Arithmetic with fractions

Learning outcome

Add, subtract, multiply and divide fractions, simplify efficiently, and evaluate bracketed expressions while keeping every denominator and divisor valid.

Understanding fraction arithmetic

Addition and subtraction require equal-sized parts. First create a common positive denominator; then add or subtract the signed numerators while keeping that denominator. Adding denominators would change the size of the parts rather than combine the given quantities.

For multiplication, multiply numerators together and denominators together. For positive fractions, taking a part of another part subdivides the original whole again, which explains the denominator product. Integer sign rules still apply when a numerator is negative.

You may cancel common factors between the overall numerator and denominator before multiplying. This reduces arithmetic and prevents unnecessarily large products. Cancellation removes factors of a product; it does not permit deleting terms joined by addition or subtraction.

Division asks how many divisor-sized amounts fit into the dividend. The reciprocal of a nonzero fraction c/d is d/c, because their product is one. Multiplying both dividend and divisor by this reciprocal makes the divisor one, so division becomes multiplication by the reciprocal.

When dividing by c/d, both c and d must be nonzero: the denominator must be valid and the divisor must not equal zero. Zero has no reciprocal. Convert mixed forms to improper fractions before multiplication or division. Respect brackets and the usual left-to-right evaluation of operations with equal priority.

Worked examples

Example 1: Calculate 5/6 - 1/4 + 1/3.

Use common denominator 12: 10/12 - 3/12 + 4/12. Work left to right: 7/12 + 4/12 = 11/12. The denominator stays fixed because all quantities now count twelfths.

Example 2: Calculate (14/15) × (25/28).

Cancel 14 with 28 to obtain 1 and 2; cancel 25 with 15 to obtain 5 and 3. The product becomes (1 × 5)/(3 × 2) = 5/6. These cancellations preserve the overall product’s value.

Example 3: Calculate (1 + 1/2) ÷ (3/4).

Convert the mixed quantity: 1 + 1/2 = 3/2. Reverse only the divisor, giving (3/2) × (4/3) = 12/6 = 2. Check: 2 × (3/4) = 3/2, so two divisor-sized portions fit exactly.

Common mistakes

Do not add or subtract denominators. Do not take the reciprocal of the first fraction in division. In sums, do not cancel individual terms against a denominator. Keep negative signs and simplify the final fraction.

Practice questions

1. Calculate 7/10 + 2/15.
2. Calculate (-4/9) × (3/8).
3. Calculate (5/12) ÷ (25/18).
4. Calculate (2/3 - 1/6) ÷ (5/4).

Worked answers

1. Use denominator 30: 21/30 + 4/30 = 25/30 = 5/6. The final cancellation divides both numbers by 5.
2. A negative times a positive is negative: (-4 × 3)/(9 × 8) = -12/72 = -1/6.
3. Multiply by the reciprocal: (5/12) × (18/25) = 90/300 = 3/10. The divisor is nonzero, so this operation is valid.
4. First, 2/3 - 1/6 = 4/6 - 1/6 = 1/2. Then (1/2) × (4/5) = 4/10 = 2/5.