One move opens most proportion and ratio questions: a common unit · Jump to the worked example →

Shown working, not just shown off

PSLE proportion and ratio questions, shown working.

The short version

PSLE proportion and ratio questions come in three familiar shapes: direct proportion, where quantities scale up or down together; packing or least-number reasoning, where one quantity is expressed in units of another; and sharing in proportion. The reliable move that opens all three is the same. When two different things measure the same space, put them both onto one common unit, then the problem becomes a single calculation.

Figures and tagging from GPA's tagged index of 709 PSLE questions, 664-basis, with the MOE Specimen reported separately · the worked example below was solved independently then checked against the verified GPA key · Mrs Eileen Toh signs off the mathematics · last reviewed 22 Jun 2026

The one idea

Most proportion and ratio questions look new. Far fewer are.

These questions reward steady habits more than flashes of insight. When we tagged and counted, the same few structures kept returning: direct proportion, packing or least-number reasoning, and sharing in proportion. The reliable move that opens almost all of them is to put two different things onto one common unit. Once both sit on the same unit, the question stops being a puzzle and becomes a calculation.

Direct proportion

Two quantities scale together, so you can step up or down by the same multiple.

Packing, least number

Express one quantity in units of another, then the count or the leftover is forced.

Sharing in proportion

Split a total into named parts, then read the value of one common unit.

This is analysis of past papers, not a forecast of the next one. Source: GPA tagged index, 664 sat-paper questions, with the MOE Specimen reported separately.

How proportion and ratio questions work

Three structures, one common-unit habit.

Almost every proportion or ratio question at PSLE sits in one of three structures. None of them needs faster arithmetic. Each needs you to choose a single unit that everything in the question can be measured in, and then the rest follows.

1

Direct proportion: scale up or down together

When two quantities rise and fall together, find what one unit is worth, then multiply or divide by the same factor. If 5 books cost a known amount, one book is that amount divided by 5, and any number of books follows. The whole skill is reducing to a single unit, then scaling.

2

Packing or least-number: one quantity in units of another

When a space can be filled by two different sizes, express one size in units of the other. If a shelf holds either large or small books, work out how many small-book units one large book is worth. Then a part-filled shelf, and the space remaining, can be measured exactly rather than guessed.

3

Sharing in proportion: split a total into named parts

When a total is shared in a given ratio, name each share in the same unit, add the units to match the total, and one common unit is found by a single division. Every individual share is then that unit times its own count. The ratio tells you the counts; the total tells you the unit.

The thread through all three: when two things measure the same space, convert both to a common unit. That single decision is what turns a question that looks new into one you have already solved.

2021 · Paper 2 · Q3 packing, common units

The shelf measured in two book sizes

Where to find it

This is a worded PSLE question, so we don’t reproduce it here. Find it in your Ten-Year Series (TYS) or the official paper — 2021 Paper 2, Q3 (the proportion question about packing a shelf) — then follow our worked solution below.

Video: a Genius Plus Academy teacher solving PSLE 2021 Paper 2 Question 3 Teacher video · 2021 P2 Q3

The lock

The shelf is one whole, but it is described two ways, in large books or in small books. Until both book sizes sit on one common unit, the part-packed shelf cannot be measured.

The key

Measure the shelf in small-book units; one large book is worth 1.5 small books.

Worked steps

  1. The shelf \(=\) 45 small-book units, and 30 large books fill it, so 1 large book \(= 45 \div 30 = 1.5\) small-book units.
  2. Already packed: \(3 \times 1.5 + 23 = 4.5 + 23 = 27.5\) units. Space left \(= 45 - 27.5 = 17.5\) units.
  3. \(17.5 \div 1.5 = 11\frac{2}{3}\), so at most 11 more large books.

Answer: 11 more large books.

What makes it click. Putting both book sizes on one common unit, the small-book unit, turns a packing puzzle into a single division.

Independently solved, matches the GPA handwritten key. Open the full worked solution →

The habit that unlocks them

Recognise the structure, then reuse the method.

The trap in proportion and ratio is treating a familiar structure as a brand-new puzzle. A shelf packed two ways, a recipe scaled up, a sum shared between three children: each can feel unprecedented, so a child starts from scratch every time and runs out of clock. Most of these questions look new. Far fewer are.

The habit that unlocks them is to pause and name the structure first. Is this direct proportion, packing, or sharing? Once it is named, the move is the same one you have used before: put the two different things onto one common unit. Recognising the structure and reusing the method, rather than reinventing it, is what turns a long, anxious question into a short, calm one.

Free for parents

The 10 PSLE Question Types, cheat sheet

One page that names every structure your child will meet, with the move that opens each. Direct proportion and packing are two of the ten. One email, no spam.

From one structure to all ten

Train structure-recognition across every question type.

This page trains direct-proportion and packing reasoning; the PSLE Math Intensive trains structure-recognition across all ten question types, with 158 worked examples.

Explore the PSLE Intensive →

Keep reading

The bigger picture these questions sit inside, and the sibling structures they keep landing near.

Pillar guide

Most-tested PSLE Math topics, counted

What 709 tagged questions show the paper rewards.

 Solutions hub

The hardest PSLE questions, shown working

The demanding tail, each one solved cleanly and checked.

 Type guide

Part-Whole questions

Name the whole in units that divide everything cleanly.

 Type guide

Percentage questions

Anchor the percentage to the right base, then scale.

Questions parents ask

What counts as a proportion or ratio question at PSLE?

In our tagging, these are the questions built on quantities that scale together or share a total: direct proportion, packing or least-number reasoning, and sharing in proportion. They look varied on the surface, but most reduce to the same move, putting two different things onto one common unit. This is analysis of past papers, not a forecast.

Why does my child find these questions so hard?

Usually because each one feels brand new, so a child starts from scratch and runs out of time. Most proportion and ratio questions look new; far fewer are. The fix is to recognise the structure first, name it as direct proportion, packing, or sharing, then reuse the common-unit method rather than reinventing it.

What is the single most useful habit to teach?

Putting two different things onto one common unit. When a shelf is described in large books and small books, or a total is shared three ways, the moment both sit on the same unit the question becomes a single calculation. Find the value of one unit, then scale or divide as the question asks.

Is the worked solution here reliable?

Yes. The example above was worked independently and then checked against the verified GPA key, here the handwritten key for 2021. Mrs Eileen Toh signs off the mathematics. You can browse more worked solutions and videos in our worked-solutions library.

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