Fraction multiplication
Multiplying a fraction by a whole number uses the same rule as any fraction multiplication — the whole number just needs a denominator first. Write the whole number over 1, multiply across, and simplify. This page walks through the steps, the cross-cancelling shortcut that keeps the numbers small, and the whole-number-times-mixed -number case.
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Interactive tool
Type the problem directly or adjust the factors below. The tool rewrites mixed numbers and whole numbers, simplifies when possible, and shows every multiplication step.
Result
6/5
= 1 1/5
Multiply fractions by multiplying across. Multiply the numerators, multiply the denominators, and simplify at the end.
Treat each fraction as equal-size pieces. Count how many pieces stay selected after multiplying, then simplify if possible.
Rewritten
3/1 x 2/5
Simplify first
3/1 x 2/5
Before final simplify
6/5
Mixed-number form
1 1/5
Formula flow
Step by step
Multiply across
Multiply across: Fraction multiplication does not need common denominators. The operation tracks part of a part, so multiplying across keeps the size relationship intact.
Worked example
3 × 2/5 = 3/1 × 2/5 = 6/5 = 1 1/5
Rewrite 3 as 3/1 so the multiply-across rule applies. The numerators multiply (3 × 2 = 6) and the denominator stays 5. The product 6/5 is greater than one whole, so the mixed-number form 1 1/5 makes the answer easier to read.
Worked example
4 × 1/8 = 4/1 × 1/8 = 4/8 = 1/2
Whole numbers love to cancel. The 4 in the numerator and the 8 in the denominator share the factor 4, so you can simplify before multiplying: 4/8 = 1/2, and the answer arrives as 1/2 without any large products.
Worked example
5 × 3/4 = 5/1 × 3/4 = 15/4 = 3 3/4
When the whole number does not cancel with the denominator, multiply straight across: 5 × 3 = 15 over 4. An improper fraction like 15/4 is correct, but 3 3/4 shows the three complete wholes, which is usually what the question is asking about.
Worked example
1/2 × 6 = 1/2 × 6/1 = 6/2 = 3
Order does not matter in multiplication, so the whole number can come second. Rewrite 6 as 6/1 and multiply across. Cancelling the 2 with the 6 first gives 1 × 3/1 = 3, and half of six is indeed three.
A fraction has a numerator and a denominator, and a whole number on its own has only one number. Writing 3 as 3/1 gives it both parts without changing the value: 3/1 still means three wholes. This is the only extra step whole numbers need, and it turns every whole-number problem into a regular fraction-times-fraction problem.
Some students prefer the shortcut of multiplying the numerator by the whole number directly: 3 × 2/5 = 6/5. That shortcut works because the denominator of 3/1 never changes anything — but writing n/1 first makes the rule visible and keeps the arithmetic organized.
Review how to write a whole number as a fractionWhole numbers cancel with denominators beautifully, because the whole number sits over 1 and any shared factor can come out before you multiply. In 4 × 1/8, the 4 and the 8 share the factor 4: cancel to get 1 × 1/2 = 1/2. In 6 × 5/9, the 6 and the 9 share the factor 3: cancel to get 2 × 5/3 = 10/3 = 3 1/3.
Cancelling first is not a different rule — it is the same simplification, done at the friendliest possible moment. The products stay small, which means fewer arithmetic mistakes and less reducing at the end. If nothing shares a factor, multiply across and simplify the product instead.
A mixed number cannot be multiplied in mixed form. Convert it to an improper fraction first, then use the same rewrite-and-multiply steps. For 3 × 2 1/2: write 2 1/2 as 5/2, then 3/1 × 5/2 = 15/2, which is 7 1/2. The whole number 3 does not distribute over the 2 and the 1/2 separately in multiplication the way it does in addition — the improper fraction keeps it in one clean product.
3 × 2 1/2
= 3/1 × 5/2
= 15/2
= 7 1/2
Multiplication by a whole number is repeated addition. Three groups of one quarter — 1/4 + 1/4 + 1/4 — make three quarters, just as three groups of five dollars make fifteen dollars. That picture explains the shortcut: 3 × 2/5 collects three copies of two fifths, which is six fifths. The denominator counts the size of the pieces, so it never changes when you make more groups; only the numerator counts them up.
This is also why 1/4 × 3 equals 3 × 1/4. One quarter of three and three groups of one quarter land on the same amount — 3/4 — even though the wording sounds different.
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