Page 1 · Introduction
The Singapore bar model
Singapore produces some of the world's strongest young mathematicians, and one of their secret weapons is the bar model — a way of drawing a word problem so the answer almost jumps out at you.
Instead of guessing whether to add, multiply or divide, you draw bars for the amounts. Confusing word problems turn into a simple picture.
The one key moveFind the value of one unit (one equal bar) first. Once you know one unit, everything else follows.
💡This module builds from simple sharing all the way to "before and after" puzzles. Take it a step at a time — the drawing does the hard work for you.
On the real testWhere this helpsThe Mathematical Reasoning section (25% of your mark, from 2025) is full of multi-step word problems. Bar modelling isn't a question "type" — it's a method for solving the ones the real exam asks, without high-school algebra.
Page 2 · How it works
Drawing your first bar
Problem: Sam has some marbles. Tom has 3 times as many. Together they have 48. How many does Sam have?
Draw it
Sam is 1 bar. Tom is 3 equal bars. Together that's 4 equal bars = 48.
4 units = 48, so 1 unit = 48 ÷ 4 = 12
Sam has 12. Tom has 3 × 12 = 36. Check: 12 + 36 = 48. ✓
✅See the power of it? A confusing word problem became "4 units = 48". Find one unit, and you've won.
Page 3 · The units method
Same idea, many disguises
Ratios, sharing, "times as many" — they're all the same units method underneath.
Red and blue counters are in ratio 2:3, 30 in total. How many red?
2 + 3 = 5 units = 30. One unit = 6. Red = 2 units = 12.
12 red
A pen and pencil cost $40; the pen costs 3 times the pencil. Pencil price?
Pencil = 1 unit, pen = 3 units, total 4 units = $40. One unit = $10.
Pencil = $10
💡Always ask: "how many units in total, and what are they worth?" That single question unlocks nearly every one of these.
Page 4 · Comparison bars
When one is "more than" another
When the problem gives a difference, draw both bars side by side and mark the extra piece.
Anna has $30 more than Ben. Together they have $100. How much has Ben?
Ben = 1 bar; Anna = same bar + $30. Together: 2 bars + $30 = $100, so 2 bars = $70, one bar = $35.
Ben $35, Anna $65
The trickTake the extra away first ($100 − $30 = $70), then split what's left equally. This handles almost every "more than" problem.
Page 5 · Fraction bars
Fractions made visual
Split the bar into equal parts and fractions become obvious.
⅗ of a class of 25 are girls. How many boys?
Split into 5 parts = 25, each part = 5. Girls = 3 parts = 15. Boys = 2 parts = 10.
10 boys
A boy spends ¼ of $20, then ½ of what's left. How much remains?
¼ of 20 = $5 spent, $15 left. ½ of $15 = $7.50 spent, $7.50 left.
$7.50
✅Fractions stop being scary when you see them as slices of a bar. Draw the slices and count.
Page 6 · Two-step problems
Building up in steps
Harder problems stack two ideas. Solve one step at a time and keep your units straight.
A shop had 80 apples. It sold ¼ in the morning and 30 in the afternoon. How many are left?
Morning: ¼ of 80 = 20 sold, 60 left. Afternoon: 60 − 30 = 30 left.
30 apples
💡Don't try to do it all at once. One step, write the result, next step. Neat steps prevent silly mistakes.
Page 7 · Practice
Level 1 · Easy
Warm up
Single-step units problems. Draw a quick bar if it helps, and tap to check.
1. Two numbers are in ratio 1:4 and add to 25. The bigger number?
1 + 4 = 5 units = 25, one unit = 5. Bigger = 4 units = 20.
2. A 36 cm ribbon is cut so one piece is twice the other. The shorter piece?
A 18 cm
B 24 cm
C 12 cm
D 9 cm
1 + 2 = 3 units = 36, one unit = 12. Shorter = 12 cm.
3. ⅖ of 40 students walk to school. How many do NOT walk?
5 parts of 8. Walk = 2 parts = 16. Don't walk = 3 parts = 24.
🎉Great start — the units method is clicking. Now two-step problems.
Page 8 · Practice
Level 2 · Medium
Two steps now
These need a comparison bar or a small chain of steps.
1. Sara has $50 more than Kim. Together they have $150. How much has Kim?
Take off the extra: 150 − 50 = 100, split in 2 = 50 each. Kim $50 (Sara $100).
2. A tank is ⅓ full. After adding 12 L it is ½ full. Full capacity?
½ − ⅓ = ⅙. That ⅙ = 12 L, so full = 12 × 6 = 72 L.
3. $120 is shared so Bo gets twice Al, and Cy gets the same as Bo. How much does Cy get?
Al = 1, Bo = 2, Cy = 2 → 5 units = 120, one unit = 24. Cy = 2 × 24 = 48.
💪The tank one is sneaky — the "jump" between fractions is the unit. Spotting that is a real skill.
Page 9 · Practice
Level 3 · Hard
Before & after
The famous Singapore "before and after" problems. Draw one bar for before, one for after — and look for what doesn't change.
1. Ann has $90 and Ben has $60. Ben gives Ann some money; now Ann has exactly twice Ben. How much did Ben give?
The total never changes: $150. After, Ann = twice Ben → 3 units = 150, one unit = 50. Ben ends with $50, so gave $60 − $50 = $10.
2. Red and blue counters are in ratio 3:2. After 10 red are removed, it's 1:2. How many blue?
Blue never changes = 2 units throughout. Red goes 3 units → 1 unit, a drop of 2 units = 10, so 1 unit = 5. Blue = 10.
3. Tim spent ½ his money on a book, then ⅓ of what was left on a pen. He had $8 left. How much did he start with?
Work backwards. After the book, half remained. He spent ⅓ of that half, leaving ⅔ of the half = $8, so the half = $12, whole = $24.
💡The secret to "before and after": find the thing that stays the same (the total, or one colour). It anchors everything.
Page 10 · Challenge
Level 4 · Challenge
Think smart
The toughest tier. These reward careful bar-drawing and a clever eye. Don't worry if they take a few attempts.
1. Ann and Ben have money in ratio 5:3. Ann gives Ben $8, and now they have equal amounts. How much did Ann have at first?
Ann 5 units, Ben 3 units. After giving $8: 5u − 8 = 3u + 8, so 2 units = 16, one unit = 8. Ann had 5 × 8 = $40.
2. A tap fills ⅓ of a tank in 10 minutes. At the same rate, how long to fill the whole tank?
A 20 min
B 30 min
C 13 min
D 40 min
If ⅓ takes 10 minutes, the whole (3 thirds) takes 3 × 10 = 30 minutes. Scaling the unit up is the move.
3. A shop had 80 pens. It sold ¼ on Monday, then 30 on Tuesday. How many are left?
Monday: ¼ of 80 = 20 sold, 60 left. Tuesday: 60 − 30 = 30 left. Careful two-step work wins.
🌟You've reached challenge level in a method many adults never learn. That's a real achievement — well done for sticking with it.
Complete
Brilliant work!
You've completed the Singapore Maths module.
You can now turn word problems into bars, find the value of one unit, handle comparison and fraction bars, and crack "before and after" puzzles by spotting what stays the same. That's a genuinely powerful toolkit for the Maths section.