Changing ratios: find what stays constant

These are the ratio questions that separate AL1 from AL3. A ratio is given, something changes, a new ratio is given — and the units from the two ratios are different sizes, so you cannot compare them directly.

The method is always the same: find the quantity that didn’t change, and make it the same number of units in both ratios.

The three constants

Nearly every changing-ratio question has one of these hiding in it.

1. One quantity is unchanged. Only one person spends or receives; the other’s amount is untouched.

2. The total is unchanged. Something is transferred between two people — nothing enters or leaves, so the total is fixed.

3. The difference is unchanged. Both quantities change by the same amount — both save $20, both spend $15 — so the gap between them stays put.

Read the question and decide which applies before writing anything. That decision is the problem.

Worked example — one quantity unchanged

The ratio of Ali’s savings to Bala’s savings was 2 : 5. Bala then spent $60, and the ratio became 2 : 3. How much did Ali save?

Only Bala spends, so Ali is unchanged.

Ali is 2 units in the first ratio and 2 units in the second. They’re already the same number, so one unit means the same thing throughout — and Bala’s change is readable straight off the drawing:

Bala before = 5u
Bala after  = 3u
Bala spent  = 2u = $60
       1u   = $30

Ali saved 2u = $60.

Check: Ali $60, Bala $150 → 60 : 150 = 2 : 5 ✓. Bala spends $60 → $90. 60 : 90 = 2 : 3 ✓

When the units don’t already match

Usually you have to force the constant quantity to the same number of units.

The ratio of red to blue counters was 3 : 4. After 18 more red counters were added, the ratio became 3 : 2. How many blue counters are there?

Blue is unchanged, but it’s 4 units in the first ratio and 2 in the second. Multiply the second ratio to make blue match — multiply by 2:

Before  red : blue = 3 : 4
After   red : blue = 3 : 2  →  ×2  →  6 : 4

Now blue is 4u in both, so units are comparable.

Red before = 3u
Red after  = 6u
Added      = 3u = 18
      1u   = 6

Blue = 4u = 24 counters.

Check: red 18, blue 24 → 18 : 24 = 3 : 4 ✓. Add 18 red → 36 : 24 = 3 : 2 ✓

Worked example — the total unchanged

Priya and Qi shared some sweets in the ratio 7 : 5. Priya gave Qi 12 sweets, and they then had equal numbers. How many sweets were there altogether?

Sweets are transferred, so the total is unchanged: 7 + 5 = 12u throughout.

Equal shares means 6u each afterwards. Priya went from 7u to 6u, so she gave away 1u.

1u = 12
Total = 12u = 144 sweets

Check: Priya 84, Qi 60. Priya gives 12 → 72 and 72 ✓

Common mistakes

• Comparing units across ratios without matching the constant. 1u in the first ratio is a different size from 1u in the second until you fix it.
• Assuming the total is constant when it isn’t. Money spent leaves the system; money given to each other doesn’t.
• Picking the wrong constant. If the numbers come out fractional, you’ve probably chosen wrongly — check the other two before grinding on.
• Using the same letter for both sets of units. Write u and p.

Quick check

The ratio of Sara’s money to Tom’s was 5 : 2. Each of them then saved another $30, and the ratio became 8 : 5. How much did Sara have at first?

Before: 5u − 2u = 3u
After:  8p − 5p = 3p

3u = 3p, so u = p
   (the units are already the same size)

Tom: 2u + 30 = 5u
          3u = 30
          1u = 10

Sara = 5u = 5 × 10