Problem 13 · 1989 AJHSME
Hard
Fractions, Decimals & Percents
proportional-scaling
97 × 53=
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Answer: A — .9 ⁄ (.7 × 53).
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Hint 1 of 3
A fraction keeps its value only when the top and the bottom are shrunk (or grown) by the SAME factor — multiplying top and bottom by the same thing changes nothing.
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Hint 2 of 3
Look at each choice as 'what got multiplied by 0.1?' The original is 9/(7×53); track how many ×0.1's land on top versus on the bottom.
Still stuck? Show hint 3 →
Hint 3 of 3
9 → .9 is one ×0.1 on top. To stay equal you need exactly one matching ×0.1 somewhere on the bottom — and 7 → .7 supplies it while 53 stays put.
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Approach: count the ×0.1 factors on top vs. bottom
- The starting fraction is 9/(7×53). Every choice turns the 9 into .9, which is one factor of 0.1 on top. For the value to be unchanged, the bottom must pick up exactly one matching factor of 0.1 too.
- Choice A is .9/(.7×53): top got ×0.1, and 7→.7 is ×0.1 on the bottom while 53 is untouched. Same ×0.1 top and bottom cancels, so .9/(.7×53) = 9/(7×53). That's the match — choice A.
- Every other choice puts a different amount of shrinking on the bottom: B and E shrink two bottom numbers (too much), C turns 53→5.3 (only ×0.1 but value still off because... ) — quick test: in C the bottom is .7×5.3, which is ×0.1 on the 7 AND ×0.1 on the 53, so the bottom shrank by ×0.01 total versus ×0.1 on top, changing the value.
- Why this transfers: to keep a fraction's value, the ×0.1 (or ×10) factors on top and bottom must balance exactly — count them like matching pairs. This 'scale top and bottom the same' rule is the engine behind every equivalent-fraction problem.
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