Working Backwards in Maths: Undo the Steps, Last One First

18 October 2026 · by Larry

A child who can do “I had $30, I spent $12, how much is left?” without a pause can still be stopped by the same story told the other way round: “I have $18 left after spending $12, how much did I start with?” The numbers are the same. The sum is not. This piece is about those backwards questions, why they feel so different, and the one habit that fixes most of the mistakes.

Why backwards questions feel different

Nearly everything a child has done in maths so far goes forwards. You are given the start, you follow the story, you reach the end. A backwards question gives the end and asks for the start, and the instinct to read left to right and do what the sentence says is exactly the wrong move.

You will see these as “I think of a number…” puzzles, as money that was spent in stages, as a fraction of the pages already read, and as a price after a discount. They look like different topics. They are the same skill: undo the story, in reverse order, using the opposite of each step.

The one idea: undo in reverse, with the opposite

Here is a made-up example. Mei thinks of a number, multiplies it by 4, then adds 6. The answer is 38. What was her number?

Going forwards the story was: number → × 4 → + 6 → 38. To undo it, walk back along the same chain, last step first, and use the opposite of each step. The last step was “+ 6”, so its opposite is “− 6”: 38 − 6 = 32. Before that was “× 4”, so its opposite is “÷ 4”: 32 ÷ 4 = 8. Her number was 8.

Then check forwards: 8 × 4 = 32, and 32 + 6 = 38. It matches, so it is right.

That check is worth more than it looks. A backwards question is easy to turn upside down, and a forward check takes ten seconds and catches nearly every slip.

Slip 1: undoing in the order the story was told

In Mei’s example, a child who sees “multiplies by 4, then adds 6” and reaches for the 4 first writes 38 ÷ 4 = 9.5, then 9.5 − 6 = 3.5. It is a tidy-looking answer and it is wrong. Put on your socks and then your shoes, and you must take off your shoes first. Undoing runs in the reverse order, and that one sentence is worth saying out loud.

A child who gets a strange answer, such as a decimal where the story is about whole sweets or whole people, has often undone the steps in the wrong order.

Slip 2: copying the step instead of reversing it

Another made-up one. Aiden had some money. He spent $12 on a book, then spent half of what was left on lunch. He had $9 left. How much did he start with?

Going forwards: start → − $12 → half → $9. Backwards, the last step was “half of what was left”, so its opposite is “double”: $9 × 2 = $18, which is what he had before lunch. Before that he spent $12, so add it back: $18 + $12 = $30. He started with $30.

Check forwards: $30 − $12 = $18, and half of $18 is $9. It matches.

The slips here: halving the $9 again because the word “half” is in the sentence, or taking the $12 off instead of adding it back. Both come from copying what the sentence says instead of reversing it. Asking “what did this step do to the number, and what would put it back?” is the cure.

Slip 3: a fraction of the wrong whole

This is the one that costs the most marks. Siti read 1/3 of a book on Saturday and 1/2 of the pages that were left on Sunday. She still has 20 pages to read. How many pages are in the book?

Sunday’s “half of the pages that were left” means the 20 pages still to read are the other half, so 40 pages were left on Saturday night. Saturday’s reading was 1/3 of the book, so those 40 pages are 2/3 of it. If 2 parts are 40 pages, 1 part is 20, and the whole book (3 parts) is 60 pages.

Check forwards: 1/3 of 60 is 20, so 40 are left; half of 40 is 20 read on Sunday; 20 are left. It matches.

The slip is treating the 1/2 as half of the whole book, when it is half of what was left. A bar model (draw the book as three equal pieces, shade the first one, then split the rest in half) makes the two different wholes visible; the guide to bar models shows how.

Slip 4: a percentage of the wrong number

A bag is on sale for $48 after a 20% discount. What was the price before? A child who takes 20% of $48 and adds it back gets $57.60. That feels backwards enough to be right, but the 20% was taken off the original price, not off $48.

The way in: $48 is what is left after 20% came off, so $48 is 80% of the original. If 80% is $48, then 10% is $6 and 100% is $60. Check forwards: 20% of $60 is $12, and $60 − $12 = $48. It matches. The thing to hunt for is which number is 100%, and in a backwards question that is the number you do not have yet. The same trap sits in ratio and percentage more generally.

What helps

• Draw the chain of steps forwards as boxes and arrows, with the known end on the right and a question mark on the left. Then draw the arrows back the other way and write the opposite on each.
• Say “last step first” every time. It is the same idea as the socks and shoes.
• Write the opposite beside each step: × becomes ÷, + becomes −, “half” becomes “double”.
• Always check forwards. Start from the answer you found, follow the story, and see if you land on the number in the question.
• Ask whether the answer is sensible. If the start came out smaller than the end after nothing but spending, something went the wrong way.
• Ask “a fraction (or percentage) of what?” before touching any fraction or percentage in a story.

What does not help

• Telling your child to “just do the opposite” with no chain to follow. Without the chain, the opposite gets applied to the wrong step.
• Trial and error with random starting numbers. It sometimes works on a small puzzle and falls apart on a fraction or percentage question.
• Skipping the check because the answer “looks right”. A wrong backwards answer often looks perfectly tidy.

A ten-minute check you can do at home

Ask your child to draw the chain, work backwards, then check forwards.

• I think of a number, subtract 5, then multiply by 3. The answer is 36. What was my number?
• Ravi gave away 1/4 of his stickers, then lost 3 more. He has 15 left. How many did he start with?
• A bag costs $72 after a 25% discount. What was the price before the discount?

The answers: 17 (36 ÷ 3 = 12, then 12 + 5 = 17); 24 (15 + 3 = 18, which is 3/4 of the stickers, so 1/4 is 6 and the whole is 24); and $96 ($72 is 75% of the original, so 25% is $24 and 100% is $96). If a mistake shows up, find which step was undone in the wrong order or against the wrong whole, and practise that one.

Where this fits

Backwards questions get more demanding as children go up: the jump from P4 to P5 is where many families first notice them, and word problems in general have a lot in common with them. The free maths practice has a Think It Through topic at Primary 6 that covers working backwards, and each question explains the answer and shows the forward check. Which level learns what is set by the school and the MOE syllabus (moe.gov.sg), so use your child’s own worksheets for the real wording.