Problem 6 · 1986 AJHSME
Medium
Fractions, Decimals & Percents
simplify-complex-fraction
2 ⁄ (1 − 2⁄3) =
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Answer: E — 6.
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Hint 1 of 2
A fraction with a fraction inside it looks scary. Untangle the inside (the bottom) into one clean number before you do anything else.
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Hint 2 of 2
Dividing by a fraction is the same as multiplying by its flip — so dividing by 1⁄3 means multiplying by 3.
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Approach: collapse the inner fraction, then flip-and-multiply
- Work from the inside out. The bottom is 1 − 2⁄3; think of 1 as 3⁄3, so 3⁄3 − 2⁄3 = 1⁄3.
- Now it's just 2 ÷ 1⁄3. Dividing by 1⁄3 asks "how many thirds fit in 2?" — and 6 thirds make 2, so the answer is 6.
- Sanity check on size: the bottom 1⁄3 is much smaller than 1, and dividing by something less than 1 always makes a number *bigger*, so an answer above 2 is expected (rules out every choice but 6).
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