Class 5 Mathematics Chapter 3 Complete Guide Book — Fractions

A fraction is a part of a whole. The bottom number, the denominator, tells you how many equal parts the whole is cut into, and the top number, the numerator, tells you how many of those parts you have. In 3/4, the whole is cut into 4 equal parts and you have 3 of them.

If the numerator is smaller than the denominator, the fraction is proper — it is less than one whole, like 3/4. If the numerator is bigger, it is improper — more than one whole, like 7/4. And a mixed number writes the same amount as whole parts plus a fraction: 7/4 is 1 whole and 3/4 left over, written 1¾.

To change an improper fraction into a mixed number, divide: 9 ÷ 4 = 2 remainder 1, so 9/4 = 2¼. The remainder becomes the new numerator, and the denominator stays the same. Going back is just as easy: 2¼ = (2 × 4 + 1)/4 = 9/4.

Different fractions can show exactly the same amount. Cut a roti into 2 parts and take 1, or cut it into 4 parts and take 2, or into 6 parts and take 3 — every time you have eaten half the roti. So 1/2 = 2/4 = 3/6. These are called equivalent fractions.

The rule is simple: multiply or divide the top and the bottom by the same number, and the value does not change. That is why 2/5 becomes 4/10 (times 2) or 6/15 (times 3), and why simplifying works in reverse: 12/16 divided top and bottom by 4 gives 3/4.

Simplifying means dividing by the HCF of the numerator and denominator until nothing common is left. Note that 6/8 is also equal to 12/16, but it is not the simplest form — only 3/4 is fully simplified, because 3 and 4 share no factor except 1. In exams, always give the simplest form unless the question asks otherwise.

You can only add fractions when the pieces are the same size — that means the denominators must match. When they already match, the work is easy: 3/8 + 2/8 = 5/8. Just add the numerators and keep the denominator the same. Never add the denominators: 3/8 + 2/8 is not 5/16.

When the denominators differ, first make them equal using the LCM. For 1/2 + 1/3, the LCM of 2 and 3 is 6, so change both into sixths: 1/2 = 3/6 and 1/3 = 2/6. Now add: 3/6 + 2/6 = 5/6. This is exactly why adding straight across as 2/5 is wrong — halves and thirds are different-sized pieces, and you cannot add different sizes until you cut them into equal sixths.

The pizza example shows it in real life. Ahmed ate 2/5 of a pizza and Sara ate 1/5. The denominators already match, so together they ate 3/5, and 2/5 of the pizza is left (because 5/5 − 3/5 = 2/5). Subtraction follows the same rule: equal denominators first, then subtract the numerators.

Multiplying fractions is the friendliest operation of all: multiply the tops together and the bottoms together. For 2/3 × 3/5, that gives 6/15, which simplifies to 2/5. You can simplify before multiplying too — the 3 on top and the 3 at the bottom cancel — which keeps the numbers small.

Multiplying by a whole number works the same way. A rope 3/4 m long, taken twice, is 3/4 × 2 = 6/4 = 1½ m. And taking half of something is also multiplication: half of 3/4 cup of sugar is 1/2 × 3/4 = 3/8 cup. That is why halving a recipe is easy once you know fraction multiplication.

Division has one golden trick: keep the first fraction, flip the second, then multiply. Dividing by a fraction is asking how many of those pieces fit inside, and flipping turns the question into a multiplication you already know. Always simplify your final answer, and check that it makes sense: dividing by a fraction smaller than 1 should give a bigger answer, not a smaller one.