The Remainder Model

Here is the sentence that trips up more PSLE students than almost any other: "He spent two-fifths of his money, then a quarter of the remainder…" The danger word is remainder. That second fraction is not a fraction of the original — it is a fraction of whatever was left over. The leftover becomes a brand-new whole that you cut up again. Get that idea, and these problems become easy marks.

Parents: let your child predict what the leftover gets cut into before revealing. The "Method tip" boxes name what a marker rewards — here, re-cutting the bar into equal small units.

By the end you'll be able to spot a "fraction of the remainder" problem, re-cut the bar, and find one small unit to solve it. Let's cut.

The leftover is a new whole

The key word is remainder. When a problem takes a fraction of the remainder, picture this: shade the first part of the bar, then look only at the leftover and divide that into the new fraction.

The clever move is to cut every part of the bar into the same size of small piece, so the whole bar ends up made of equal small units. Once every unit is the same size, you are back to a normal find-one-unit problem.

Re-cutting the bar

Try the classic. Mrs Tan baked some muffins. She sold two-fifths in the morning, then four-ninths of the remainder in the evening, and sold 120 more in the morning than the evening. How many did she bake?

Cut the bar into fifths, then re-cut the leftover three-fifths so every piece is the same small unit. The whole bar becomes 45 equal small units: morning is 18, evening is 12, and 15 are left. Now step one small unit until "morning − evening" equals 120.

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Muffins: re-cut the remainder

🤔 Predict first: After selling 2/5 in the morning, what fraction is the remainder?

2/53/5

Morning is 18 units and evening is 12 units, so the difference is 18 − 12 = 6 units = 120, giving 1 unit = 20. The whole bar is 45 units = 45 × 20 = 900 muffins.

The branching shortcut

Some students prefer a branching method instead of re-cutting a bar — a little tree where each fraction branches off a piece and the rest carries on. It gives the same answer; it is just tidier when there are two or more remainders in a row. Use whichever you can draw cleanly.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. Ben read 1/3 of a book, then 1/2 of the remainder the next day. The 1/2 applies to…

   The whole bookThe part left after the first dayHalf the whole book

2. After spending 2/5, what fraction of the money is the remainder?

   2/53/55/2

3. A bar is cut into fifths, then the 3 leftover fifths are each split into 3. How many equal small units is the whole bar now?

   8 units15 units9 units

4. A boy spent 1/2 of his money, then 1/4 of the remainder. What fraction of his original money was the second amount?

   1/41/81/2

5. In a remainder problem the whole bar is 12 equal units and 1 unit = 15. What is the total amount?

   1802712

6. Why is spending 1/4 then 1/3 of the remainder NOT the same as spending 1/4 + 1/3?

   Because fractions cannot be addedBecause the 1/3 acts on the smaller leftover, not the wholeBecause 1/4 + 1/3 is more than 1

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