Fractions
How to Multiply Fractions
To multiply fractions, multiply the numerators together, multiply the denominators together, and simplify. No common denominator is needed. This guide shows why that works using an area model, how to cancel before you multiply, how to multiply fractions with whole numbers and mixed numbers, and how to set up the word problems where fraction multiplication hides behind the word “of”.
How to multiply fractions
Multiplying fractions is the simplest of the four fraction operations. Top times top gives the new numerator; bottom times bottom gives the new denominator. Then reduce the result if you can.
Example: 3/5 × 2/7
6 and 35 share no factor, so 6/35 is already in lowest terms.
Example: 5/6 × 3/10
The product is correct either way, but 15/60 is not finished until it is reduced. The next section shows how to avoid that cleanup entirely.
If you want to check an answer, the fraction calculator multiplies any two fractions, mixed numbers or whole numbers and shows the working. Unlike adding, which needs the denominators to match first (see how to add fractions), multiplication works on the fractions exactly as they are written.
Why multiplying across works: the area model
The word of is the key. 2/3 × 3/4 means “two thirds of three quarters”. You can see the answer by drawing it.
- Draw a square and divide it into 4 equal columns. Shade 3 of them. That shaded part is 3/4 of the square.
- Now divide the square into 3 equal rows. Shade 2 of the rows in a second color. That is 2/3 of the square.
- The grid now has
3 × 4 = 12small cells. The cells shaded twice, inside both the 3 columns and the 2 rows, number2 × 3 = 6.
So two thirds of three quarters is 6/12, which is 1/2. Look at where each number came from. The denominator 12 is the number of cells in the grid, which is the product of the two denominators. The numerator 6 is the number of overlap cells, which is the product of the two numerators. That is the whole rule, drawn out.
The model also explains why the answer is smaller than both fractions. You are taking a part of a part, so you end up with less than either part on its own. Multiplication in either order gives the same overlap, which is the commutative property at work: 2/3 × 3/4 = 3/4 × 2/3. The guide to commutative and associative properties covers why order and grouping never change a product.
Cancel before you multiply (cross-cancelling)
Before multiplying, look for any numerator that shares a factor with any denominator. Divide both by that factor. This is called cancelling, or cross-cancelling when the pair sits in different fractions. It keeps the numbers small and means the answer comes out already simplified.
Example: 9/14 × 7/12
Pair each numerator with a denominator it shares a factor with.
Without cancelling, the product is 63/168. Getting from there to 3/8 means finding that the GCF of 63 and 168 is 21, which is harder than spotting a 3 and a 7.
Why is this allowed? Because all the numbers end up multiplied together in one big fraction anyway: (9 × 7) / (14 × 12). Inside a single fraction you can divide any factor on top and any factor underneath by the same number. Cancelling is just simplifying early, before the multiplication makes the numbers big.
The rules for cancelling
- Always pair a numerator with a denominator. Two numerators never cancel with each other, and neither do two denominators.
- The pair can be in the same fraction or in different fractions.
- You can cancel the same number more than once, in stages, until no numerator shares a factor with any denominator.
- Only cancel across a multiplication sign. In a division problem, flip the second fraction first; in an addition problem, never cancel at all.
Spotting shared factors is the same skill as reducing a fraction. If you need practice, how to simplify fractions covers divisibility tests and the greatest common factor, and the GCF calculator finds the largest factor two numbers share.
Multiplying fractions with whole numbers
Write the whole number as a fraction over 1, then multiply as usual. Since the bottom is 1, the shortcut is: multiply the whole number by the numerator and keep the denominator.
Example: 3/5 × 4
This is also repeated addition: 3/5 + 3/5 + 3/5 + 3/5 = 12/5. Four lots of three fifths is twelve fifths.
Example: 5 × 7/10, cancelling first
Multiplying first gives 35/10, which reduces to the same 7/2.
Finding a fraction of an amount
“Two thirds of 18” is 2/3 × 18. The quickest route is usually to divide by the denominator first, then multiply by the numerator: 18 ÷ 3 = 6, and 6 × 2 = 12. That is cancelling in disguise, and it keeps every step in whole numbers.
| Problem | Divide by the bottom | Multiply by the top | Answer |
|---|---|---|---|
| 2/3 of 18 | 18 ÷ 3 = 6 | 6 × 2 | 12 |
| 3/8 of 16 | 16 ÷ 8 = 2 | 2 × 3 | 6 |
| 3/4 of $60 | 60 ÷ 4 = 15 | 15 × 3 | $45 |
| 3/8 of $240 | 240 ÷ 8 = 30 | 30 × 3 | $90 |
When the whole number does not divide evenly by the denominator, the answer is a fraction or mixed number, as with 3/5 × 4 above. That is normal. The improper fraction calculator turns a result like 12/5 into 2 2/5.
Multiplying mixed numbers
Convert every mixed number to an improper fraction, then cancel and multiply. To convert, multiply the whole part by the denominator and add the numerator: 1 2/3 becomes (1 × 3 + 2)/3 = 5/3.
Example: 1 2/3 × 2 1/4
A quick estimate confirms the size: roughly 1.7 × 2.25 is a little under 4.
Example: 3 × 2 1/2
Here there is also a shortcut through the distributive property: 3 × 2 1/2 = 3 × 2 + 3 × 1/2 = 6 + 1 1/2 = 7 1/2. That works whenever only one of the two numbers is mixed.
Why you cannot multiply the parts separately
The tempting mistake is to multiply whole by whole and fraction by fraction. For 2 1/3 × 1 1/2 that gives 2 × 1 = 2 and 1/3 × 1/2 = 1/6, so 2 1/6. The right answer is 7/3 × 3/2 = 7/2 = 3 1/2.
The area model shows what went missing. A rectangle 2 1/3 wide and 1 1/2 tall splits into four pieces, not two:
The parts method keeps only the first and last pieces and throws away the two cross pieces in the middle. Converting to improper fractions includes all four automatically. The mixed number calculator handles the conversion and the arithmetic in one go.
Multiplying three or more fractions
The rule extends without change: multiply all the numerators, multiply all the denominators. Cancelling pays off even more here, because the products grow fast.
Because multiplication is associative, you can also multiply two at a time in any order. 2/3 × 9/10 = 3/5, then 3/5 × 5/6 = 1/2.
Does multiplying fractions make the answer bigger or smaller?
It depends on whether you multiply by something bigger or smaller than 1, not on whether a fraction is involved.
- Multiply by a number less than 1 and the result shrinks.
12 × 3/4 = 9. - Multiply by exactly 1 and nothing changes.
3/4 × 4/4 = 12/16, which is still 3/4. This is how equivalent fractions are made. - Multiply by a number greater than 1 and the result grows.
3/4 × 1 1/2 = 3/4 × 3/2 = 9/8.
Two proper fractions multiplied together always give something smaller than both. That gives you a free check: if 4/5 × 2/3 came out larger than 2/3, something went wrong. (It is 8/15, about 0.53.)
Fraction multiplication word problems
Most real fraction multiplication arrives in words. Look for “of”, “times”, “each” or an area or rate.
Recipe: part of a measure
A recipe needs 3/4 cup of milk, and you are making two thirds of the recipe.
Recipe: scaling up
The recipe needs 2 1/3 cups of flour, and you are making 1 1/2 batches.
Area: a garden bed
A bed measures 3 1/2 feet by 2 3/4 feet.
Nothing cancels here, since 7 and 11 are prime and the bottom is all 2s.
Fraction of a fraction of a group
In a class of 30, 3/5 of the students walk to school, and 2/3 of the walkers live within a mile.
Rate × time
You walk at 2 1/4 miles per hour for 1 1/3 hours.
A part of a part of a whole is the same structure as a percentage of a percentage; percentages are fractions over 100, and they multiply the same way.
Products of common fractions
| Problem | Before simplifying | Answer | Decimal |
|---|---|---|---|
| 1/2 × 1/2 | 1/4 | 1/4 | 0.25 |
| 1/2 × 2/3 | 2/6 | 1/3 | ≈ 0.3333 |
| 2/3 × 3/4 | 6/12 | 1/2 | 0.5 |
| 3/4 × 4/5 | 12/20 | 3/5 | 0.6 |
| 3/8 × 2/3 | 6/24 | 1/4 | 0.25 |
| 5/6 × 3/5 | 15/30 | 1/2 | 0.5 |
| 4/9 × 3/8 | 12/72 | 1/6 | ≈ 0.1667 |
| 2/5 × 5/8 | 10/40 | 1/4 | 0.25 |
| 7/10 × 5/14 | 35/140 | 1/4 | 0.25 |
| 3/4 × 3/4 | 9/16 | 9/16 | 0.5625 |
| 2/3 × 6 | 12/3 | 4 | 4 |
| 5/8 × 12 | 60/8 | 7 1/2 | 7.5 |
| 1 1/2 × 1 1/2 | 9/4 | 2 1/4 | 2.25 |
| 2 1/2 × 4 | 20/2 | 10 | 10 |
The middle column is what you get by multiplying straight across. The gap between it and the answer is exactly the work that cancelling saves.
Common mistakes
- Finding a common denominator first. It is not wrong, but it is wasted work.
1/2 × 1/3is 1/6 directly. - Keeping the denominator the same. Writing
2/5 × 3/5 = 6/5treats multiplication like addition. The answer is 6/25. - Multiplying a whole number into both top and bottom.
4 × 2/3is 8/3, not 8/12. Writing 4 as 4/1 prevents this. - Multiplying mixed numbers part by part. Convert to improper fractions first; the parts method drops the cross terms.
- Cancelling top with top. Cancelling always pairs a numerator with a denominator.
- Cancelling in a division before flipping. Flip first, as how to divide fractions explains, then cancel.
- Leaving the answer unsimplified. 15/60 is correct but unfinished. Reduce to 1/4.
Frequently asked questions
What is 1/2 times 1/3?
1/6. Multiply the numerators, 1 × 1 = 1, and the denominators, 2 × 3 = 6. In words, half of a third is a sixth: split a third into two equal pieces and each one is a sixth of the whole. The answer is smaller than both fractions, as it should be when two proper fractions are multiplied.
How do you multiply fractions with whole numbers?
Write the whole number over 1 and multiply across. That means multiplying the whole number by the numerator and keeping the denominator: 4 × 3/5 = 12/5 = 2 2/5. If the whole number shares a factor with the denominator, cancel first. For 5 × 7/10, cancel the 5 and 10 to get 7/2, or 3 1/2.
Do you need a common denominator to multiply fractions?
No. Common denominators are only needed for adding and subtracting, where the pieces must be the same size before you can count them together. To multiply, go straight across: top times top and bottom times bottom. Finding a common denominator first still gives the right answer, but it adds work and bigger numbers.
What is 3/4 of 20?
15. “Of” means multiply, so work out 3/4 × 20. The easy way is to divide 20 by the denominator, 4, to get 5, then multiply by the numerator, 3, to get 15. Multiplying first, 60/4, gives the same 15. A quarter of 20 is 5, so three quarters is three times that.
How do you multiply mixed numbers?
Turn each mixed number into an improper fraction, cancel any shared factors, then multiply across. 1 2/3 × 2 1/4 becomes 5/3 × 9/4; cancel the 3 into the 9 to get 5/1 × 3/4 = 15/4, which is 3 3/4. Never multiply the whole parts and the fraction parts separately.
What is cross-cancelling in fractions?
Dividing a numerator in one fraction and a denominator in the other by a shared factor before you multiply. In 9/14 × 7/12, the 9 and 12 share 3 and the 7 and 14 share 7, leaving 3/2 × 1/4 = 3/8. It gives the same answer as multiplying first and reducing later, with smaller numbers.
Why does multiplying fractions give a smaller number?
Because multiplying by a number less than 1 means taking only part of something. Half of 10 is 5, which is less than 10. Two proper fractions multiplied together always give a result smaller than both, since you take a part of a part. Multiply by a fraction greater than 1, such as 3/2, and the result grows instead.
How do you multiply three fractions?
Multiply all three numerators together and all three denominators together, then simplify. 2/3 × 9/10 × 5/6 gives 90/180, which is 1/2. Cancelling first is easier when there are several fractions, because any numerator can cancel with any denominator. You can also multiply two at a time, in any order.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/how-to-multiply-fractions/