Problem 6 · 1988 AJHSME
Medium
Fractions, Decimals & Percents
ratio-of-powers
(.2)³ ⁄ (.02)² =
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Answer: E — 20.
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Hint 1 of 2
Don't compute the tiny decimals. Notice that .2 is exactly 10 times .02 — so a .2 over a .02 is just a clean 10. How many of those can you pair up?
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Hint 2 of 2
Split one factor of .2 off the top so the rest pairs as (.2 ⁄ .02)² — each pair becomes 10.
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Approach: pair matching factors so .2 ⁄ .02 = 10
- There are three .2's on top and two .02's on the bottom. Peel one .2 aside, then pair the other two .2's with the two .02's: (.2)³ ⁄ (.02)² = .2 × (.2 ⁄ .02) × (.2 ⁄ .02) = .2 × 10 × 10.
- = .2 × 100 = 20.
- Why this transfers: when a fraction is built from the same digits at different decimal scales, don't grind out the decimals — pair top and bottom into clean powers of 10. Counting the factors of 10 is far safer than tracking tiny zeros.
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