Mathematics Tutor Singapore | PSLE Working Backwards

Mathematics Tutor Singapore | PSLE Maths word problems sometimes tell you the final amount, then describe what was spent or given away earlier. A child may try to guess the starting number repeatedly. A more systematic approach is to reverse the story—one stage at a time—while keeping clear which fraction refers to which amount.

Did you know that if half of a collection is left at the final stage, reversing that stage means doubling the final number? If thirty objects were removed just before that, add thirty next. Working backwards uses ordinary inverse arithmetic; the challenge is preserving the correct order.

This Singapore Primary 5–6 Maths tutor guide develops working backwards as a problem-solving heuristic using fractions, fixed amounts, money, remainders and forward checks. The examples are original learning problems, not a claim that a particular invented puzzle will appear in a PSLE examination.

Browse the working-backwards word-problem chapters

Working backwards means reversing one stage at a time
A timeline separates the changing amounts
Inverse operations are the engine of the method
Worked example: spend one-fourth then one-third of what remains
Why two fractions may refer to different wholes
Worked example: remove 20, use a fraction, remove 15
Three-stage sweets puzzle with 25 left
Savings, spending and what remains
How to undo a fraction of a remainder
When backwards reasoning meets ratio
Choose backwards, bar models or a forward equation
Forward checks are not optional
Nine common backwards-method mistakes
Sixteen original working-backwards questions and answers
A 10-session PSLE problem-solving practice plan
How a Maths tutor diagnoses the first wrong stage
Frequently asked PSLE working-backwards questions
MOE syllabus context and related Mathematics routes

Working backwards means reversing one stage at a time

Imagine a number machine. It starts with an unknown number, adds five and then doubles the result. The final number is thirty. We can recover the starting number by undoing the last action first: divide thirty by two to obtain fifteen, then subtract five to obtain ten.

The forward check is equally simple: start with ten, add five to make fifteen and double to obtain thirty. The reverse process works because addition and subtraction undo each other, and multiplication by a nonzero number and division undo each other.

Forward action	Reverse action	Illustration
Add 5	Subtract 5	10→15; 15→10
Subtract 7	Add 7	20→13; 13→20
Multiply by 3	Divide by 3	6→18; 18→6
Take 2/5 of a quantity	Multiply by 5/2	40→16; 16→40

Reverse the operation and the order of stages.

This sounds easy in a short number machine, yet PSLE word problems conceal the same actions within a story. The pupil’s first task is not calculation. It is to identify the chronological stages and state exactly what changed at each one.

Find the final amount supplied in the story.
Mark each operation in the order it happened.
Name the current whole before every fractional operation.
Reverse the last stage first, then the previous stage.
Keep intermediate quantities and units on a table or bar diagram.
Check the recovered starting amount by following the story forwards.
Reject a proposed answer that violates a nonnegative or whole-object condition.

A timeline separates the changing amounts

A useful diagram reads from left to right as the story unfolds, and right to left when we reverse it. For example: original amount → spend one-quarter → spend one-third of the remainder → final amount. The original whole and the remaining amount after Stage 1 are not necessarily equal.

I draw boxes for the quantities between stages. The words “of what remains” are written above the second arrow, because that phrase changes the base of the fraction. A strong solution never applies both fractions automatically to the original amount.

A timeline also prevents a confusing mixed-up calculation. If the story says “give away 20, then halve what remains, then give away 15”, the backwards operations are “add 15, double, add 20”—not necessarily in the original order.

Inverse operations are the engine of the method

The inverse of removing twenty objects is adding twenty back. The inverse of multiplying the quantity by three is dividing by three. The inverse of retaining three-quarters of a quantity is dividing the retained amount by three-quarters, or multiplying by four-thirds.

Forward step	If final stage amount is y	Original stage amount
Add 12	y=x+12	x=y−12
Take away 12	y=x−12	x=y+12
Keep 3/4	y=(3/4)x	x=(4/3)y
Keep 2/5	y=(2/5)x	x=(5/2)y

Fractions in a backwards calculation describe the retained share of the immediate earlier amount.

A pupil who reads “one-quarter was spent” must recognise that three-quarters was retained. Reversing from the remainder requires dividing by three-quarters, not by one-quarter. This is the most important conceptual step for many two-stage fraction stories.

Worked example: spend a quarter, then a third of what remains

A fictional student starts with an unknown amount of pocket money. They spend one-quarter of it on a book. Next they spend one-third of the remaining money. They have $90 left. How much did they have originally?

Stage 2 spent one-third of the amount available then, so the final $90 represents two-thirds of the amount after Stage 1. Reverse it: $90 ÷ (2/3) = $90×3/2 = $135 after Stage 1.

Stage 1 spent one-quarter of the original amount, leaving three-quarters. Reverse this: $135 ÷ (3/4) = $135×4/3 = $180 originally.

Story stage	Forward relation	Work backwards
Original money	Unknown	$180 recovered
After one-quarter spent	Retain 3/4 of original	$135 = (3/4)×$180
After one-third of remainder spent	Retain 2/3 of $135	$90 = (2/3)×$135
Final	$90 given	Begin reversal here

Reverse the last fractional stage before the first.

Check forwards: one-quarter of $180 is $45, leaving $135. One-third of $135 is $45, leaving $90. The starting amount and both stage amounts agree with every sentence.

An elegant alternative is to combine the retained shares: (3/4)×(2/3)=1/2 of the original remains. Thus $90 is half the original and the start is $180. This shortcut is correct here, but I teach the two-stage interpretation before asking pupils to multiply fractions.

Two fractions may refer to different wholes

The first fraction “one-quarter of the original” uses the initial amount as its whole. The second “one-third of the remainder” uses the changed amount after the first operation. Applying one-third directly to the original would produce a different story.

Suppose the original amount is indeed $180. One-quarter of it is $45. One-third of the original would be $60, whereas one-third of the $135 remainder is $45. Those two quantities are not equal.

Phrase	Reference whole	At start $180
1/4 of original	$180	$45
1/3 of remainder after first spend	$135	$45
1/3 of original	$180	$60

The reference whole is decided by the words, not by the most prominent number.

I ask students to underline “remaining”, “of those left” and “of the original” with different markings. A correct fraction calculation on the wrong reference whole still gives the wrong model.

Worked problem: remove 20, use a fraction, then remove 15

A box originally contains an unknown number of counters. First, twenty counters are taken out. Next, one-third of the counters then remaining are used. Finally, another fifteen counters are removed. There are twenty-five left. How many were in the box originally?

Reverse the last action: add fifteen to the final twenty-five, giving forty counters before Stage 3. The fraction stage left two-thirds of the amount available just before it. Divide forty by two-thirds: forty×3/2=60 counters just before Stage 2.

Reverse the first action: add twenty to sixty to obtain 80 counters originally.

Backwards step	Calculation	Amount before that stage
Undo final removal of 15	25+15	40
Undo spending 1/3 of remainder	40÷(2/3)	60
Undo first removal of 20	60+20	80

Undo fixed subtraction and fractional reduction in reverse order.

Check forwards: 80−20=60. One-third of sixty is twenty, leaving forty. Forty−15=25. The story ends at the given amount, so the starting answer is consistent.

Three-stage sweets puzzle: twenty-five left

A fictional sweet jar begins with an unknown number of sweets. One-fifth are given away. Thirty more sweets are then given away. Finally half of the sweets then remaining are given away, leaving twenty-five. What was the original number?

Reverse the final half: immediately before it, there must have been 25×2=50 sweets. Add back the fixed thirty, obtaining 80 sweets immediately after the first stage.

The first stage gave away one-fifth of the original, leaving four-fifths. Therefore four-fifths of the original is eighty. The full original is 80×5/4=100 sweets.

Check: one-fifth of one hundred is twenty, leaving eighty. Subtract thirty, leaving fifty. Give away half of fifty, leaving twenty-five. Every count is a feasible whole number.

Savings, spending and what remains

A learner saves an unknown sum of money. They spend 40% of it, then add $24 from a gift, and have $84. Work backwards: remove the gift, 84−24=60. The $60 represents 60% of the original, so original=60÷0.6=$100.

Check forwards: 40% of $100 is $40, leaving $60. Add the $24 gift and reach $84. The gift was an increase, so its inverse is subtraction; the spending was a reduction, so its inverse is division by the remaining percentage.

This illustration is a bridge from Primary percentage foundations. It does not rely on an actual savings product or real-world rate. If the context were a financial service, extra fees, interest or taxes could change the model.

How to undo a fraction of a remainder

Suppose five-sevenths of a quantity remains after some items were removed. If the final number is forty-five, the amount just before the removal is 45÷(5/7)=45×7/5=63.

The quick unit method is also transparent: five equal units correspond to forty-five, so one unit is nine and seven units total sixty-three. This is one reason bar models remain valuable even when a calculator is available.

What if “five-sevenths were removed” rather than “five-sevenths remain”? Only two-sevenths remains. The inverse multiplier becomes seven-halves rather than seven-fifths. A small language change can make a large numerical difference.

When working backwards meets ratio

A ratio describes parts of a total. If the final group contains boys and girls in ratio 2:3 and there are twenty girls, then each ratio unit represents 20÷3, which is not a whole number. That result signals either a context allowing fractional quantities or a problem whose stated integer counts are incompatible.

For a feasible example, a class has boys:girls=2:3 and thirty girls. One unit equals ten pupils, so boys number twenty and the total is fifty. If ten new pupils joined after this stage, the earlier total would be forty; reversing an addition is straightforward only when the ratio is clearly associated with the right stage.

I teach this as a lesson in keeping conditions attached to the correct moment. A ratio from the end of a story does not necessarily describe the group before transfers or arrivals.

Choose backwards, bar models or a forward equation

Working backwards is effective when the ending amount is known and operations are reversible stage by stage. A bar model is useful when fractions of a whole or changing subgroups need to be seen. A forward equation may be more efficient if the relationships are simple enough.

For the eighty-counter example, a forward equation using N as the start is (2/3)(N−20)−15=25. Add fifteen, multiply by three-halves, add twenty and recover N=80. The algebra performs the same reverse operations in another notation.

Method	Why it helps	Best initial learner
Backwards timeline	Shows reverse order	Student comfortable with four operations
Bar model	Shows fractions of changing wholes	Student needing visual parts
Forward equation	Compresses several stages	Student ready for algebra
Guess and check	Can verify candidates systematically	Student evaluating feasibility

The best method is justified and checkable, not necessarily the longest.

Forward checks are not optional

A successful reverse calculation should always be followed by a forward check. Write the proposed starting amount, perform Stage 1 exactly as described, update the reference amount, and continue until the final stated total appears.

This check is more than repeating arithmetic. It tests whether the model respected each word in the question. If a fraction was meant to apply to what remained but the forward check instead uses the original, the two interpretations will not agree.

A plausible-looking whole number is not sufficient when the story involves indivisible objects. Each stage must also produce valid counts. The solved value should not require half a pupil or a negative number of boxes unless the question has made a divisible measure explicit.

Nine common backwards-method mistakes

First wrong move	Likely issue	Focused repair
Reverses the first story step first	Order not reversed	Draw three chronological boxes
Undoing a subtraction by subtracting again	Inverse operation confused	Test a short number machine
Uses spent fraction instead of retained fraction	Remainder relationship missed	Shade what is left
Applies second fraction to original	Changing reference whole ignored	Label every intermediate whole
Halves instead of doubles to undo halving	Inverse multiplication wrong	Try a simple eight-to-four example
Leaves fixed gift or fee in wrong stage	Timeline not interpreted	Mark adds and removals
Starts with final 25 and guesses freely	No systematic strategy	Create reverse-action table
Finds fractional pupils despite whole-person story	Feasibility ignored	Check all stage counts
Skips checking recovered start forwards	Verification omitted	Reconstruct given final total

One incorrect relationship can change every subsequent amount.

Two children may get the same wrong start for completely different reasons. One can manipulate fractions but ignores the reference whole. Another interprets the story correctly but divides by the wrong fraction. A tutor should identify the first failed step.