How to solve balance scale puzzles
Balance scales: walk the chain of ratios
'2 ▲ balance 3 ■' is a weight ratio in disguise. Chain the ratios from the shape you are given to the shape that is asked — and mind the direction at every link.
Updated
A few pan balances between kinds of shapes, then a question: so-many of one shape balance how many of another? The premises form a chain — each shares a shape with the next — and substitution walks you from the asked shape to the target. Every answer comes out a whole number; if yours doesn't, a link went wrong.
Read a balance as a ratio — the right way round
If 2 ▲ balance 3 ■, then each ▲ is the HEAVIER shape: one ▲ weighs 3⁄2 of a ■. Fewer on a pan means heavier apiece. Reading the ratio upside-down is this family's signature mistake, and the upside-down final answer is usually among the options.
Substitute through the shared shape
Convert step by step: turn the asked count of the first shape into its worth in the second, then the second into the third. Keep exact fractions mid-chain — 5 ▲ may be 7½ ■ on the way to 15 ● — and let the whole number appear at the end, as it must.
Or assign weights
Give the first shape a convenient weight, derive every other shape's weight from the premises, then divide the two weights in question. It is the same arithmetic with less to hold in your head — use whichever form you can check faster.
The strategy
- Find the premise containing the asked shape; start there.
- Write each balance as 'one X = so-much Y' before chaining.
- Check the direction at every link: fewer on the pan = heavier apiece.
- Confirm the final count is whole — a fraction means a flipped or skipped link.
Common traps
- The inverted ratio — the single most common miss.
- Chaining two shapes that never share a premise directly.
- Dropping a factor mid-chain and landing one step off — the near-miss options are built for exactly that.
- Answering in the starting shape's count instead of the asked shape's.
Speed · ~70s/item: Roughly 40–110 seconds by chain length. One careful pass through 2–4 links beats any amount of staring at the picture.
Try it yourself: 4 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
Every shape of the same kind weighs the same. Using the balanced scales, how many shapes are needed to balance the last one?
Every scale is balanced. How many ● balance the shapes on the last scale?
Reveal the answer
The answer is D — 6.
Show why — the worked reasoning
Every shape of the same kind weighs the same. Using the balanced scales, how many shapes are needed to balance the last one?
Every scale is balanced. How many ▲ balance the shapes on the last scale?
Reveal the answer
The answer is E — 9.
Show why — the worked reasoning
Every shape of the same kind weighs the same. Using the balanced scales, how many shapes are needed to balance the last one?
Every scale is balanced. How many ■ balance the shapes on the last scale?
Reveal the answer
The answer is E — 3.
Show why — the worked reasoning
Every shape of the same kind weighs the same. Using the balanced scales, how many shapes are needed to balance the last one?
Every scale is balanced. How many ▲ balance the shapes on the last scale?
Reveal the answer
The answer is B — 9.
Show why — the worked reasoning
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Questions people ask
Will balance scales questions be on the free test?
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