How to Calculate with Fractions — Complete Guide

What is a fraction?

A fraction represents a part of a whole. It has two parts: the numerator (top number, how many parts you have) and the denominator (bottom number, how many equal parts the whole is divided into). So 3/4 means "3 out of 4 equal parts."

Fractions come in three varieties:

• Proper fractions: numerator is less than the denominator (e.g. 2/5). Value is between 0 and 1.
• Improper fractions: numerator is greater than or equal to the denominator (e.g. 7/3). Value is 1 or greater.
• Mixed numbers: a whole number combined with a proper fraction (e.g. 2 and 1/3). Equivalent to an improper fraction.

How to add and subtract fractions — step by step

The golden rule: you can only add or subtract fractions that share the same denominator. If they don't, you need to create one first.

1. Check the denominators. If they are the same, skip to step 3. If not, continue.
2. Find the Least Common Denominator (LCD). The LCD is the smallest number that both denominators divide into evenly. For 1/4 and 1/6, the LCD is 12. A quick method: multiply both denominators (4 × 6 = 24), then simplify — or simply list multiples of each until you find one in common.
3. Convert each fraction. Multiply the numerator and denominator of each fraction by whatever factor gets you to the LCD. Example: 1/4 becomes 3/12 (×3), and 1/6 becomes 2/12 (×2).
4. Add or subtract the numerators. The denominator stays the same. 3/12 + 2/12 = 5/12. For subtraction: 3/12 − 2/12 = 1/12.
5. Simplify the result. Divide numerator and denominator by their Greatest Common Divisor (GCD). If the GCD is 1, the fraction is already in simplest form.

Example: 2/3 + 3/4

• LCD of 3 and 4 is 12.
• 2/3 = 8/12 and 3/4 = 9/12.
• 8/12 + 9/12 = 17/12.
• 17/12 is already in simplest form. As a mixed number: 1 and 5/12.

How to multiply fractions — step by step

Multiplication is the easiest fraction operation because you do not need a common denominator.

1. Multiply the numerators. The product of the top numbers becomes your new numerator.
2. Multiply the denominators. The product of the bottom numbers becomes your new denominator.
3. Simplify. Find the GCD of the result and divide both parts.

Example: 3/5 × 2/7 = (3×2)/(5×7) = 6/35. Already simplified (GCD is 1).

Tip — cross-cancel first: Before multiplying, simplify diagonally to keep numbers smaller. For 4/9 × 3/8 — the 4 and 8 share a GCD of 4 (reduce to 1 and 2), and 3 and 9 share a GCD of 3 (reduce to 1 and 3). Result: 1/3 × 1/2 = 1/6, much easier than simplifying 12/72.

How to divide fractions — step by step

Dividing by a fraction is the same as multiplying by its reciprocal (flip the second fraction).

1. Keep the first fraction unchanged.
2. Flip the second fraction (swap numerator and denominator). This is the reciprocal.
3. Multiply the two fractions as shown above.
4. Simplify.

Example: 3/4 ÷ 2/5

• Flip 2/5 → 5/2.
• 3/4 × 5/2 = 15/8.
• As a mixed number: 1 and 7/8.

How to simplify fractions

A fraction is in its simplest form (lowest terms) when the numerator and denominator share no common factor other than 1. To simplify:

1. Find the GCD of numerator and denominator using the Euclidean algorithm (repeatedly divide and take remainders until you reach 0).
2. Divide both numbers by the GCD.

Example: Simplify 18/24. GCD(18, 24): 24 = 1×18 + 6, 18 = 3×6 + 0 → GCD is 6. So 18÷6 = 3 and 24÷6 = 4. Result: 3/4.

Common mistakes to avoid

• Adding denominators: 1/2 + 1/3 is NOT 2/5. You must find a common denominator first.
• Forgetting to simplify: Always check if the result can be reduced.
• Dividing incorrectly: Always flip the second fraction, not the first, when dividing.
• Mixed number errors: Convert mixed numbers to improper fractions before calculating. 1 and 1/2 = 3/2, not 1.5 in fractional arithmetic.

Frequently asked questions

How do you add fractions with different denominators?

Find the Least Common Denominator (LCD), convert each fraction, then add the numerators. Example: 1/2 + 1/3 — LCD is 6, so 3/6 + 2/6 = 5/6.

How do you multiply fractions?

Multiply numerators together and denominators together, then simplify. No common denominator needed. Example: 2/3 × 3/4 = 6/12 = 1/2.

What is a mixed number?

A combination of a whole number and a proper fraction. Example: 7/3 = 2 and 1/3 (because 7 ÷ 3 = 2 remainder 1).

How do you simplify a fraction?

Divide both numerator and denominator by their GCD. Example: 12/18 ÷ 6 = 2/3.

Conclusion

Fraction arithmetic follows a short set of consistent rules. Addition and subtraction require a common denominator; multiplication and division do not. Simplifying after every calculation keeps numbers manageable. Use our free Fraction Calculator to check your work instantly or handle complex calculations on the fly.

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