🍰 Fraction Isles · Fractions & Proportion
Multiplying & Dividing Fractions
Multiply fractions directly and divide by multiplying by the reciprocal, understanding why the rules work.
In short
- To multiply, multiply the numerators and multiply the denominators — no common denominator is ever needed.
- Reading "x" as "of" explains the rule, and the area model shows why the denominators multiply.
- Common factors may be cancelled between any numerator and any denominator before multiplying, but never across a plus or minus.
- Dividing by a fraction means multiplying by its reciprocal: keep, change, flip — and only the divisor is flipped.
- Multiplying by less than 1 shrinks a number; dividing by less than 1 grows it. Use this to check every answer.
Multiplying means "of"
Multiplication of fractions is easier than addition: no common denominator is needed at all.
Multiply the numerators, multiply the denominators.
2/3 x 4/5 = (2 x 4)/(3 x 5) = 8/15
The reason is worth seeing. Read "x" as the word of: 2/3 x 4/5 means "two thirds of four fifths". Draw a square, shade 4/5 of it going across, then shade 2/3 of that going down. The overlap is 8 small rectangles out of 15 — an area model of the rule.
Multiplying by a whole number is the same rule with the whole number written over 1:
3/8 x 5 = 3/8 x 5/1 = 15/8 = 1 7/8
Only the numerator ends up multiplied, which makes sense: five copies of a piece is five times as many pieces of the same size.
Cancel before you multiply
You can divide out a common factor between any numerator and any denominator before multiplying, because multiplication mixes them all together anyway.
9/10 x 5/12: the 5 and the 10 share 5, and the 9 and the 12 share 3.
9/10 x 5/12 = (3 x 1)/(2 x 4) = 3/8
Cancelling first keeps the numbers small and means less simplifying at the end. Doing it afterwards gives the same answer: 45/120 = 3/8.
Be careful about what cancelling is allowed. It is a shortcut for simplifying a single product, so it works across a multiplication sign — never across a plus or minus.
Another useful prediction: multiplying by a fraction less than 1 makes a number smaller, and multiplying by a fraction greater than 1 makes it larger. If 2/3 x 4/5 came out bigger than 4/5, something went wrong.
Dividing: how many fit inside?
"6 / (3/4)" asks: how many three-quarters fit inside 6? Each whole holds 4 quarters, so it holds 4/3 of a three-quarter measure, and 6 wholes hold 8 of them.
The rule that gets there is: keep, change, flip. Keep the first fraction, change the division to a multiplication, and flip the second — replace the divisor by its reciprocal.
6 / (3/4) = 6 x 4/3 = 24/3 = 8
The reciprocal of a/b is b/a, because a/b x b/a = 1. Dividing by a number and multiplying by its reciprocal are the same operation, exactly as dividing by 2 is the same as multiplying by 1/2.
Only the divisor is flipped. Flipping the first fraction gives the reciprocal of the right answer.
Predict the size again: dividing by a number less than 1 gives an answer larger than you started with, because small measures means you need lots of them.
Mixed numbers and word problems
Mixed numbers cannot be multiplied part by part. 2 1/2 x 3 is not 6 1/2 — it is 15/2, which is 7 1/2.
Always convert to improper fractions first, then multiply or divide, then convert back if a mixed number is wanted.
2 1/2 x 1 1/3 = 5/2 x 4/3 = 20/6 = 10/3 = 3 1/3
In word problems, look for the phrasing that names the operation:
- "of" a quantity — multiply. Two thirds of 24 guards is 2/3 x 24 = 16.
- "how many ... fit into", "how many portions", "shared into pieces of size" — divide.
- "each ... uses ..., how much for n of them" — multiply.
Whichever you choose, check the direction with the size rule: did the answer move the way it should have?
Worked examples
Example 1
8/9 x 3/4 = ?
- No common denominator is needed — this is multiplication, so numerators go with numerators.
- Cancel first: 3 and 9 share 3, leaving 1 and 3; 8 and 4 share 4, leaving 2 and 1.
- The product becomes (2 x 1)/(3 x 1) = 2/3.
- Check the long way: 8 x 3 = 24 and 9 x 4 = 36, and 24/36 = 2/3.
- Size check: 3/4 is less than 1, so the answer should be smaller than 8/9 — and 2/3 is.
Example 2
(5/6) / (2/9) = ?
- Keep the first fraction, change the sign to multiplication, and flip the divisor: 5/6 x 9/2.
- Cancel across the multiplication: 9 and 6 share 3, leaving 3 and 2.
- Now multiply: (5 x 3)/(2 x 2) = 15/4.
- 15/4 = 3 3/4.
- Size check: 2/9 is much smaller than 5/6, so plenty of them fit inside — an answer above 1 is right.
Example 3
A cauldron needs 6 cups of moonwell water, and the only measure holds 3/4 of a cup. How many measures are needed?
- "How many 3/4-cup measures fit into 6 cups" is a division: 6 / (3/4).
- Flip the divisor and multiply: 6 x 4/3.
- 6 x 4 = 24 and 24 / 3 = 8.
- Eight measures are needed.
- Check: 8 x 3/4 = 24/4 = 6 cups, exactly the amount required.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 55 x 1/5 = ?
Answer: 1
- Write 5 as 5/1: 1/5 x 5/1.
- Multiply the numerators and the denominators: 1 x 5 = 5, and 5 x 1 = 5.
- = 5/5
- Simplify: 1.
Problem 2
Difficulty 3 of 51/10 x 1/2 = ?
Answer: 1/20
- Numerators: 1 x 1 = 1.
- Denominators: 10 x 2 = 20.
- 1/10 x 1/2 = 1/20
- 1 and 20 share no factor, so 1/20 is the answer.
Problem 3
Difficulty 4 of 511/12 / 1/6 = ?
Answer: 5 1/2
- Keep the first fraction, change the division to multiplication, and flip the second: 11/12 x 6/1.
- Numerators: 11 x 6 = 66. Denominators: 12 x 1 = 12.
- = 66/12
- 11/12 / 1/6 = 5 1/2.
Common mistakes
- Multiplying straight across when the question is a division, instead of flipping the divisor first.
- Flipping the first fraction rather than the divisor, which gives the reciprocal of the right answer.
- Looking for a common denominator before multiplying, which is needed for addition but not here.
- Multiplying a fraction by a whole number and multiplying the denominator too, so the fraction never changes.
- Multiplying mixed numbers part by part instead of converting them to improper fractions first.
What you should be able to do
- Multiply fractions and mixed numbers, simplifying before multiplying.
- Divide fractions using the reciprocal and explain why it works.
- Interpret "of" as multiplication in fraction word problems.
- Predict whether a product will be larger or smaller than the starting number.
Where this fits in the curriculum
Common Core
- 5.NF.B.4
Grade 5 — Multiply a fraction by a whole number or by another fraction.
- 5.NF.B.5
Grade 5 — Interpret multiplication as scaling: predict whether a product is larger or smaller than the starting number.
- 6.NS.A.1
Grade 6 — Divide a fraction by a fraction, and explain why multiplying by the reciprocal works.