Problem 24 · 1990 AJHSME
Stretch
Algebra & Patterns
substitutionbalance
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Answer: C — 3.
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Hint 1 of 2
A balance scale is just an equals sign you can see. Write down what each picture says: 3 triangles + 1 diamond = 9 circles, and 1 triangle = 1 diamond + 1 circle.
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Hint 2 of 2
The first balance has triangles in the way, but the second tells you exactly what one triangle is *made of*. Swap every triangle for 'a diamond and a circle' — substitution — and only diamonds and circles are left.
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Approach: substitute the simpler balance to clear out the triangles
- Read the scales as equations: 3T + D = 9C, and T = D + C. You want how many circles equal two diamonds (2D).
- The second balance lets you replace each triangle by 'one diamond + one circle.' Swap all three triangles in the first balance: 3(D + C) + D = 9C, i.e. 4D + 3C = 9C.
- Take the 3C off both sides (remove 3 circles from each pan): 4D = 6C. Cut everything in half: 2D = 3C. So two diamonds balance 3 circles.
- *Why this transfers:* when one unknown is given in terms of others, substitute it in to wipe that unknown out — and on a balance you can always add or remove the same thing from both pans without tipping it.
Another way — stay concrete — double the simpler balance:
- From the right scale, 1 triangle = 1 diamond + 1 circle, so 3 triangles = 3 diamonds + 3 circles.
- The left scale says 3 triangles + 1 diamond = 9 circles. Replace the 3 triangles: (3 diamonds + 3 circles) + 1 diamond = 9 circles, so 4 diamonds + 3 circles = 9 circles.
- Remove 3 circles from each side: 4 diamonds = 6 circles. Halve it: 2 diamonds = 3 circles — exactly the two diamonds in the question.
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