How to add, subtract, multiply and divide fractions

A fraction like $\tfrac{3}{4}$ has a top number, the numerator, and a bottom number, the denominator. The bottom says how many equal parts make a whole; the top says how many you have. Each operation has its own rule.

Adding and subtracting

You can only add or subtract fractions that share a denominator. If they differ, rewrite them over a common denominator first, then add or subtract the numerators and keep the denominator.

Multiplying

Multiplication is the easy one. Multiply the numerators together and the denominators together. No common denominator needed.

Dividing

To divide by a fraction, flip the second fraction and multiply. The flipped fraction is called the reciprocal.

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Simplifying the result

A fraction is in lowest terms when the top and bottom share no common factor. Divide both by their greatest common factor. For $\tfrac{8}{12}$, both divide by $4$, giving $\tfrac{2}{3}$.

Common mistakes

• Adding denominators. $\tfrac{1}{4} + \tfrac{2}{3}$ is not $\tfrac{3}{7}$; only the numerators add, over a common denominator.
• Forgetting to simplify. An answer like $\tfrac{8}{12}$ is correct but not finished; reduce it to $\tfrac{2}{3}$.
• Flipping the wrong fraction when dividing. Only the divisor gets flipped.

Match the rule to the operation, then simplify at the end. That is all four cases covered.