Fraction Arithmetic
Three pictures cover the four operations: same-size pieces for + and −, an overlap grid for ×, and "how many fit?" for ÷. The rules all drop out of the pictures — no magic to memorise.
National curriculum KS3: use the four operations applied to proper and improper fractions and mixed numbers1 · Adding needs same-size pieces
You cannot add thirds to quarters any more than you can add metres to minutes. First cut both fractions into the same size pieces — a common denominator — then just count pieces. Slide the fractions and watch both bars get re-cut.
Finding the common size: use the lowest common multiple of the two bottoms. For 3 and 4 it is 12: thirds become twelfths (× 4) and quarters become twelfths (× 3). Whatever you do to the bottom, do to the top.
2 · Multiplying: a fraction of a fraction
2/3 × 4/5 means "two thirds of four fifths". On a grid, shade 4/5 one way and 2/3 the other way: the overlap is the answer. The shortcut falls out: multiply the tops, multiply the bottoms.
3 · Dividing: how many fit?
6 ÷ 1/2 asks "how many halves fit in 6?" — twelve. Dividing by a fraction makes the answer bigger, which surprises everyone once. The rule: Keep the first fraction, Flip the second, Change ÷ to ×.
= 3/4 × 5/2
= 15/8 = 1 7/8
each whole holds 2 → 6 × 2 = 12
so ÷ 1/2 is the same as × 2
Worked examples
Work out 1/3 + 1/4
4/12 + 3/12 = 7/12
Work out 5/6 − 1/2
Work out 2/3 × 6/7, simplified
Work out 4/5 ÷ 2/3
Quick check
What you've learned
To add or subtract, re-cut both fractions into the same size pieces (the LCM of the bottoms), then count tops — never add tops and bottoms. To multiply, multiply tops and multiply bottoms, then simplify. To divide, keep the first fraction, flip the second, change ÷ to ×. Dividing by a fraction smaller than 1 gives a bigger answer, and that is normal.