Problem 3 · 1995 AJHSME
Medium
Fractions, Decimals & Percents
fraction-operations
Which of the following operations has the same effect on a number as multiplying by 34 and then dividing by 35?
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Answer: E — multiplying by 5/4.
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Hint 1 of 2
Dividing by a fraction is the same as multiplying by its flip. So 'divide by 3/5' can become 'multiply by 5/3' — now everything is multiplication.
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Hint 2 of 2
Two multiplications in a row are really just one. Combine them into a single multiplier and see which answer it matches.
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Approach: turn dividing into multiplying by the flip, then merge
- The insight: 'divide by 3/5' is the same as 'multiply by 5/3' (flip the divisor). Now both steps are multiplications, which combine cleanly.
- So the effect is × 34 × 53. The 3's cancel, leaving × 54 — i.e. multiplying by 5/4.
- Why this transfers: any string of ×'s and ÷'s by fractions collapses to one fraction — flip every divisor, then multiply across and cancel.
Another way — test with an easy number:
- Pick 12 (divisible by 4 and 3). Multiply by 3/4: 12 → 9. Divide by 3/5: 9 ÷ 3/5 = 9 × 5/3 = 15.
- So 12 became 15 — that's × 5/4 (since 12 × 5/4 = 15). Matches multiplying by 5/4.
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