External Change
What stays the same when amounts are added or removed from outside, and how to use that invariant.
⏱ 5 min · 🎯 4 things to master
You already know that when one person gives money to another, the total stays the same. But what if money is spent at a shop or earned? Then money leaves or enters from outside, so the total changes. These are external-change problems, and the skill is the same powerful question as always: what stays the same? — only now the answer is not the total.
Parents: let your child predict what stays the same before revealing. The "Method tip" boxes name the move a marker rewards — finding the quantity that does not change.
By the end you'll be able to spot when the total changes and use the invariant that remains. Let's find it.
When the total changes
When money is added from outside (earned, received) or removed to outside (spent, donated), the combined total is different before and after. So you cannot anchor on a — that trick only works for transfers between two people.
🤔 Predict first: Tom has $40 and Sara has $30. They each spend $10 at the canteen. What happens to their total?
Both change equally — the difference stays
Here is the most common PSLE version. Joseph had $900 and Ruth had $500. They each bought the same watch. After that, Joseph had 3 times as much as Ruth. How much was the watch? Because they spent the same amount, the difference between them never changed — it is 900 − 500 = $400 before and after. After the purchase Joseph is 3 units and Ruth is 1 unit, so the difference is 2 units = $400. Step one unit until the gap is $400.
Both buy the same watch
Predict first: They each spend the SAME amount. What stays the same?
With 1 unit = $200, Ruth has $200 left after buying. She started with $500, so the watch cost 500 − 200 = $300. (Check Joseph: 900 − 300 = $600 = 3 × $200.)
What if only one changes?
If only one person spends or receives, then the other person is unchanged — and that untouched amount becomes your anchor. The rule never changes: read the story, decide what is invariant (the total, the difference, or one quantity), and build your model around it.
🤔 Predict first: Ken has $60 and Lee has $40. Only Ken spends $15. Whose amount is unchanged?
Watch out — these are easily mixed up
Quick recap
🎯 Mastery check
Answer all 6 — your progress is saved on this device.
Two friends have $80 in total. Each spends $12 at the bookshop. What is their total now?
Amir has $50 and Ben has $30. They each receive the same gift of $20. What stays the same?
After both spend the same amount, one has 3 units and the other has 1 unit, with a difference of $60. What is one unit?
Mei has $70 and Lily has $90. Only Mei spends some money. Whose amount is the safe anchor?
Raj had $120 and Sam had $40. Each bought the same toy. After, Raj had 5 times as much as Sam. What is the difference between them?
In the question above, how much did the toy cost?