How do you multiply fractions?

To multiply fractions, multiply the numerators to get the new numerator, multiply the denominators to get the new denominator, then simplify. For example, 2/3 × 3/4 = 6/12, which simplifies to 1/2. You do not need a common denominator. Write a whole number over 1, and turn a mixed number into an improper fraction before you multiply.

How do you multiply fractions with whole numbers?

A fraction is multiplied by a whole number by writing the whole number over 1 and multiplying straight across. For 3 × 2/5, write 3/1 × 2/5 = 6/5, which is 1 1/5. The whole number multiplies only the numerator, because 3 × 2/5 means three copies of 2/5, and 2/5 + 2/5 + 2/5 = 6/5.

How do you multiply mixed fractions?

Mixed fractions are multiplied by turning each one into an improper fraction, multiplying straight across, then turning the answer back into a mixed number. For 2 1/2 × 1 1/3, write 5/2 × 4/3 = 20/6 = 10/3, which is 3 1/3. Multiplying the whole parts and the fraction parts separately gives the wrong answer.

Do you need a common denominator to multiply fractions?

A common denominator is not needed to multiply fractions; it is needed only to add or subtract them. Multiply the numerators and the denominators as they are, so 2/5 × 3/7 = 6/35. Finding a common denominator first gives the same answer after simplifying, but it adds work and bigger numbers.

Why does multiplying fractions make the answer smaller?

Multiplying by a fraction less than 1 makes the answer smaller because you are taking only part of the number. The product 1/2 × 3/4 means half of 3/4, which is 3/8, and 3/8 is less than 3/4. Multiplying by a fraction greater than 1, such as 3/2, makes the answer larger.

How to Multiply Fractions: Steps and Examples

Q: When do you cross multiply fractions?

Cross multiplication is used to compare two fractions or to solve a proportion, and it is a different job from multiplying fractions. To compare 3/4 and 5/7, work out 3 × 7 = 21 and 4 × 5 = 20, so 3/4 is larger. The shortcut for multiplying is cross-cancelling, which divides a numerator and the opposite denominator by a shared factor.

Three quarters of the brownie pan is left, and your child's grade 5 homework asks what half of that leftover part is. Knowing how to multiply fractions turns the question into one line: multiply the tops, multiply the bottoms, and simplify. Multiplying fractions is easier than adding them, because there is no common denominator to find, and the same three steps cover whole numbers and mixed numbers too.

What does multiplying fractions mean?

Multiplying fractions means finding a fraction of an amount, so the × sign can be read as "of". The question 1/2 × 3/4 asks for half of three quarters, and the answer is 3/8.

The Common Core standard 5.NF.B.4 asks fifth graders to "apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction". It states the rule in one line: "(In general, (a/b) × (c/d) = ac/bd.)"

Why the answer can get smaller

A product of fractions gets smaller when you multiply by a fraction less than 1, because you take only part of the amount. Standard 5.NF.B.5 asks students to explain "why multiplying a given number by a fraction less than 1 results in a product smaller than the given number".

An area model shows the rule at work. Draw a square, split it into 3 rows and 4 columns, shade 3 of the 4 columns for 3/4 and 2 of the 3 rows for 2/3. The overlap covers 2 × 3 = 6 of the 3 × 4 = 12 small boxes, so 2/3 × 3/4 = 6/12 = 1/2.

An area model for 2/3 times 3/4: a square split into 3 rows and 4 columns, with 3 columns shaded blue, 2 rows shaded orange and the 6 overlapping boxes green, showing 2/3 × 3/4 = 6/12 = 1/2.

How to multiply fractions, step by step

Multiplying fractions takes three steps (multiply the numerators, multiply the denominators, simplify) plus a set-up step for whole numbers and mixed numbers. The OpenStax Prealgebra chapter on multiplying fractions puts it the same way: "To multiply fractions, we multiply the numerators and multiply the denominators."

Write every number as a fraction. A whole number goes over 1, so 4 becomes 4/1, and a mixed number becomes an improper fraction, so 2 1/2 becomes 5/2.
Cancel common factors first, if you can. Look for a numerator and a denominator that share a factor, and divide both by it. In 4/9 × 3/8, 4 and 8 share 4, and 3 and 9 share 3.
Multiply the numerators. The product of the tops is the new top: in 2/3 × 3/4, 2 × 3 = 6.
Multiply the denominators. The product of the bottoms is the new bottom: 3 × 4 = 12, so the product is 6/12.
Simplify the answer. Divide the top and bottom by their greatest common factor: 6/12 divided by 6 is 1/2.
Turn an improper answer into a mixed number. If the top is bigger than the bottom, divide: 10/3 is 3 remainder 1, so 3 1/3.

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Cancel first: the cross-cancelling shortcut

Cross-cancelling means dividing a numerator and the opposite denominator by the same factor before you multiply, which keeps the numbers small. OpenStax calls it "removing common factors earlier".

Take 4/9 × 3/8. Divide 4 and 8 by 4 to get 1 and 2, divide 3 and 9 by 3 to get 1 and 3, and the problem becomes 1/3 × 1/2 = 1/6. Multiplying first gives 12/72, which also simplifies to 1/6 but needs a bigger common factor, and the guide to GCF and LCM shows how to spot shared factors fast.

Cross-cancelling 4/9 × 3/8: the 4 and the 8 are struck through and divided by 4, the 3 and the 9 are struck through and divided by 3, giving 1/3 × 1/2 = 1/6, with 12/72 = 1/6 as a check.

Four ways to multiply fractions compared

The four common methods for multiplying fractions give the same answer, and each fits a different moment. Children usually meet repeated addition and the area model first, then move to the straight-across rule and cancelling.

Method	How it works	Example	Best for	Limits
Straight across	Top × top, bottom × bottom, then simplify	2/3 × 3/4 = 6/12 = 1/2	Every problem	Big numbers before simplifying
Area model	Shade one fraction in rows, the other in columns, count the overlap	2/3 × 3/4: 6 of 12 boxes	Seeing why the rule works	Slow with large denominators
Cross-cancelling	Divide a top and the opposite bottom by a shared factor first	4/9 × 3/8 = 1/3 × 1/2 = 1/6	Larger numbers, fewer steps	Needs quick factor facts
Repeated addition	Add the fraction as many times as the whole number says	3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5	A whole number times a fraction	Does not work for fraction × fraction

Cross-multiplying is a different tool

Cross-multiplying compares two fractions or solves a proportion, and it is a separate tool from multiplying fractions. To compare 3/4 and 5/7, work out 3 × 7 = 21 and 4 × 5 = 20, so 3/4 is the larger fraction.

To solve 2/3 = x/12, cross-multiplying gives 3 × x = 2 × 12 = 24, so x = 8. The What Works Clearinghouse practice guide on fractions recommends that teachers "help students understand why procedures for computations with fractions make sense", which is the job the area model does before any shortcut.

How to multiply fractions with whole numbers and mixed numbers

A fraction is multiplied by a whole number by writing the whole number over 1 and multiplying straight across. Standard 4.NF.B.4 uses the example "3 × (2/5) as 6 × (1/5)", which is 6/5, and gives the rule "n × (a/b) = (n × a)/b."

The whole number multiplies only the top. The standard's own word problem asks how much roast beef 5 people need at 3/8 of a pound each: 5 × 3/8 = 15/8, which is 1 7/8 pounds, between 1 and 2.

How to multiply mixed numbers

Mixed numbers are multiplied by converting each one to an improper fraction first. Multiply the whole number by the denominator, add the numerator, and keep the denominator: 2 1/2 becomes (2 × 2 + 1)/2 = 5/2, and 1 1/3 becomes (1 × 3 + 1)/3 = 4/3.

Then multiply: 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3. Cross-cancelling the 2 and the 4 first gives 5/1 × 2/3 = 10/3, the same answer with smaller numbers.

Standard 5.NF.B.6 asks students to "solve real world problems involving multiplication of fractions and mixed numbers". A recipe is the usual case: half of a batch that needs 3/4 cup of sugar takes 1/2 × 3/4 = 3/8 cup, and the explainer on fractions for parents covers improper fractions and mixed numbers from the start.

How to check that fraction multiplication has landed

Five short questions show whether a child or an adult can multiply fractions with understanding.

The "of" meaning: Ask what 1/2 × 3/4 means in words. "Half of three quarters" is the answer to listen for, and the product is 3/8.
Cancelling: Give 3/5 × 10/21. Cancelling 5 with 10 and 3 with 21 gives 1/1 × 2/7 = 2/7.
A whole number: Ask for 4 × 5/6. The answer is 20/6 = 10/3 = 3 1/3, and an answer of 20/24 means the 4 went into the bottom too.
A mixed number: Ask for 1 3/4 × 2/7. The answer is 7/4 × 2/7 = 14/28 = 1/2.
Size sense: Ask, without working it out, whether 7/8 × 15 is more or less than 15. The answer is less, because 7/8 is less than 1.

A five-minute weekly check

A five-minute check once a week, with the working written down, shows more than a long page of one problem type. Mix one question of each kind and look at the steps, because most errors come from the set-up of whole numbers and mixed numbers, not from the multiplication.

The national curriculum in England follows the same order. The mathematics programmes of study ask Year 5 pupils to "multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams", and Year 6 pupils to "multiply simple pairs of proper fractions, writing the answer in its simplest form".

Fast multiplication facts carry every step, and the multiplication worksheet for 5th graders gives printable practice with an answer key. The missing factor game trains the "what times what" thinking that cancelling needs.

Common mistakes when multiplying fractions

Four mistakes cause most wrong answers when multiplying fractions, and each has a one-line fix.

Finding a common denominator first: Turning 1/2 × 1/3 into 3/6 × 2/6 gives 6/36, which is 1/6 only after a long simplification. Multiply the denominators as they are: 1/2 × 1/3 = 1/6.
Adding the denominators: Writing 1/2 × 1/3 = 1/5 mixes up the rules for adding and multiplying. The denominators multiply: 2 × 3 = 6, so the answer is 1/6.
Multiplying the whole number into the top and the bottom: Writing 3 × 2/5 = 6/15 leaves the fraction unchanged, because 6/15 is 2/5. Put the whole number over 1 so it reaches only the top: 6/5.
Multiplying mixed numbers part by part: Working 2 1/2 × 1 1/3 as 2 × 1 plus 1/2 × 1/3 gives 2 1/6, far from the right answer of 3 1/3. Convert both to improper fractions before you multiply.

Edge case: fractions with different denominators

Fractions with different denominators multiply the same way as fractions with equal ones. The product 2/5 × 3/7 is 6/35, and 5/6 × 3/10 cancels to 1/2 × 1/2 = 1/4, so a different denominator is never a reason to change the method.

Dividing fractions is the operation that adds a step: you flip the second fraction and multiply. The guide on how to divide fractions builds on the same rule.

Your fraction multiplication checklist and next steps

Read × as "of": 1/2 × 3/4 is half of three quarters.
Set up: put whole numbers over 1 and turn mixed numbers into improper fractions.
Cancel: divide any top and any bottom by a shared factor.
Multiply straight across: top × top, bottom × bottom, no common denominator.
Simplify: reduce the answer and turn an improper fraction into a mixed number.
Estimate: a factor less than 1 makes the answer smaller.

Parents of fifth graders will find grade-level practice on the math game for 5th graders page, and adults rebuilding their number sense can start with the math game for adults. The guide on teaching long multiplication helps when numerators and denominators grow past the times tables.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty that matches the level to each player, and daily practice in 33 languages. Families can race each other in 2–5 player real-time battles, and 22 of the games play free in the browser with no sign-up.