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Solving Problems Involving Ratios
Rung 3 of 4 · The traps

When the Ratio Plays Tricks

Two slips catch nearly everyone: reading “a:b” as a fraction, and being handed one share instead of one part. Spot them once and they stop fooling you.


CompareFlip between the two traps. Each shows the slip beside the truth — hit “why?” for the reasoning.
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The parts method is reliable — but two misreadings sneak past people constantly. Both come from confusing what the numbers in a ratio actually stand for.

Trap One: "a : B" Is Not "a Out of B"

If the ratio is 3 : 2, the first share is not 3/2 of the whole — and it's not 3/2 at all, since that's more than one. The denominator of the fraction is the total parts, which is 3 + 2 = 5. So the first share is 3/5 of the whole and the second is 2/5. The toy puts the slip right next to the truth so you can see the missing parts.

Say it plainly: the bottom of the fraction is a + b, never just b. The ratio counts parts; the fraction is "your parts out of all the parts".

Trap Two: Given One Share, Not One Part

Sometimes a question hands you a share and asks for the rest. "In the ratio 3 : 2, the first share is 18 — find the other." The slip is to treat 18 as the value of one part and multiply it up. But 18 is 3 parts, so first undo that: 18 ÷ 3 = 6 per part. Now the other share is 2 × 6 = 12, and the whole is 5 × 6 = 30. This is the "unitary" move — always get back to one part before you scale.

Your Shares Must Add Back

Both traps get caught by the same safety net from rung 1: the shares have to add back to the whole. If you'd said "18 is one part, so the whole is 5 × 18 = 90", that's a giant red flag — the first share alone was only 18. Whenever an answer feels too big or too small, add the shares and see if they rebuild the total.

Us, Thinking Out Loud

Why is the fraction "3 out of 5", not "3 out of 2"?

How would you spot, fast, that an answer can't be right?