Dividing Fractions

Study Sheet

Dividing Fractions

Multiply by the reciprocal

Why Flip and Multiply

Concept
The Big Idea
Six wholes cut into halves: 12 halves fit in 6.

Dividing asks “how many of these fit?” Dividing by cd\dfrac{c}{d} is the same as multiplying by dc\dfrac{d}{c}, because that undoes the division.

Dividing asks "how many of these fit into that?" 6÷12=126 \div \tfrac12 = 12 because twelve halves make six. The shortcut — multiply by the reciprocal, flipping the second fraction — always gives the same answer as this counting, which is why it works even when the numbers are ugly.

Example
Finding 12÷14\dfrac{1}{2} \div \dfrac{1}{4}
12two ¼
Two quarters (heavy and light shading) fill one half.

Flip and multiply: 12×41=2\dfrac{1}{2} \times \dfrac{4}{1} = 2. And indeed two quarters fit inside a half.

Example
Finding 34÷18\tfrac34 \div \tfrac18
3468
Three quarters cover exactly six eighths.

How many eighths fit in three quarters? 34=68\tfrac34 = \tfrac68, so six eighths fit: 34÷18=6\tfrac34 \div \tfrac18 = 6. Flip and multiply agrees: 34×81=244=6\tfrac34 \times \tfrac81 = \tfrac{24}{4} = 6.

Side note
Where it goes wrong

Flipping the first fraction instead of the second, or flipping both. Only the divisor (the number you divide BY) is inverted.

Try it
Try it: how many fit?

Find 2÷132 \div \tfrac13 and 56÷16\tfrac56 \div \tfrac16. Explain each as "how many fit?".

Show the answer

Each whole holds 33 thirds, so 22 wholes hold 6\mathbf{6}: 2×31=62 \times \tfrac31 = 6. Five sixths contain 5\mathbf{5} sixths: 56×61=306=5\tfrac56 \times \tfrac61 = \tfrac{30}{6} = 5.

In Plain Terms & More Examples

Concept
In plain terms

To divide by a fraction, flip it (its reciprocal) and multiply. Dividing asks how many of the second fraction fit inside the first.

Example
Finding 34÷12\dfrac{3}{4} \div \dfrac{1}{2}
34halves
One whole half (heavy shading) and half of another (light shading): 1 1/2 halves.

Flip and multiply: 34×21=64=32\dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4} = \dfrac{3}{2}.

Example
Finding 23÷16\dfrac{2}{3} \div \dfrac{1}{6}

Flip and multiply: 23×61=123=4\dfrac{2}{3} \times \dfrac{6}{1} = \dfrac{12}{3} = 4.

Side note
Where it goes wrong

Expecting division to make things smaller. 34÷12=32\tfrac34 \div \tfrac12 = \tfrac32 is bigger than 34\tfrac34, and that is right: you are counting how many halves fit, and a small piece fits more than once. Dividing by a number less than 11 always gives more than you started with.

Try it
Try it: sevenths of quarters

Find 78÷14\dfrac78 \div \dfrac14.

Show the answer

Flip and multiply: 78×41=288=72=312\dfrac78 \times \dfrac41 = \dfrac{28}{8} = \dfrac72 = \mathbf{3\tfrac12}. Check: 14=28\tfrac14 = \tfrac28, and 28\tfrac28 fits into 78\tfrac78 three and a half times.

Mixed Numbers and Whole Numbers

Concept
Convert, flip the second, multiply

Change mixed numbers to improper fractions, keep the first, flip the divisor, multiply. A whole number divisor 44 is 41\tfrac41, so dividing by 44 is multiplying by 14\tfrac14.

Example
Finding 212÷342\tfrac12 \div \tfrac34
21212313
Groups of three quarters: 3 whole groups and 1/3 of a group.

52×43=206=103=313\tfrac52\times\tfrac43 = \tfrac{20}{6} = \tfrac{10}{3} = 3\tfrac13. Sense check: 34\tfrac34 fits into 2122\tfrac12 a bit more than 33 times. ✓

Example
Dividing by a whole number: 23÷4\tfrac23 \div 4
234 shares
Two thirds is 8 twelfths; one share (heavy shading) is 2 twelfths = 1/6.

Share 23\tfrac23 equally among 44. Dividing by 44 is multiplying by 14\tfrac14: 23×14=212=16\tfrac23 \times \tfrac14 = \tfrac{2}{12} = \tfrac16. Each share is 16\tfrac16.

Side note
Where it goes wrong

Flipping a mixed number piece by piece. The reciprocal of 1121\tfrac12 is not "1211\tfrac21". Convert first: 112=321\tfrac12 = \tfrac32, whose reciprocal is 23\tfrac23. Check: 32×23=1\tfrac32 \times \tfrac23 = 1.

Try it
Try it: two ways

Compute 56÷5\tfrac{5}{6} \div 5 and 6÷236 \div \tfrac{2}{3}.

Show the answer

56×15=16\tfrac56\times\tfrac15 = \mathbf{\tfrac16}. 6×32=96\times\tfrac32 = \mathbf{9} (nine two-thirds make six).

Word Problems

Concept
Two questions division answers

Division can ask how many groups? ("how many 12\tfrac12-cup servings are in 3123\tfrac12 cups?") or how much in one group? ("if 23\tfrac23 of a tank is 1212 gallons, how much is a whole tank?"). Both are answered by dividing.

In the first kind you know the size of a group and want the number of groups; in the second you know how much fills part of a group and want one whole group. The same fraction division answers both, and a quick picture of the story shows what to divide by what.

Example
Servings
3 1/2 cups is 7 half-cups.

A 3123\tfrac12-cup bag of rice makes servings of 12\tfrac12 cup: 72÷12=7\tfrac72 \div \tfrac12 = 7 servings.

Example
A whole tank
66612 gallons
Two of three equal parts hold 12, so each part is 6.

23\tfrac23 of a tank holds 1212 gallons. How much does the whole tank hold? 12÷23=12×32=1812 \div \tfrac23 = 12 \times \tfrac32 = 18 gallons. In the picture: two thirds are 1212, so one third is 66, and three thirds are 1818.

Side note
Where it goes wrong

Dividing in the wrong order. "How many 12\tfrac12-cup servings are in 3123\tfrac12 cups?" is 312÷12=73\tfrac12 \div \tfrac12 = 7. Writing 12÷312=17\tfrac12 \div 3\tfrac12 = \tfrac17 answers a different question: what part of the bag one serving is.

Try it
Try it: the board

A board 4124\tfrac12 feet long is cut into pieces 34\tfrac34 foot long. How many pieces?

Show the answer

92×43=366=6\tfrac92\times\tfrac43 = \tfrac{36}{6} = \mathbf{6} pieces.

Common Denominators Also Work

Concept
Same pieces, divide the counts

When two fractions have the same denominator, dividing them is dividing the tops: 68÷18=6÷1=6\tfrac68 \div \tfrac18 = 6 \div 1 = 6. So any two fractions can be divided by first giving them a common denominator.

With equal denominators the pieces are the same size, so "how many of these pieces fit into those?" is just "how many 11s fit into 66?". This method also explains flip-and-multiply: the two always give the same answer.

Example
Finding 56÷112\tfrac56 \div \tfrac{1}{12}

Twelfths: 56=1012\tfrac56 = \tfrac{10}{12}, so the question is how many 112\tfrac{1}{12} fit into 1012\tfrac{10}{12}: 10÷1=1010 \div 1 = 10.

Example
Finding 23÷34\tfrac23 \div \tfrac34
2334
8 twelfths against 9 twelfths: 8/9 of a fit.

Twelfths: 812÷912\tfrac{8}{12} \div \tfrac{9}{12}. Divide the tops: 8÷9=898 \div 9 = \tfrac89. The answer is less than 11 because 34\tfrac34 is bigger than 23\tfrac23: it does not fit even once.

Example
Checking with flip-and-multiply

23÷34=23×43=89\tfrac23 \div \tfrac34 = \tfrac23 \times \tfrac43 = \tfrac89. The same answer, as it must be.

Side note
Where it goes wrong

Keeping the common denominator in the answer. 68÷18\tfrac68 \div \tfrac18 is 66, not 68\tfrac68. The answer counts how many pieces fit, so here it is a plain number.

Try it
Try it: sixths and thirds

Find 56÷13\dfrac56 \div \dfrac13 using a common denominator.

Show the answer

13=26\dfrac13 = \dfrac26, so 56÷26=5÷2=212\dfrac56 \div \dfrac26 = 5 \div 2 = \mathbf{2\tfrac12}. Flip and multiply agrees: 56×31=156=212\dfrac56 \times \dfrac31 = \dfrac{15}{6} = 2\tfrac12.

Why It Works

Dividing by ab\tfrac ab must undo multiplying by ab\tfrac ab. Multiplying by ba\tfrac ba does exactly that, since ab×ba=1\tfrac ab\times\tfrac ba = 1. So x÷ab=x×bax \div \tfrac ab = x\times\tfrac ba — the reciprocal rule is just "division is the inverse of multiplication" written for fractions. Check with 6÷12=6×21=126 \div \tfrac12 = 6\times\tfrac21 = 12 ✓.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.