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A friendly flat-vector scene of two ratio bars with one section locked the same size in both, beside a padlock, in IllumiTutor navy and amber on an off-white background.

The Unchanged Quantity

Spotting the quantity that stays the same in a ratio problem and rescaling both ratios to match it.

⏱ 5 min · 🎯 4 things to master

When a ratio "changes", something usually causes it — and often one of the two quantities is left completely alone. That unchanged quantity is your secret weapon. If you rescale the before-ratio and the after-ratio so the unchanged quantity is drawn the same size in both, then whatever changed lines up perfectly and reveals the value of one unit.

Parents: let your child spot which quantity is untouched before revealing. The "Method tip" boxes name the move a marker rewards — making the unchanged quantity equal in both ratios.

By the end you'll be able to spot an unchanged quantity and rescale both ratios to match it. Let's lock it down.

Find what does not change

Read the story and ask: which quantity is not touched? In "Ali and Billy" problems where only Billy spends money, Ali is unchanged. That untouched amount must be the same number before and after — even though, annoyingly, the two ratios describe it with different numbers of units.

🤔 Predict first: Ali to Billy is 5 : 6. Then Billy spends some money and the ratio becomes 3 : 2. Whose amount did NOT change?

Rescale to match it

The before-ratio says Ali is 5 units; the after-ratio says Ali is 3 units — but it is the same money, so make those equal. The smallest number both 5 and 3 divide into is 15. Rescale: before becomes 15 : 18, and after becomes 15 : 10. Now Ali is 15 in both, locked.

Look at Billy: he went from 18 units to 10 units, a drop of 8 units. If Billy spent $16, then 8 units = $16, so 1 unit = $2. Billy now has 10 units = $20.

🤔 Predict first: Before is 15 : 18 and after is 15 : 10 (Ali locked at 15). Billy dropped from 18 to 10. How many units did Billy lose?

Which "constant" is it?

This sits in a family. If a transfer keeps the total the same, anchor the total. If both change equally, the difference is the same. If one quantity is untouched, lock that quantity. Spot the invariant first, then rescale to match it.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

  1. The ratio of red to blue beads is 4 : 5. Some blue beads are removed and it becomes 4 : 3. Which colour is unchanged?

  2. To lock a quantity that is 5 units in one ratio and 3 units in the other, rescale both to…

  3. After locking, one quantity drops from 18 units to 10 units. How many units changed?

  4. If that drop of 8 units equals $24, what is one unit?

  5. A is 2 units and B is 3 units. Later A is unchanged and the ratio is 4 : 5. To lock A, rescale the first ratio so A is…

  6. Why must you lock the unchanged quantity to the SAME number of units in both ratios?