“She spent one third of her money, then one quarter of what was left.” That tiny phrase—what was left—can change the entire solution. Your child may multiply fractions confidently and still get this PSLE Mathematics word problem wrong because the second fraction no longer refers to the original amount. For parents searching for PSLE fraction of the remainder Maths tuition in Bukit Timah, the difficulty is rarely about learning another fraction formula. It is about tracking which whole each fraction describes.
The core aim of Bukit Timah Mathematics tuition for fraction-of-remainder word problems is to teach Primary 5 and Primary 6 students to distinguish fractions of the original whole from fractions of a changing remainder, represent successive changes clearly, and work backwards when the final amount is known. A sound PSLE Maths solution should say what each fraction represents, maintain the correct reference amount through every step and verify the answer by reversing the process. This strengthens number sense, multi-step problem solving and the transition to secondary algebra rather than relying on one memorised heuristic.
Once the quantities are labelled, a daunting story becomes a sequence of understandable steps. The child no longer needs to guess whether to multiply or divide; the model tells them which relationship the operation must express.
What does ‘fraction of the remainder’ mean?
A fraction of the remainder is taken from the quantity that exists after an earlier change, rather than from the original quantity. If one third of 120 books is sold, 80 books remain. One quarter of the remainder is one quarter of 80, which is 20 books, not one quarter of the original 120.
The two amounts differ because the reference whole differs. The original 120 is 100% of the initial collection. After the first sale, the current collection is 80, so one quarter of that smaller group represents 20. Good tuition makes this explicit before applying arithmetic.
The most useful question: what is the whole right now?
When reading a multi-step fraction problem, encourage the learner to underline words such as originally, remaining, of what was left and afterwards. These words identify the changing reference quantity. A fraction is meaningful only when we know which amount is divided into the equal parts.
A tutor may draw a sequence of bars, one for the original collection and another for the smaller remaining collection. The second bar should not silently become the same length as the first. If the child writes “one quarter” without saying one quarter of which amount, the reasoning is still incomplete.
Worked example 1: a direct two-stage fraction question
A child has 120 stickers. They give away one third, then give away one quarter of the stickers remaining. How many are left? First, one third of 120 is 40, so 80 stickers remain after the first gift.
The second gift is one quarter of 80, which is 20. The final number of stickers is 80 − 20 = 60. The answer can also be obtained by multiplying the remaining fractions: 120 × (2/3) × (3/4) = 60.
The two routes are equivalent because two thirds remains after the first stage and three quarters of that remainder survives the second. The fractions refer to successive stages. The child should be able to explain why multiplying them gives the fraction of the original collection remaining.
Worked example 2: both fractions are spent, but on different bases
A person begins with S$150. They spend two fifths of the original amount, then spend one third of the money left. How much is left? First spending: 2/5 × 150 = S$60. Remaining after the first stage: S$150 − S$60 = S$90.
Second spending: 1/3 × 90 = S$30. Final amount = 90 − 30 = S$60. A common wrong route is to subtract 2/5 + 1/3 directly from one whole. Those fractions have different reference amounts, so they cannot be combined as two parts of the same original quantity without first converting the second fraction.
To compare on the original base, the second spending is 1/3 of 3/5 = 1/5 of the original amount. The total spent is 2/5 + 1/5 = 3/5, leaving 2/5 of S$150, or S$60. This extension reveals the structure behind the multi-stage calculation.
Worked example 3: find the starting amount by working backwards
A girl spends one quarter of her savings. Then she spends two fifths of the amount remaining. She has S$54 left. How much did she have at first? Work backwards from the final amount. After spending two fifths, three fifths of the intermediate remainder survived.
If three fifths equals S$54, the intermediate amount is 54 ÷ (3/5) = S$90. That S$90 represented three quarters of the original savings, so the original amount is 90 ÷ (3/4) = S$120.
Check forwards. One quarter of S$120 is S$30, leaving S$90. Two fifths of S$90 is S$36, leaving S$54. Every condition is satisfied. Checking the sequence in forward order gives the student a reliable way to detect a mistake made during reversal.
Worked example 4: use an algebraic model
The previous problem can also be represented with an equation. Let the original savings be x. After the first spending, 3x/4 remains. After the second, 3/5 of that remains, so the final amount is (3/5)(3x/4) = 9x/20.
Since 9x/20 = 54, multiply both sides by 20/9 to obtain x = 120. The symbolic route and the working-backwards route must agree, because both record the same two changes. This is a useful bridge from Primary fraction models to Secondary algebra when the learner is ready.
Do not rush a child into letters if they do not yet know what the successive fractions represent. The algebra compresses the relationship; it does not explain an incorrect reading of the problem. The PSLE bar models versus algebra guide explores that distinction.
Worked example 5: three stages with a known final remainder
A shop sells one fifth of its original inventory in the morning. In the afternoon it sells one quarter of the remaining items. At closing, 120 items are left. How many items were there at opening?
After the morning sale, four fifths of the inventory remains. After the afternoon sale, three quarters of that amount remains. Therefore the closing inventory is (4/5)(3/4) = 3/5 of the original total. The starting inventory was 120 ÷ (3/5) = 200 items.
Check: one fifth of 200 is 40, leaving 160. One quarter of 160 is another 40, leaving 120. Notice that the two stages happened to sell the same number of items even though the fractions differed. That numerical coincidence is a result of this setup, not a general rule.
Worked example 6: fractions of a remaining group
In a class of 90 pupils attending a school activity, one third leave at noon. Then two fifths of those still present leave at the next break. How many remain? One third of 90 is 30, leaving 60 pupils. Two fifths of 60 is 24, leaving 36 pupils.
The final group represents (2/3)(3/5) = 2/5 of the original 90. Since 2/5 × 90 = 36, the proportional check agrees. If the child calculated two fifths of 90 for the second departure, they ignored the change in the group after noon.
Worked example 7: a fraction of the remainder and a fixed amount
A family has an unknown number of postcards. They give away one quarter of them, then give away another 18 postcards. The final number is 42. How many were there initially? Undo the fixed removal first: immediately before the 18 postcards were given away, there must have been 42 + 18 = 60 postcards.
The 60 postcards represent three quarters of the original amount, so the original number was 60 ÷ (3/4) = 80 postcards. Check: one quarter of 80 is 20, leaving 60; giving away 18 more leaves 42. The order of reversing the two changes is crucial.
A student who divides 42 by three quarters before adding 18 has undone the operations in the wrong order. A timeline can reveal which change happened most recently and should therefore be reversed first.
Worked example 8: a final percentage rather than a number
A school club spends one third of its funds, then spends one half of the remainder. What percentage of the original funds is left? After the first spending, two thirds remain. Half of that is (1/2)(2/3) = 1/3 of the original funds.
One third expressed as a percentage is 33⅓%. The child should not subtract 1/3 + 1/2 and conclude that only one sixth remains, because the half refers to a smaller, later amount. The question’s changing base determines the result.
Why multiplying the remaining fractions works
If a first stage leaves a fraction a of the original quantity, and a second stage leaves a fraction b of the first remainder, then the final amount is a × b of the original. This follows because the second fraction acts on the result of the first change. Multiplication expresses the successive scaling.
But students should not blindly multiply every fraction in the story. Some fractions refer to the original whole, some to an intermediate remainder and some to a particular subgroup. The correct operation follows only after the student identifies the reference amounts.
Bar model versus a flow diagram
A bar model can show the original whole divided into equal units, then identify the remaining section and subdivide it further. A flow diagram offers another view: start amount → multiply by 3/4 → multiply by 3/5 → final amount. To work backwards, reverse the arrows and divide by the corresponding fractions.
For a child who loses track of stages, a two- or three-row table may be the clearest representation: original amount, amount after first change, final amount. Teaching should help the student choose a clear route rather than require one elaborate drawing for every problem.
A counterexample: two fractions of the original
Not every two-stage problem uses a changing reference amount. Suppose a child spends one quarter of the original amount in the morning and another one fifth of the original amount in the afternoon. These are both fractions of the same original whole, so the fraction spent is 1/4 + 1/5 = 9/20, leaving 11/20.
This is different from spending one quarter of the original and then one fifth of the remainder. In that second case, the fraction left is (3/4)(4/5) = 3/5. The wording changes the mathematics. Comparing the two helps prevent students from applying one memorised shortcut to every multi-stage story.
The six most common mistakes
- Using the original whole twice: calculating the second fraction from the original amount when it refers to the remainder.
- Adding fractions across different bases: subtracting unlike percentages or fractions directly from one whole.
- Reversing steps in the wrong order: dividing before undoing the final fixed addition or removal.
- Misidentifying the known final amount: treating a remainder as a total or as an amount spent.
- Skipping a consistency check: failing to replay the original sequence with the proposed starting quantity.
- Following a worksheet template: recognising only questions with the familiar title instead of interpreting new wording.
The first incorrect line points to the real teaching need. A child who has not understood the original whole requires conceptual explanation. A child who knows the meaning but makes a fraction division error needs numerical practice. A child who solves guided examples but cannot start a fresh one needs independent method-selection tasks.
What Primary 4, 5 and 6 learners should practise
For Primary 4 pupils, begin with straightforward fractions of whole-number groups and concrete representations. For Primary 5, connect fraction changes to ratio and percentage and gradually introduce mixed problems. For Primary 6 and PSLE preparation, practise unknown starting amounts, repeated changes and word problems without chapter labels.
The teaching sequence should follow the MOE Primary Mathematics syllabus and the child’s actual school materials. The examples here illustrate mathematical relationships; they are not predictions of the exact questions on a future PSLE paper.
A three-week repair plan
- Week 1: identify original and remaining quantities; solve direct one- and two-stage fraction examples using labelled bars.
- Week 2: reverse the changes to recover unknown starting amounts; compare a fraction of the original with the same fraction of the remainder.
- Week 3: mix fresh problems, including a fixed amount removed and a percentage remainder; check whether the child can start and verify the answer without prompts.
These three weeks are an illustrative cycle, not a guarantee of mastery. Some children need more time on equivalent fractions, multiplication or fraction division. It is better to stabilise the foundational concept than rush to a difficult question set the learner cannot yet interpret.
How small-group Mathematics tuition supports accurate diagnosis
The immutable eduKateSG three-student Mathematics tutorial reference describes 1.5-hour weekly teaching near Sixth Avenue MRT, with attention to individual workings and targeted correction. In a close-feedback group, a tutor can see whether one student ignored the word “remaining”, another reversed the stages incorrectly and a third needs a more demanding algebraic extension.
The class size is useful only when those observations change the next teaching task. Parents should ask whether the child can solve a fresh fraction-of-remainder question independently a few days after the original explanation.
Frequently asked questions
Why do we multiply fractions in a fraction-of-remainder problem?
Because the second fraction is taken from a quantity already reduced by the first stage. Multiplication combines successive scaling relationships. The student must still identify each reference amount correctly.
Is working backwards always better than drawing a bar model?
No. A flow diagram can make reversed operations clear, while a bar model can make equal parts and changing wholes visible. The best method depends on the student and the story.
What is the biggest mistake in these PSLE word problems?
Many incorrect solutions apply the second fraction to the original amount rather than the remainder. Mark the stages, label the quantities and test the final answer by replaying the changes.
Can my child use algebra to solve fraction remainder questions?
Yes, when the learner understands the reference amounts and equation balance. An expression such as (3/5)(3x/4) = 54 can express the relationship compactly, but letters do not substitute for mathematical understanding.
How do I know Maths tuition has repaired this weakness?
Give an unfamiliar two- or three-stage story without naming the heuristic. The learner should label the changing whole, select a method, obtain the answer and check both the intermediate and final amounts independently.
The core aim is tracking the same whole as it changes
Fraction-of-remainder problems are not a collection of traps designed to catch children. They reward careful reading of quantities and a disciplined understanding of parts and wholes. When a pupil knows which whole is changing and can reconstruct the sequence, the arithmetic becomes a tool rather than a guess.
For further learning, explore before-and-after PSLE Maths word problems, fractions, ratio and percentage foundations and the twelve-week PSLE Maths revision guide. To enquire about current 3-pax tuition, contact eduKate Singapore with the child’s original worked problems.