Formula
To multiply fractions: multiply numerator × numerator and denominator × denominator, then simplify. No common denominator is needed — multiplication is the simplest of the four fraction operations.
How to use this calculator
Enter the numerator and denominator of each fraction. The calculator multiplies them instantly, showing the simplified result, mixed number (where applicable), decimal, and percentage — along with a three-step breakdown of the multiplication and simplification.
Multiplying fractions formula
Multiply numerators together and denominators together:
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Unlike addition and subtraction, multiplication requires no common denominator. The fraction is computed directly and then simplified by dividing both parts by their GCD.
Worked examples
Example 1: result simplifies
Multiply 2/3 by 3/5:
$$\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15} = \frac{2}{5}$$
Numerators: 2 × 3 = 6. Denominators: 3 × 5 = 15. GCD(6, 15) = 3. Simplified: 2/5.
Example 2: larger numbers
Multiply 3/4 by 8/9:
$$\frac{3}{4} \times \frac{8}{9} = \frac{3 \times 8}{4 \times 9} = \frac{24}{36} = \frac{2}{3}$$
Numerators: 3 × 8 = 24. Denominators: 4 × 9 = 36. GCD(24, 36) = 12. Simplified: 2/3.
Cross-cancellation shortcut
Cross-cancellation simplifies before multiplying by dividing a numerator and a diagonal denominator by a shared factor. This keeps intermediate numbers small:
$$\frac{3}{4} \times \frac{8}{9}: \quad \frac{\cancel{3}^{1}}{\cancel{4}_{1}} \times \frac{\cancel{8}^{2}}{\cancel{9}_{3}} = \frac{1 \times 2}{1 \times 3} = \frac{2}{3}$$
In 3/4 × 8/9: the 3 (first numerator) and 9 (second denominator) share a factor of 3 → divide both by 3. The 4 (first denominator) and 8 (second numerator) share a factor of 4 → divide both by 4. The remaining product is 1/1 × 2/3 = 2/3, with no further simplification needed.
Cross-cancellation is optional — it produces the same answer as multiplying then simplifying, but avoids large intermediate values that are harder to work with by hand.
Multiplying a fraction by a whole number
Write the whole number as a fraction with denominator 1, then apply the standard formula:
$$5 \times \frac{2}{3} = \frac{5}{1} \times \frac{2}{3} = \frac{10}{3} = 3\frac{1}{3}$$
This is equivalent to adding the fraction to itself that many times: 5 × 2/3 = 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 10/3.
Multiplying mixed numbers
Convert each mixed number to an improper fraction before multiplying:
$$1\frac{1}{2} \times 2\frac{2}{3} = \frac{3}{2} \times \frac{8}{3} = \frac{3 \times 8}{2 \times 3} = \frac{24}{6} = 4$$
1½ → 3/2 (1 × 2 + 1 = 3, over 2). 2⅔ → 8/3 (2 × 3 + 2 = 8, over 3). Multiply: 3/2 × 8/3. Cross-cancel: 3s cancel and the 2 and 8 simplify to 1 and 4. Result: 4/1 = 4.
Negative fractions
The sign rules for multiplying fractions follow standard integer rules:
- Positive × positive = positive
- Negative × negative = positive
- Positive × negative = negative
- Negative × positive = negative
$$\frac{-2}{3} \times \frac{3}{4} = \frac{-2 \times 3}{3 \times 4} = \frac{-6}{12} = \frac{-1}{2}$$
Count the negative signs: one negative among the two fractions → negative result. Two negatives → positive result.
Reference table
| Problem | Raw product | GCD | Simplified |
|---|---|---|---|
| 1/2 × 1/3 | 1/6 | 1 | 1/6 |
| 2/3 × 3/4 | 6/12 | 6 | 1/2 |
| 3/5 × 5/6 | 15/30 | 15 | 1/2 |
| 4/7 × 7/8 | 28/56 | 28 | 1/2 |
| 5/6 × 3/10 | 15/60 | 15 | 1/4 |
| 7/8 × 4/5 | 28/40 | 4 | 7/10 |
Common mistakes
Finding a common denominator first
A common denominator is needed for addition and subtraction — not multiplication. Adding a common denominator step to a multiplication problem introduces unnecessary work and often produces errors. Multiply straight across.
Multiplying only the numerators
Both numerator and denominator must be multiplied. 2/3 × 3/5 ≠ 6/3 and ≠ 2/5 (before simplification). The denominator product is 3 × 5 = 15, giving 6/15 = 2/5 after simplification.
Not converting mixed numbers to improper fractions
Multiplying 1½ × 2⅔ by treating whole parts and fractions separately (1 × 2 + ½ × ⅔ = 2 + 1/3) gives a wrong answer. The whole and fractional parts of each number are not independent factors. Convert to improper fractions first.
Forgetting to simplify
The raw product is often not in its simplest form. Always check whether numerator and denominator share a common factor. Use cross-cancellation beforehand to avoid large numbers that are harder to simplify after.
Frequently asked questions
How do you multiply fractions?
Multiply numerator × numerator to get the new numerator, multiply denominator × denominator to get the new denominator, then simplify by dividing both parts by their GCD.
Do you need a common denominator to multiply fractions?
No. A common denominator is only needed for addition and subtraction. Multiplication works by multiplying straight across.
What is cross-cancellation?
A simplification shortcut that divides a numerator and a diagonal denominator by a shared factor before multiplying. It gives the same final answer but avoids large intermediate numbers.
How do you multiply a fraction by a whole number?
Write the whole number as a fraction over 1, then multiply as normal. 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3⅓.
How do you multiply mixed numbers?
Convert each mixed number to an improper fraction first, then multiply normally.
What is the sign rule for multiplying fractions?
Same signs (both positive or both negative) give a positive result. Different signs give a negative result. Count the negatives: even number → positive; odd number → negative.