Fractions

All four operations — including dividing a fraction by a fraction (no calculator). Mixed numbers, improper fractions, and fraction word problems.

÷ fraction ÷ fractionmixed numbersfraction of a set P1 · no calc P2 · calc
Stage 1 of 6 — Overview
What PSLE fractions tests & common pitfalls
Fractions on the PSLE: All four operations on fractions — including the two new P6 division types: fraction ÷ whole number and whole number / fraction ÷ fraction. Plus word problems involving fraction of a set, fraction of remainder, finding the whole from a part, and fractions combined with ratio or percentage.

Exam note: Fraction computations are tested in Paper 1 without a calculator. Word problems appear in both papers — shorter ones in Paper 1 (2 marks), longer structured ones in Paper 2 (3, 4 or 5 marks).
+ / − : find common denominator first × : multiply straight across, cancel first ÷ a fraction = × by its reciprocal "of remainder" → apply to what's LEFT Find the whole = part ÷ fraction
⚠ Common mistakes to avoid
  • "Of remainder" ≠ "of total". If a question says "14 of the remainder", apply 14 to what is LEFT, not to the original total.
  • Flip the divisor, not the dividend. 34 ÷ 6 = 34 × 16 — flip the second number only.
  • Regroup before subtracting mixed numbers. For 415 − 223, since 15 < 23, borrow 1 whole from the 4 first.
  • Always simplify the final answer. 68 should be 34. 1712 should be 1512.
  • "Find the whole" → divide, not multiply. If 35 of a number is 24, the number is 24 ÷ 35 = 40, not 24 × 35.
Stage 2 of 6 — Computation
Add, subtract, multiply (Paper 1, no calculator)
Q1 — Adding unlike denominators Paper 1 · no calc 2 marks

Calculate 34 + 56. Give your answer as a mixed number in its simplest form.

Common denom.LCM of 4 and 6 = 12
Convert34 = 912 , 56 = 1012
Add912 + 1012 = 1912
Simplify1912 = 1712
Answer: 1 712

Always convert the final improper fraction to a mixed number unless the question says otherwise. Check whether 7⁄12 can simplify further — here 7 and 12 share no common factor, so it's already simplest.

Q2 — Subtracting mixed numbers with regrouping Paper 1 · no calc 2 marks

Calculate 415 − 223.

Common denom.LCM of 5 and 3 = 15
Convert415 = 4315 , 223 = 21015
RegroupSince 315 < 1015, borrow 1 whole from 4:   4315 = 31815
Subtract31815 − 21015 = 1815
Answer: 1 815

The regrouping step is the most common error point. Always check whether the first fraction is smaller than the second before subtracting — if yes, borrow 1 whole (= 15⁄15 here) from the integer part.

Q3 — Multiplying fractions (with cancelling) Paper 1 · no calc 2 marks

Calculate 815 × 512. Give your answer in its simplest form.

Cancel8 and 12 share factor 4 → 8 becomes 2, 12 becomes 3
Cancel5 and 15 share factor 5 → 5 becomes 1, 15 becomes 3
Multiply23 × 13 = 2 × 13 × 3 = 29
Answer: 29

Always cancel BEFORE multiplying. Cancelling 40⁄180 at the end works but creates unnecessarily big numbers. Cancel any numerator with any denominator (they don't have to be in the same fraction).

Stage 3 of 6 — Computation
Division — the three new P6 types
The Standard P6 syllabus introduces three division types that weren't in P5 — all without a calculator:
1. Proper fraction ÷ whole number  ·  2. Whole number ÷ proper fraction  ·  3. Proper fraction ÷ proper fraction.
The single rule for all three: flip the divisor, then multiply.
Q4 — Dividing a fraction by a whole number ⭐ P6 new Paper 1 · no calc 2 marks

Calculate 34 ÷ 6. Give your answer in its simplest form.

Rewrite 66 = 61
Flip & ×34 ÷ 61 = 34 × 16
Cancel3 and 6 share factor 3: 14 × 12
Multiply= 18
Answer: 18

Think of it as "splitting 34 into 6 equal parts". Each part is 18. The answer must be smaller than 34 — a useful sanity check.

Q5 — Dividing a whole number by a fraction ⭐ P6 new Paper 1 · no calc 2 marks

Calculate 8 ÷ 25.

Flip & ×8 ÷ 25 = 8 × 52
Cancel8 and 2 share factor 2: 4 × 51
Multiply= 4 × 5 = 20
Answer: 20

Think: "How many 25s are in 8?" The answer must be bigger than 8, since we're counting small pieces. Dividing by a fraction smaller than 1 always makes the answer larger.

Q6 — Dividing a fraction by a fraction ⭐ P6 new Paper 1 · no calc 2 marks

A rope of length 45 m is cut into pieces, each of length 215 m. How many pieces of rope are there?

SetupNumber of pieces = 45 ÷ 215
Flip & ×= 45 × 152
Cancel= 4 × 155 × 2 = 6010 = 6
Answer: 6 pieces

The question asks "how many pieces", so the answer must be a whole number. If you get a fraction, you've set up the division the wrong way round. Always flip the divisor (the number after ÷).

Stage 4 of 6 — Fraction of a quantity
Fraction of a set, of a remainder, & finding the whole
Q7 — Simple fraction of a set Paper 1 · no calc 1 mark · MCQ

A bag contains 45 marbles. 35 of the marbles are blue. How many marbles are blue?

MethodBlue = 35 × 45
Compute= 3 × 455 = 3 × 9 × 55 = 3 × 9 = 27
Answer: 27 blue marbles

"Fraction of" always means multiply. For whole-number calculations, cancel the denominator with the whole number first (45 ÷ 5 = 9), then multiply by the numerator.

Q8 — Fraction of remainder Paper 1 · no calc 2 marks

Ali had 240 stickers. He gave 38 of his stickers to his sister and 15 of the remainder to his friend. How many stickers did Ali have left?

240 stickers at first Sister (90) Remainder = 150 Of the 150 remainder: Friend (30) Left = 120
Step 1To sister: 38 × 240 = 90
Step 2Remainder: 240 − 90 = 150
Step 3To friend: 15 × 150 = 30
Step 4Left: 150 − 30 = 120
Answer: 120 stickers

Key trap: Do NOT apply 15 to the original 240 — apply it to 150, the remainder after Step 2.

Q9 — Finding the whole given a part Paper 1 · no calc 2 marks

35 of the members of a swim club are boys. If there are 24 boys, how many members are there altogether?

Total = 5 units 24 boys (3 units) Girls (2 units) 3 units = 24 → 1 unit = 8 → 5 units = 40
Set up35 of total = 24
3 units= 24  →  1 unit = 24 ÷ 3 = 8
Total= 5 units = 5 × 8 = 40
Answer: 40 members

"35 of the whole is 24" means the whole splits into 5 equal units, and 3 of those units make 24. Find 1 unit by dividing, then scale up to 5 units.

Stage 5 of 6 — Word problem patterns
5 classic PSLE structures (Paper 2)
Most Paper 2 fraction questions follow one of 5 patterns: (1) work backwards from the remainder, (2) before-and-after with transfer, (3) constant total, (4) constant difference, (5) repeated identity. Recognising the pattern is half the battle — once you spot it, the method follows.
Q10 — Pattern 1: Work backwards from the remainder Paper 2 · calculator 4 marks

Mrs Tan had a bag of flour. She used 23 of it to bake bread and 14 of the remainder to bake cookies. She was left with 112 kg of flour. How much flour did Mrs Tan have at first?

Total flour (12 small units) Bread (2⁄3 = 8 units) Cookies Left = 1½ kg (1 unit) (3 units)
After bread13 of total remained
Cookies14 × 13 = 112 of total
Left over13 − 112 = 412 − 112 = 312 = 14 of total
Solve14 of total = 1½ kg  →  Total = 1½ × 4 = 6 kg
Answer: 6 kg

The trick: express EVERY quantity as a fraction of the original whole first. Once you know what fraction of the whole = 1½ kg, you can scale up to the whole.

Q11 — Pattern 2: Before and after (transfer between two people) Paper 2 · calculator 4 marks

Alex had 57 as many stamps as Ben. Alex then gave 15 stamps to Ben. After that, Alex had half as many stamps as Ben. How many stamps did Alex have at first?

BEFORE — ratio 5 : 7 (total = 12 parts of whole) Alex = 5⁄12 Ben = 7⁄12 AFTER — ratio 1 : 2 (total = 3 parts of whole) Alex = 1⁄3 = 4⁄12 Ben = 2⁄3 = 8⁄12 Alex lost 5⁄12 − 4⁄12 = 1⁄12 of total = 15 stamps Total = 180, Alex at first = 5⁄12 × 180 = 75
Key ideaTotal stamps (Alex + Ben) stays the same — stamps are only transferred, not created or destroyed.
BeforeAlex : Ben = 5 : 7 → Alex = 512 of total
AfterAlex : Ben = 1 : 2 → Alex = 13 = 412 of total
ChangeAlex lost 512 − 412 = 112 of total = 15 stamps
Total= 15 × 12 = 180 stamps
Alex first= 512 × 180 = 75 stamps
Answer: 75 stamps

When items are only transferred between two people, the combined total stays constant. Express both the before and after amounts as a fraction of this constant total — then use the change to find the total.

Q12 — Pattern 3: Constant total (items switching category) Paper 2 · calculator 3 marks

In a class of 60 students, 35 of them study Chinese and the rest study Malay. Some students later switched from Chinese to Malay. After the switch, only 14 of the 60 students study Chinese. How many students switched?

Before: 3⁄5 × 60 = 36 Chinese, 24 Malay Chinese 36 Malay 24 After: 1⁄4 × 60 = 15 Chinese, 45 Malay Ch 15 Malay 45
BeforeChinese = 35 × 60 = 36
AfterChinese = 14 × 60 = 15
Switched= 36 − 15 = 21
Answer: 21 students

The class size (60) stays the same — only labels change. Compute the "before" and "after" Chinese counts directly, then find the difference.

Q13 — Pattern 4: Constant difference (both change by the same amount) Paper 2 · calculator 5 marks

Sarah had 34 as many stickers as Tom. After each of them bought another 10 stickers, Sarah had 56 as many stickers as Tom. How many stickers did Tom have at first?

BEFORE — Sarah : Tom = 3 : 4 (difference = 1 unit) Sarah (3 u) Tom (4 u) AFTER (+10 each) — Sarah : Tom = 5 : 6 (diff = 1 part) Sarah (5 parts) Tom (6 parts) 1 unit = 1 part Sarah: 3u + 10 = 5 parts 10 = 2 parts → 1 part = 5
Key ideaWhen both people add the SAME number, the DIFFERENCE between them stays the same.
Before diffSarah : Tom = 3 : 4  →  difference = 1 unit
After diffSarah : Tom = 5 : 6  →  difference = 1 part
Equal1 unit = 1 part (both represent the same difference)
Sarah3 units + 10 = 5 parts → 3 parts + 10 = 5 parts → 10 = 2 parts → 1 part = 5
Tom first= 4 units = 4 × 5 = 20 stickers
Answer: 20 stickers

Check: Sarah had 15, Tom had 20. After +10 each: Sarah 25, Tom 30. 25:30 = 5:6 ✓

Q14 — Pattern 5: Repeated identity (equal fractions of different wholes) Paper 2 · calculator 4 marks

Mary has some money and Jane has some money. 13 of Mary's money is equal to 14 of Jane's money. If they have $105 in total, how much does Mary have?

Mary's money (3 equal units) u u u Jane's money (4 equal units, same size) u u u u 7 units = $105 1 unit = $15 Mary = 3 × $15 = $45
Key idea13 of Mary = 14 of Jane means: if Mary's money is split into 3 equal units AND Jane's money is split into 4 equal units, each "unit" has the SAME dollar value.
Total units= 3 (Mary) + 4 (Jane) = 7 units
Solve7 units = $105 → 1 unit = $15
Mary= 3 units = 3 × $15 = $45
Answer: Mary has $45

When two different fractions give equal amounts, the ratio of the wholes matches the ratio of the denominators. Here Mary : Jane = 3 : 4.

Stage 6 of 6 — Combined topics & what's next
Fractions with ratio & decimals
Paper 2 questions often combine fractions with another topic — ratio, percentage, or decimals. The plan is the same: identify which topic to use first, find one quantity, then use the second topic to find the rest.
Q15 — Fractions with ratio Paper 2 · calculator 3 marks

The ratio of boys to girls in a class is 3 : 5. 25 of the boys wear glasses, and there are 6 boys who wear glasses. How many students are in the class altogether?

Boys — 2⁄5 wear glasses = 6 boys glasses = 6 no glasses Total boys = 15 Boys : Girls = 3 : 5 (3 units = 15 → 1 unit = 5) Boys 15 Girls 25
Boys25 of boys = 6 → Total boys = 6 ÷ 25 = 6 × 52 = 15
RatioBoys : Girls = 3 : 5 → 3 units = 15 → 1 unit = 5
Girls= 5 units = 5 × 5 = 25
Total= 15 + 25 = 40 students
Answer: 40 students

Two-topic problems need a plan: "find the fraction → use the ratio" OR "use the ratio → then apply the fraction". Here, use the fraction first to find boys, then apply the ratio to get girls.

Q16 — Fractions with decimals Paper 2 · calculator 3 marks

Mrs Lim bought 2.4 kg of flour. She used 13 of it to bake a cake. She packed the remaining flour into small bags of 0.2 kg each. How many bags of flour did she pack?

2.4 kg total flour Cake: 0.8 kg Remaining: 1.6 kg → bags of 0.2 kg 1.6 ÷ 0.2 = 8 bags
Cake13 × 2.4 = 0.8 kg
Remaining= 2.4 − 0.8 = 1.6 kg
Bags= 1.6 ÷ 0.2 = 8
Answer: 8 bags

Fractions and decimals can be mixed freely in Paper 2. When multiplying a fraction by a decimal, it's often easiest to divide first: 2.4 ÷ 3 = 0.8, then multiply if needed.

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