Working Backwards: The PSLE Math Strategy for Tricky Problems
When a PSLE Math problem feels stuck, working backwards from the answer often unlocks it in seconds. Learn the 4-step method with 5 worked examples.
Working Backwards: The PSLE Math Strategy for Tricky Problems
Some PSLE Math problems look impossible from the front — too many unknowns, too many steps. But flip the question around and start from the end, and the same problem becomes a five-line answer. This is “working backwards”, and it is one of the most powerful heuristics on Paper 2.
Why Working Backwards Works
In a normal math problem, you read forward: given some starting amount → something happens → find the ending amount. You solve it by walking forward, step by step.
But many PSLE Paper 2 problems are written in reverse order: they tell you the ending amount and ask you to find the starting amount. Trying to solve those forward means juggling unknowns. Working backwards turns each unknown into a number you can actually calculate.
| Forward Problem | Backwards Problem |
|---|---|
| ”Sarah had $360. She spent ¼ on a bag, then ⅔ of the rest on shoes. How much is left?" | "Sarah spent ¼ on a bag, then ⅔ of the rest on shoes. She has $90 left. How much did she start with?” |
| Easy: just multiply | Looks hard: but reverse it and it becomes equally easy |
💡 The Mental Switch
Forward thinking asks: “What happens next?” Backwards thinking asks: “What had to be true just before this step?” Train yourself to ask the second question whenever you get stuck.
When to Use Working Backwards
Working backwards is not the right tool for every problem. Look for these five signal patterns:
| Signal | Example Phrase |
|---|---|
| 1. Question asks for the original / starting amount | ”How much did he have at first?“ |
| 2. The final amount is given as a number | ”He had $90 left.” |
| 3. A chain of operations happened in order | ”He spent… then gave… then bought…“ |
| 4. Each step depends on the remainder of the previous | ”Then ⅔ of what was left…“ |
| 5. Reverse arithmetic is easy | The operations can be cleanly undone |
If three or more of these match, working backwards will usually be faster than algebra or guess-and-check.
The 4-Step Working Backwards Method
ℹ️ The Method
Step 1. Write the final amount as your starting point.
Step 2. Identify the last operation that happened (the one closest to the end).
Step 3. Undo that operation (multiply ↔ divide, add ↔ subtract).
Step 4. Repeat for each earlier operation, moving step by step toward the start.
The key insight: you do not need to know all the values upfront. Each backwards step gives you the next number to work with.
The “Undo” Table
| Forward Operation | Backwards Operation |
|---|---|
| Add | Subtract |
| Subtract | Add |
| Multiply by | Divide by |
| Divide by | Multiply by |
| Spent , kept | Divide by to recover the previous total |
That last row is the one PSLE students miss most often. We will use it in Examples 1 and 4.
Example 1: Classic Fraction Remainder
Example 1: Sarah's Shopping
Problem:
Sarah spent of her money on a bag, then of the remainder on shoes. She had $90 left. How much money did she have at first?
Working backwards from $90:
Step 1 (undo the shoes). She spent of the remainder on shoes, so the \frac13$ of the remainder.
Step 2 (undo the bag). She spent of her original money, so \frac34$ of the original.
✅ Sarah had $360 at first.
💡 Notice What We Avoided
We never had to write a single equation with . We never had to guess. Each number we computed was a real value (not an unknown), so checking is easy: , and . ✓
Example 2: Before-After Transfer
Example 2: David's Marbles
Problem:
David gave 18 marbles to his sister. Then he lost half of what he had left. Then he bought 5 more marbles. He now has 23 marbles. How many marbles did David have at first?
Working backwards from 23 marbles:
Step 1 (undo “bought 5 more”). Subtract 5.
Step 2 (undo “lost half”). Half was lost, so 18 is the other half. Multiply by 2.
Step 3 (undo “gave 18 to sister”). Add 18 back.
✅ David had 54 marbles at first.
⚠️ The Order Matters — Reverse It Carefully
The forward order was: gave → lost → bought. The backwards order is the exact reverse: un-bought → un-lost → un-gave. Many students try to “undo” the operations in the original (forward) order — that gives the wrong answer every time.
Example 3: Money Across Three Days
Example 3: Mei's Allowance
Problem:
On Monday, Mei spent half her money on a book. On Tuesday, she spent \frac13 of what she had left on a notebook. She had \8 left at the end. How much money did she have on Monday morning?
Working backwards from $8:
Step 1 (undo Wednesday). She spent of what she had, so $8 = of what she had on Wednesday morning.
Step 2 (undo Tuesday). Add back the $12 she spent on lunch.
Step 3 (undo Monday). She spent half her money, so $24 is the other half.
✅ Mei had $48 on Monday morning.
Example 4: Multi-Stage Fraction Remainder
Example 4: Three Boxes of Cookies
Problem:
A baker sold of his cookies in the morning. He sold of the remainder in the afternoon. He gave 6 of what was left to his neighbour and was left with 18 cookies. How many cookies did he bake?
Working backwards from 18 cookies:
Step 1 (undo the gift). Add the 6 cookies given away.
Step 2 (undo the afternoon). He sold of the remainder, so 24 = of what he had after the morning.
Step 3 (undo the morning). He sold of the original, so 36 = of the original.
✅ The baker baked 60 cookies.
💡 Always Check by Going Forward
Once you have the answer, walk it through the forward direction to check. Start with 60: morning sold, leaving 36. Afternoon sold, leaving 24. Gave away 6, leaving 18. ✓ Matches the question — the answer is correct.
Example 5: A “Doubled Plus” Pattern
Example 5: Three Stalls
Problem:
Mr Lim went to a market with some money. At the first stall, he spent half his money plus 4. At the third stall, he spent half of what was left plus 0 left. How much money did he start with?
Working backwards from $0:
Step 1 (undo Stall 3). He spent half + $4 and ended with $0. So just before the +$4 step, he had $0 + $4 = $4. That $4 must have been the other half of what he had at Stall 3.
Step 2 (undo Stall 2). Same pattern: $8 + $4 = $12, then $12 × 2 = $24.
Step 3 (undo Stall 1). $24 + $4 = $28, then $28 × 2 = $56.
✅ Mr Lim started with $56.
ℹ️ Why This One Is So Hard Forward
Try solving Example 5 going forward: “Let x be his money. Then he had left, then …” You end up with nested fractions and a messy equation. Backwards, it is a string of additions and doublings — the kind of arithmetic you can do without a pencil.
When NOT to Use Working Backwards
Working backwards is powerful, but it is not always the fastest tool. Skip it when:
| Situation | Better Method |
|---|---|
| The final amount is unknown (you are asked to find it) | Work forward |
| There is only one operation | Just compute directly |
| The problem involves two unknowns at once (e.g. before-after ratio with two people) | Bar model with constant total / difference |
| Each step requires non-reversible operations (e.g. “rounded to the nearest 10”) | Algebra or guess-and-check |
| Speed, distance, time problems with movement | Speed table + formula |
A useful rule of thumb: if the question gives you an amount at the end and asks for an amount at the start, try working backwards first. If it gives you an amount at the start and asks for an amount at the end, work forward.
5 Common Mistakes
❌ Mistake 1: Reversing Operations in the Wrong Order
Many students undo operations in the same order they happened. They must be undone in reverse order — last operation first. Forward: A → B → C. Backwards: undo C → undo B → undo A.
❌ Mistake 2: Using Multiplication Instead of Division for Fractions
When the problem says “spent of the remainder”, the amount left = of that remainder. To recover the remainder, divide by (or multiply by 3) — do not multiply by . Many students do this in the wrong direction and get a smaller answer than the original.
❌ Mistake 3: Confusing 'Half of Total' with 'Half That Was Left'
“He spent half plus $4 at each stall” means half of what he currently had, not half of the original. The amount he had keeps shrinking, and so does each “half”. Always re-read the question to see which whole the fraction refers to.
❌ Mistake 4: Forgetting the '+/− Constant' Step
In Example 5, the operation was “spent half plus $4”. The ”+$4” must be undone (subtracted) before undoing the “half” (multiplying by 2). Skip the +$4 step and the entire chain breaks.
❌ Mistake 5: Not Checking by Going Forward
Always re-substitute your answer into the original question and walk forward. Working backwards is reliable, but a small arithmetic slip can throw the whole chain off. The forward check takes 30 seconds and catches almost all errors.
5-Day Practice Plan
Work through one type per day. Time yourself — aim for under 5 minutes per problem by Day 5.
| Day | Focus | Practice |
|---|---|---|
| Day 1 | Single-step undo (add/subtract, multiply/divide) | 5 simple “started with x” problems |
| Day 2 | Two-stage fraction remainder | 5 problems like Example 1 |
| Day 3 | Mixed operations (give away, spend, buy) | 5 problems like Example 2 |
| Day 4 | Three-stage problems with fractions | 5 problems like Examples 3 & 4 |
| Day 5 | Mixed exam-style + the “half plus constant” pattern | Past PSLE Paper 2 questions |
💡 A Practice Tip Most Students Miss
After solving each problem, write the forward version of your work as a one-line check. This trains both forward and backward thinking from the same problem — and it is the easiest way to catch silly errors before the marker does.
Quick Reference Card
The 4-Step Working Backwards Method
- Start with the final amount given.
- Identify the last operation that happened.
- Undo it (reverse the arithmetic).
- Repeat — earlier operations next, in reverse order.
Operation Reversals
- Add ↔ Subtract
- Multiply by ↔ Divide by
- “Spent ” → divide remaining amount by
- “Half plus ” → subtract , then multiply by 2
Always check forward before moving on.
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