Equivalent Fractions

الطريقة، خطوة بخطوة

1. 1

   Multiply top and bottom by the same number

   1/2 = (1×3)/(2×3) = 3/6

   The rule is that both parts must be treated identically. Multiplying only the numerator makes the fraction three times larger; multiplying both is multiplying by 3/3, which is 1, and multiplying by 1 changes nothing.

2. 2

   Divide top and bottom to simplify

   18/24 = (18÷6)/(24÷6) = 3/4

   Simplifying is the same move in reverse. Dividing by the highest common factor of 18 and 24 — which is 6 — gets to lowest terms in one step. Dividing by any smaller common factor works too; it just takes more steps.

3. 3

   Recognise lowest terms

   3/4 — HCF of 3 and 4 is 1, so it cannot simplify further

   A fraction is in its lowest terms when the numerator and denominator share no factor above 1. Exam questions almost always expect this form, and a correct answer left unsimplified frequently loses the final mark.

4. 4

   Use equivalence to compare fractions

   3/5 vs 5/8 → 24/40 vs 25/40 → 5/8 is larger

   Two fractions cannot be compared directly unless the denominators match. Rewriting both over a common denominator — here 40, the LCM of 5 and 8 — makes the comparison a matter of reading two numerators.

5. 5

   Check by cross-multiplying

   Is 6/9 = 8/12? 6 × 12 = 72, 9 × 8 = 72 ✓

   If two fractions are equivalent, the cross products are equal. This is a fast check that avoids simplifying either fraction, and it works regardless of how ugly the numbers are.

You are multiplying by 1

The reason the rule works is that 3/3, 6/6 and 100/100 are all equal to 1, and multiplying any number by 1 leaves it unchanged. So multiplying 1/2 by 3/3 gives 3/6, which looks different and is worth exactly the same. Naming the disguised 1 turns an arbitrary procedure into an obvious one.

It also explains, without a separate rule, why you cannot add the same number to top and bottom. Adding 3 to both parts of 1/2 gives 4/5, which is not one half — because adding 3 to both is not multiplying by 1, and nothing guarantees the value survives.

• Multiply or divide both parts by the same number
• That is multiplying by a disguised 1, so the value holds
• Adding the same number to both parts does not work
• Lowest terms: numerator and denominator share no factor above 1

Simplifying in one step

Dividing by the highest common factor gets a fraction to lowest terms immediately. Dividing by 2 repeatedly gets there eventually and takes longer, and each extra step is another chance for an arithmetic slip. For students who know their HCF work, this is where it pays off.

Where the HCF is not obvious, halving repeatedly is a perfectly legitimate route and no marks distinguish between them. The advice is to spend five seconds looking for a big common factor before defaulting to the slow method.

This is the skill the rest of fractions stands on

Adding and subtracting fractions requires rewriting them over a common denominator, which is equivalence. Comparing and ordering fractions requires the same. Simplifying an answer requires it. Later, cancelling in algebraic fractions is exactly this rule applied to expressions instead of numbers.

A student who is shaky here will struggle with every fraction topic that follows and it will look like several separate problems. It is usually one.

How we teach equivalent fractions

We start with a diagram — the same strip divided into halves and then into sixths — because seeing that three of the six pieces cover exactly the same length as one of the two makes equivalence a fact about quantity rather than a rule about digits.

Then we name the disguised 1 explicitly. It is one sentence, and it is what makes the add-to-both-parts error visibly wrong rather than merely forbidden.