
How much in one group? You know how much fills part of a group and want the amount for one whole group. 'Two thirds of a bag of soil weighs 10 pounds. How much does the whole bag weigh?' Here $10 \div \tfrac{2}{3} = 10 \times \tfrac{3}{2} = 15$ pounds. The bag is heavier than 10 pounds, which makes sense because 10 pounds is only part of it.
Both questions use the same arithmetic. The skill is spotting that a story is a division at all. Look for these clues: 'how many ... fit', 'how many ... can be made', 'how much for one whole', 'how far in one hour', and 'what fraction of ... is'. When the story gives a part and asks for the whole, you divide by the fraction. When it gives the whole and asks for a part, you multiply by it.
A rule you cannot explain is easy to misuse, so here is why this one works. Start with two fractions that share a bottom. $\tfrac{9}{10} \div \tfrac{3}{10}$ asks how many 3-tenths fit into 9-tenths. Tenths are just the unit here, like apples, so the answer is $9 \div 3 = 3$. When the bottoms match, divide the tops.
Any two fractions can be given the same bottom. Take $\tfrac{2}{3} \div \tfrac{4}{5}$. Using fifteenths, $\tfrac{2}{3} = \tfrac{10}{15}$ and $\tfrac{4}{5} = \tfrac{12}{15}$, so the answer is $10 \div 12 = \tfrac{10}{12} = \tfrac{5}{6}$.
Now look at the multiplication $\tfrac{2}{3} \times \tfrac{5}{4} = \tfrac{10}{12}$. It gives the same $\tfrac{10}{12}$. That is no accident: the 10 is $2 \times 5$ and the 12 is $3 \times 4$ in both methods. Flipping the divisor and multiplying is a quick way of doing the common-bottom method without writing the common bottom.
There is a second reason too. Division undoes multiplication. The quotient must be the number that, multiplied by the divisor, gives the dividend. Check it: $\tfrac{5}{6} \times \tfrac{4}{5} = \tfrac{20}{30} = \tfrac{2}{3}$. It works, so $\tfrac{5}{6}$ is the quotient.
Keep, flip, multiply works only on fractions, so turn everything into a fraction first.
Never flip a mixed number piece by piece. The reciprocal of $2\tfrac{3}{4}$ is not $2\tfrac{4}{3}$; it is $\tfrac{4}{11}$, the reciprocal of the single fraction $\tfrac{11}{4}$.
When the answer is a fraction with a big top, such as $\tfrac{23}{6}$, you can leave it as it is or change it back to a mixed number: $23 \div 6 = 3$ remainder 5, so $3\tfrac{5}{6}$. In a story, the mixed number is usually easier to picture: $3\tfrac{5}{6}$ pieces means 3 whole pieces and most of a fourth.
Before you calculate, predict. A prediction catches most mistakes, and it takes a few seconds.
| Divisor | Quotient compared with the dividend | Example |
|---|---|---|
| greater than 1 | smaller | $6 \div \tfrac{3}{2} = 4$ |
| equal to 1 | the same | $6 \div \tfrac{5}{5} = 6$ |
| less than 1 | bigger | $6 \div \tfrac{2}{3} = 9$ |
You can also estimate with friendly numbers. $5\tfrac{7}{8} \div 1\tfrac{1}{10}$ is close to $6 \div 1 = 6$, a little less because the divisor is a little more than 1. If your exact answer is 60 or 0.6, you know at once to look for the slip. Another quick test: dividing by $\tfrac{1}{2}$ always doubles, and dividing by $\tfrac{1}{4}$ always multiplies by 4.
To divide by a fraction:
To check:
In the United States a half-gallon carton of milk holds 64 fluid ounces, which is 8 cups, because a cup is 8 fluid ounces. Suppose a family pours $\tfrac{3}{4}$-cup glasses for the children. How many glasses does one carton fill?
$$8 \div \frac{3}{4} = 8 \times \frac{4}{3} = \frac{32}{3} = 10\tfrac{2}{3}.$$
So a carton fills 10 full glasses, with $\tfrac{2}{3}$ of a glass left, which is $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{1}{2}$ cup. If the children get bigger glasses of $1\tfrac{1}{4}$ cups, the carton fills only $8 \div \tfrac{5}{4} = \tfrac{32}{5} = 6\tfrac{2}{5}$ glasses. Bigger servings mean fewer of them, exactly as dividing by a bigger number should give a smaller answer. Shoppers use this every week to decide how many cartons to buy.
A standard outdoor running track is 400 meters around the inside lane, which is very close to $\tfrac{1}{4}$ mile, so coaches often call four laps a mile. A runner training for a 5K wants to run $2\tfrac{1}{2}$ miles on the track. How many laps is that?
$$2\tfrac{1}{2} \div \frac{1}{4} = \frac{5}{2} \times \frac{4}{1} = \frac{20}{2} = 10 \text{ laps}.$$
On an indoor track the lap is shorter, often $\tfrac{1}{8}$ mile (about 200 meters). The same $2\tfrac{1}{2}$ miles is then $\tfrac{5}{2} \times 8 = 20$ laps, twice as many, because the lap is half as long. Counting laps is division by a fraction, and coaches do it in their heads by multiplying by the reciprocal: four laps a mile outdoors, eight indoors.
Flipping the wrong fraction. Only the divisor, the second fraction, is turned over. $\tfrac{3}{4} \div \tfrac{2}{5}$ is $\tfrac{3}{4} \times \tfrac{5}{2}$, not $\tfrac{4}{3} \times \tfrac{5}{2}$ and not $\tfrac{4}{3} \times \tfrac{2}{5}$.
Expecting division to shrink. $4 \div \tfrac{1}{2}$ is 8, not 2. Dividing by a half is not the same as dividing in half; dividing in half is dividing by 2.
Flipping a mixed number in pieces. Change $1\tfrac{1}{2}$ to $\tfrac{3}{2}$ first; its reciprocal is $\tfrac{2}{3}$.
Dividing tops and bottoms straight across. $\tfrac{6}{8} \div \tfrac{3}{4} = \tfrac{2}{2}$ happens to work, but $\tfrac{2}{3} \div \tfrac{4}{5}$ straight across gives $\tfrac{0.5}{0.6}$, a mess. Keep, flip, multiply always works.
Mixing up the two stories. If a part is known and the whole is wanted, divide by the fraction; if the whole is known and a part is wanted, multiply.
Divide $\tfrac{5}{6}$ by $\tfrac{2}{9}$. First predict the size of the answer.
$\tfrac{2}{9} < 1 \;\Rightarrow\; \text{answer} > \tfrac{5}{6}$
A divisor smaller than one whole fits many times.
Write the reciprocal of the divisor.
$\tfrac{2}{9} \to \tfrac{9}{2}$
Only the second fraction is turned upside down.
Multiply the first fraction by that reciprocal.
$\tfrac{5}{6} \times \tfrac{9}{2} = \tfrac{45}{12}$
Dividing by a number is multiplying by its reciprocal.
Simplify by dividing top and bottom by 3.
$\tfrac{45}{12} = \tfrac{15}{4} = 3\tfrac{3}{4}$
3 is the greatest common factor of 45 and 12; then 15 divided by 4 is 3 remainder 3.
Check by multiplying the quotient by the divisor.
$\tfrac{15}{4} \times \tfrac{2}{9} = \tfrac{30}{36} = \tfrac{5}{6}$
Multiplying back gives the dividend, so the quotient is right.
A rope is $3\tfrac{1}{3}$ yards long. How many pieces $1\tfrac{1}{4}$ yards long can be cut from it? Write the division.
$3\tfrac{1}{3} \div 1\tfrac{1}{4}$
The question asks how many pieces of one size fit into a longer length.
Change the dividend to a single fraction.
$3\tfrac{1}{3} = \tfrac{3 \times 3 + 1}{3} = \tfrac{10}{3}$
Three wholes are nine thirds, and one more third makes ten.
Change the divisor to a single fraction.
$1\tfrac{1}{4} = \tfrac{1 \times 4 + 1}{4} = \tfrac{5}{4}$
A mixed number cannot be flipped until it is one fraction.
Multiply by the reciprocal of the divisor.
$\tfrac{10}{3} \times \tfrac{4}{5} = \tfrac{40}{15}$
Keep the first fraction, flip the second, multiply.
Simplify and write as a mixed number.
$\tfrac{40}{15} = \tfrac{8}{3} = 2\tfrac{2}{3}$
Divide top and bottom by 5; then 8 thirds is 2 wholes and 2 thirds.
Answer the question in words.
$2 \text{ whole pieces, and } \tfrac{2}{3} \text{ of a piece left over}$
Only whole pieces are useful here, and the leftover is two thirds of $1\tfrac{1}{4}$ yards, which is $\tfrac{5}{6}$ yard.
Three quarters of a pound of cheese fills $\tfrac{2}{3}$ of a container. How many pounds fill the whole container? Decide what kind of question this is.
$\text{a part is known, the whole is wanted}$
The cheese fills only part of the container, so the whole container holds more.
Write the division.
$\tfrac{3}{4} \div \tfrac{2}{3}$
The amount for one whole is the amount divided by the fraction it fills.
Turn the divisor upside down.
$\tfrac{2}{3} \to \tfrac{3}{2}$
The reciprocal of two thirds is three halves.
Multiply the two fractions.
$\tfrac{3}{4} \times \tfrac{3}{2} = \tfrac{9}{8}$
Tops: 3 times 3; bottoms: 4 times 2.
Write the answer as a mixed number.
$\tfrac{9}{8} = 1\tfrac{1}{8} \text{ pounds}$
Eight eighths make one whole, with one eighth left over.
Check by taking two thirds of the answer.
$\tfrac{9}{8} \times \tfrac{2}{3} = \tfrac{18}{24} = \tfrac{3}{4}$
Two thirds of a full container must be the three quarters of a pound we started with.
Check that the size makes sense.
$1\tfrac{1}{8} > \tfrac{3}{4}$
A full container must hold more than a part of it.
Change the mixed number to a fraction.
$2\tfrac{2}{5} = \tfrac{12}{5}$
Two wholes are ten fifths, plus two more.
Multiply by the reciprocal of the lap.
$\tfrac{12}{5} \times \tfrac{5}{3} = \tfrac{60}{15}$
Keep, flip, multiply.
Simplify the product.
$10 \div \dfrac{2}{5} = 25$, which is larger than 10. Why does dividing by this fraction give a larger answer?
A baker has $2\tfrac{1}{7}$ cups of frosting and puts $\dfrac{3}{7}$ cup on each cake. How many cakes can she frost? Complete the worked solution.
Write the mixed number as a fraction: whole number times the bottom, plus the top.
$2 \times 7 + 1 =$ u pieces of size $\dfrac{1}{7}$ cup
Each whole cup holds 7 pieces of size $\dfrac{1}{7}$, and the fraction part adds 1 more.
Turn the amount per cake upside down.
$\dfrac{3}{7} \to \dfrac{7}{3}$
Dividing by a fraction means multiplying by its reciprocal.
Multiply the top of the frosting fraction by 7, the top of the reciprocal.
$\text{top} \times 7 =$ t, over $7 \times 3 = 21$
Tops multiply with tops, and bottoms with bottoms.
Divide the top of the product by its bottom.
$\text{top} \div 21 =$ q cakes
A fraction bar means divide, and here the division comes out exactly.
Check by multiplying back.
$\text{cakes} \times \dfrac{3}{7} = \text{all the frosting}$
The quotient times the divisor must give back the dividend.
Divide $\dfrac{1}{8} \div \dfrac{5}{8}$. Write the product before simplifying, then the quotient in lowest terms.
Before simplifying the product is top/bot, and in lowest terms the quotient is lt/lb.
A bag holds 10 cups of oatmeal. One serving is $\dfrac{2}{5}$ cup. How many servings are in the bag?
Before dividing, the product is top/2, so the bag holds answer servings.
Complete the table. Write each quotient as a fraction in lowest terms, giving its top and its bottom. (A whole number has a bottom of 1.)
| Top | Bottom | |
|---|---|---|
| 4 ÷ 1/3 | ||
| 1/3 ÷ 4 | ||
| 7/10 ÷ 7/3 |
After a storm, 8 gallons of water fill $\dfrac{2}{3}$ of a rain barrel. How many gallons does the full barrel hold, and how many more gallons would fill it to the top?
The full barrel holds full gallons, and more more gallons would fill it.
A box turtle moves at a steady speed and covers $\dfrac{1}{6}$ of a mile in $\dfrac{2}{7}$ of an hour. At that speed, how many miles would it travel in one whole hour? Give the answer as a fraction.
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
After a storm, 10 gallons of water fill $\dfrac{2}{9}$ of a rain barrel. How many gallons does the full barrel hold, and how many more gallons would fill it to the top?
The full barrel holds full gallons, and more more gallons would fill it.
You can divide fractions and mixed numbers and check the answer. Without looking: why is $4 \div \tfrac{1}{2}$ equal to 8, and how would you check it?
16. Your turn: how many $\tfrac{3}{5}$-mile laps make $2\tfrac{2}{5}$ miles?, step 3
$\tfrac{60}{15} = 4 \text{ laps}$
15 goes into 60 four times.