Cancel before you multiply

Most children multiply fractions out and then try to simplify a horrible number at the end. Cancelling first turns the same question into mental arithmetic — and there is exactly one rule to remember.

The rule

Always pair a TOP number with a BOTTOM number. Never two tops.

That is the whole thing. Everything else follows from it.

Worked example

Notice what was cancelled: the 3 on top of the first fraction against the 9 on the bottom of the second, and the 8 on top of the second against the 4 on the bottom of the first. Both pairings went diagonally — top with bottom.

The alternative is to multiply first: 3 × 8 = 24, 4 × 9 = 36, then simplify 24/36 down to 2/3 by spotting that 12 divides both. That works, but you have to notice 12, and under exam time pressure children either miss it or stop at 6/9.

Your turn

Again: the 5 on top paired with the 10 on the bottom, and the 9 on top paired with the 6 on the bottom. Two diagonal pairings, and the answer arrives with no simplifying needed.

The mistake nearly every child makes

Given 3/4 × 8/9, a child in a hurry sees the 3 and the 9 and cancels them — correctly. Then they see the 4 and the 8 and cancel those too, also correctly, because 4 is on the bottom and 8 is on the top.

But give them 2/5 × 4/7 and some will try to cancel the 2 against the 4 — both are on top. That is not allowed, and it changes the answer:

• Cancelling two tops would give 1/5 × 2/7 = 2/35.
• The correct answer is 2/5 × 4/7 = 8/35.

Four times out. Worth knowing why.

Why it works

When you multiply fractions, every top number multiplies together and every bottom number multiplies together. So 3/4 × 8/9 is really (3 × 8) / (4 × 9) — one big fraction.

Simplifying any fraction means dividing the top and the bottom by the same number. In this single big fraction, the 3 happens to be on top and the 9 happens to be on the bottom, so dividing both by 3 is ordinary simplifying. It does not matter that they came from different fractions originally — multiplication does not care about order.

That also explains why cancelling two tops is wrong. Dividing two numbers that are both on top divides the top twice and the bottom not at all, which shrinks the whole fraction.

So cancelling early is not a different method. It is the same simplifying, done while the numbers are still small.

One warning: division is not the same

For 3/4 ÷ 8/9 you must flip the second fraction first — 3/4 × 9/8 — and only then look for cancels. Cancelling before flipping gives a wrong answer, and it is a habit that is hard to unlearn once it sets in.

Why this matters in a timed test

In the NSW Selective test the Mathematical Reasoning section gives 35 questions in 40 minutes — under 70 seconds each. The Opportunity Class test is the same rate. At that pace, a method that removes a simplifying step at the end is not a nicety; it is where the time for the hard questions comes from.