PRIMARY 4 MATHEMATICS LEARNING GUIDE · BATCH 2 · GUIDE 6
Fraction problems become difficult when the learner stops knowing what the whole is. The symbols may look familiar, but the meaning can shift: one fraction may refer to a set of objects, another to a length, another to what remains after an earlier action, and a mixed number may represent more than one whole.
This guide focuses on fraction word problems, reference wholes, improper fractions, mixed numbers, fractions of sets, reverse fraction structures and multi-step reasoning. The purpose is to make the reference quantity explicit before any fraction procedure begins.
Series route: return to the Primary 4 Mathematics Learning Hub. For the core fraction and decimal foundation, see Fractions, Decimals and Number Relationships.
1. The Reference Whole
In the statement “3/5 of the class”, the whole is the class. In “3/5 of the ribbon”, the whole is the ribbon. In “3/5 of the remaining water”, the whole is no longer the original amount unless the remaining amount equals the original.
A fraction has meaning only relative to a whole.
Before calculating, write a short label: whole class, whole ribbon, remaining amount, original set. This single habit prevents many multi-step fraction errors.
2. Fraction of a Set
There are 36 pupils. 2/3 join a programme. How many pupils join?
One third = 36÷3 = 12.
Two thirds = 12×2 = 24 pupils.
The denominator partitions the whole set into three equal groups. The numerator selects two of those groups.
3. Reverse Fraction of a Set
If 2/3 of a class is 24 pupils, how many pupils are in the class?
Two units = 24, so one unit = 12.
Three units = 36 pupils.
This reverse form checks whether the learner understands the equal-part structure. Multiplying 24 by 2/3 again would move in the wrong direction.
4. Improper Fractions and Mixed Numbers in Context
A jug contains 7/4 litres. This is more than one litre because four quarters make one whole litre.
7/4 = 1 3/4 litres.
Mixed numbers are useful when a quantity naturally consists of complete wholes plus an additional part. Improper fractions are useful when arithmetic with a common denominator is easier.
The learner should be able to move between both representations deliberately.
5. Converting Mixed Numbers by Meaning
Convert 2 3/5 to an improper fraction.
Two wholes contain 10 fifths. Add 3 fifths to obtain 13/5.
This explains the usual shortcut 2×5+3. The shortcut is valid because it counts how many fifths are contained in the whole quantity.
6. Adding Fractions in a Word Problem
A student reads 1/4 of a book on Monday and 2/5 on Tuesday. What fraction of the book is read altogether?
The whole is the same book on both days, so the fractions can be combined after creating equal-sized parts.
1/4 = 5/20 and 2/5 = 8/20.
Total read = 13/20.
The common denominator preserves the common whole.
7. Remaining Fraction
If 13/20 of the book has been read, the fraction remaining is:
1−13/20 = 20/20−13/20 = 7/20.
“Remaining” problems often involve subtracting from one whole. Make the whole explicit as 1 before working symbolically.
8. Fraction of a Remaining Amount
A tank has 60 litres of water. First, 1/3 of the water is used. Then 1/4 of the remaining water is used. How much water remains?
First use: 1/3 of 60 = 20 litres. Remaining = 40 litres.
Second use: 1/4 of 40 = 10 litres. Final remaining = 30 litres.
The second fraction refers to the remaining 40 litres, not the original 60 litres.
When the reference whole changes, the fraction calculation must change with it.
9. Two Fractions of the Same Original Whole
A class spends 1/4 of a fund on books and 2/5 of the same original fund on equipment. What fraction is left?
Since both fractions refer to the same original fund:
1/4+2/5 = 5/20+8/20 = 13/20.
Remaining = 7/20.
This looks similar to the previous problem but the reference whole does not change. The language determines the structure.
10. Fraction and Whole Number Together
A basket contains 48 oranges. 3/8 are used. Then 7 more oranges are added. How many oranges are in the basket?
3/8 of 48 = 18.
Remaining after use = 48−18 = 30.
After adding 7 = 37 oranges.
The fraction action changes the whole-number quantity, and the later operation acts on the new amount.
11. Mixed Number Addition in Measurement
A rope is 1 3/4 m long. Another rope is 2 1/2 m long. Find the total length.
1 3/4 + 2 2/4 = 4 1/4 m.
Keep the unit attached. A mixed number is still a measurement, not merely a symbolic object.
12. Mixed Number Subtraction
A 4 1/3 m plank has 1 5/6 m cut off. Find the remaining length.
Convert to sixths:
4 2/6 − 1 5/6.
Regroup one whole: 3 8/6 − 1 5/6 = 2 3/6 = 2 1/2 m.
Regrouping is the fraction version of exchanging one whole for equivalent smaller units.
13. Fraction of a Mixed Number Quantity
A container holds 2 1/2 litres. Half the liquid is poured out. How much is poured out?
2 1/2 = 5/2 litres.
Half of 5/2 is 5/4 = 1 1/4 litres.
At Primary 4, such a question may be easier with a visual model: divide 2 1/2 litres into two equal shares. Representation should support understanding rather than force unnecessarily advanced symbolism.
14. Comparing Fraction Quantities
Which is greater: 3/4 of 40 or 2/3 of 45?
3/4 of 40 = 30.
2/3 of 45 = 30.
The quantities are equal even though the fractions and original wholes differ.
A fraction alone does not determine the resulting amount; the reference whole matters.
15. Same Fraction, Different Whole
3/5 of 20 = 12.
3/5 of 50 = 30.
The fraction is identical, but the amount changes because the whole changes.
This is a useful misconception check for students who associate a fraction with one fixed numerical result.
16. Different Fractions, Same Amount
1/2 of 24 = 12.
1/3 of 36 = 12.
3/5 of 20 = 12.
Different fraction-whole combinations can produce the same amount. This helps students distinguish the fraction relationship from the resulting quantity.
17. Bar Models for Fraction Problems
If 3/5 of a quantity is 42, draw five equal units and label three units as 42.
One unit = 14.
Five units = 70.
The bar model makes the whole visible even though the whole amount is initially unknown.
18. Fraction Problems With Difference
3/4 of a number is 18 more than 1/2 of the same number. Find the number.
Use quarters: 1/2 = 2/4.
The difference between 3/4 and 2/4 is 1/4.
Therefore 1/4 of the number = 18.
Whole = 18×4 = 72.
This problem becomes simple once both fractions are expressed relative to the same whole and same-sized units.
19. Fraction Problems With Total
A and B share a quantity. A receives 2/5 of the total. B receives the rest. If B receives 36, find the total.
B receives 3/5 of the total.
3 units = 36, so 1 unit = 12.
5 units = 60.
The visible fraction in the question belongs to A, but the known amount belongs to B. The learner must first find B’s fraction of the whole.
20. Multi-Step Fraction Problem
A box contains 80 counters. 3/8 are red. Of the remaining counters, 2/5 are blue. The rest are green. How many are green?
Red = 3/8 of 80 = 30.
Remaining after red = 50.
Blue = 2/5 of 50 = 20.
Green = 50−20 = 30.
Again, the second fraction refers to a new reference whole: the remaining 50 counters.
21. Estimation With Fractions
Before finding 5/6 of 42, notice that 5/6 is close to 1, so the answer should be somewhat below 42. The exact answer 35 is plausible.
Before subtracting 1 5/6 from 4 1/3, estimate about 2 from about 4.3, giving roughly 2.3. The exact answer 2.5 is plausible.
Fraction estimation develops magnitude sense rather than treating fractions as symbol manipulation only.
22. Common Fraction Errors
| Error | Weak link | Repair question |
|---|---|---|
| Uses original whole for every fraction step | Reference whole not updated | What amount does this fraction refer to now? |
| 2/3 of 24 found as 24×3÷2 | Denominator/numerator roles reversed | How many equal parts make the whole? |
| 1/3+1/4=2/7 | Part sizes not made equal | Are thirds and quarters the same-sized unit? |
| 2 3/5 converted to 5/8 | Mixed-number meaning lost | How many fifths are in two wholes? |
| Known remainder matched to wrong fraction | Complement not identified | If A gets 2/5, what fraction does B get? |
23. Practice Laboratory
- Find 3/4 of 36.
- If 3/4 of a set is 27, find the whole set.
- Convert 9/4 to a mixed number.
- Convert 2 3/8 to an improper fraction.
- A child reads 2/5 of a book on Monday and 1/4 on Tuesday. What fraction is read altogether?
- What fraction of the book remains?
- A tank has 72 L. 1/3 is used, then 1/4 of the remaining amount is used. Find the final amount.
- A fund spends 1/5 on books and 3/10 on equipment, both from the original fund. What fraction remains?
- Add 1 2/3 and 2 1/6.
- Subtract 1 3/4 from 4 1/2.
- Compare 2/3 of 30 with 3/5 of 35.
- 3/5 of a number is 24. Find the number.
- 3/4 of a number is 15 more than 1/2 of it. Find the number.
- A receives 3/8 of a total. B receives the rest and gets 35. Find the total.
- A box has 96 items. 1/4 are red. Of the remainder, 1/3 are blue. How many are neither red nor blue?
24. Explained Answers
1. 36÷4×3 = 27.
2. 3 units=27, one=9, whole=36.
3. 2 1/4.
4. 16/8+3/8=19/8.
5. 2/5+1/4=8/20+5/20=13/20.
6. 7/20.
7. 1/3 of72=24, remain48; 1/4 of48=12; final=36 L.
8. 1/5=2/10, spent5/10, remain=1/2.
9. 1 4/6 +2 1/6=3 5/6.
10. 4 2/4−1 3/4 =3 6/4−1 3/4=2 3/4.
11. 2/3 of30=20; 3/5 of35=21, so the second is greater by 1.
12. 3 units=24, one=8, whole=40.
13. Difference between 3/4 and1/2 is1/4; whole=15×4=60.
14. B gets5/8. Five units=35, one=7, total eight units=56.
15. Red=24, remain72; blue=24; neither=48.
25. Teaching Routine: Ask “Fraction of What?”
For every fraction word problem, require the learner to complete the sentence “This fraction is of ______.” If the blank cannot be filled confidently, do not calculate yet.
Then identify whether the whole stays constant or changes after an earlier action. Use bars or number lines when the reference whole is hidden. Rotate the problem by changing the unknown from part to whole or from whole to part.
This creates a durable sequence: name whole → partition → select → update whole if needed → calculate → check.
26. Where This Goes Next
Primary 5 ratio and percentage depend on reference-whole control. A percentage is meaningful only relative to a base quantity. A ratio compares quantities using multiplicative units. Students who already ask “fraction of what?” are better prepared to ask “percentage of what?” and “ratio of which quantities?” later.
Continue to Decimal Measurement, Place Value and Rounding Accuracy →
Sources and Boundaries
Curriculum scope is aligned with the MOE Primary Mathematics Syllabus, updated October 2025. All teaching examples and diagnostic sequences are independently written by eduKate Publishing.
Editorial control: Wintour House V1.0 · CivDJ · eduKate Publishing.
