Problem-solving method

What stays the same, explained for parents

The family of ratio problem sums where everything seems to move at once. There is one question that unlocks all of them, and it comes before any arithmetic.

Used from
P5
Taught at
P5–P6

What this method is

A whole family of questions where a ratio changes partway through. They look impossible because everything seems to move at once, and the way in is always the same: before calculating anything, work out what did not change.

That single decision picks the method, and the rest is counting units. Total, gap, or one person's share. Occasionally nothing, which is its own answer and its own approach.

When to reach for it

These are the words in a question that point at this method. You do not need to be able to solve it to spot them.

  • “gave 13\frac{1}{3} of his share to”

    An internal transfer. Nothing entered or left the pair, so the total is what held still.

  • “each of them bought a bag at the same price”

    Both sides moved by the same amount, so the gap between them is unchanged even though both numbers are new.

  • “In 3 years' time”

    Ages. The gap between two people never changes, so this is a constant difference question every single time.

  • “Billy spent $16”

    Read what is NOT in that sentence. If Ali did not spend anything, Ali's share is the fixed thing to match units against.

  • “Ali saved $60 and Billy spent $150”

    Different amounts in different directions, so nothing is constant. This is the hard class, and naming it as such is half the battle.

Watch it work

A sum of money was shared between Aini and Bala in the ratio 3 : 4. Bala gave 13\frac{1}{3} of his share to Aini. What is the new ratio of Aini's share to Bala's share?

The same 21 small units in both pictures, split 9 and 12 before, 13 and 8 after. Nothing entered or left, so the money only moved across. The violet block is what Bala handed over.

  1. Ask what stayed the same

    money moved between them, so the total is unchanged

    Bala gave to Aini. Nothing came in from outside and nothing left, so the pot is the same size before and after. This decision is the method, and it costs no arithmetic.

  2. Draw the before, and make the units divide

    Aini 3, Bala 4, total 7 → re-cut each into 3 → Aini 9, Bala 12, total 21

    A third of 4 units is not a whole unit, so the units are the wrong size. Cutting every one into 3 fixes it and changes no amount at all.

  3. Move the transfer

    13\frac{1}{3} of Bala's 12 = 4 units, from Bala to Aini

    The only event in the question. Note it comes out of Bala's 12 and lands in Aini's 9, rather than being taken off the total.

  4. Read the after

    Aini 9 + 4 = 13, Bala 12 − 4 = 8, so 13 : 8

    Both pictures still total 21, which is the check. If the two totals disagree in a transfer question, something has entered or left that should not have.

The one it gets confused with

This is the useful part of the page. Four questions, and the one your child can answer tells you which method they are looking at.

  • Constant difference

    Did both sides change by the same amount, or is it about ages? Then the gap held still, not the total, and the gap is what you match units against.

  • Constant part

    Did only one of them change? Then the other person's share is the fixed thing. Match its unit count across both pictures.

  • Everything changed

    Did they change by different amounts, in different directions? Then nothing is constant. Work in units of one person and use the second fact to pin them down.

  • Units and parts

    Does the ratio not change at all? Then there is no before and after to reconcile. Find one unit and scale.

How can you tell it has clicked?

Hand them a changing-ratio question and ask what did not change, before any working. A child who answers that in one sentence has the method, even if the arithmetic afterwards goes wrong.

The thing to watch for is a ratio equation with a cross-multiplication under it. It sometimes lands on the right answer, it is not what Primary 6 is taught to do, and it skips the only step worth learning here.

One thing to try tonight

Before any arithmetic, ask one thing: what didn't change? Total, gap, or one person's share.

You do not need to be able to solve the question to ask it. If they cannot answer, read it again together and look only for whether anything entered or left the group.

Where your child will meet it

Want the bigger picture on how these methods fit together? Maths heuristics, explained for parents

See which methods your child reaches for

StudyLah names the method in every worked solution, so you can see whether the problem was the method or the arithmetic.

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