Problem 1 · 1998 AJHSME
Easy
Algebra & Patterns
substitution
For x = 7, which of the following is the smallest?
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Answer: B — 6/(x+1).
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Hint 1 of 2
You don't have to rank all five — a fraction is smallest when it has the biggest bottom sitting under a small top. Which choice has the largest denominator?
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Hint 2 of 2
Sorting fractions: with the same small numerator (6), the one with the biggest denominator wins for smallest. Choices D and E are bigger than 1, so they can't be smallest — ignore them.
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Approach: biggest bottom under a small top = smallest fraction
- Don't grind all five. First notice D = 7/6 and E = 8/6 are each more than 1, while A, B, C all have top 6 under a bottom of 6 or more, so they're at most 1 — the smallest must be among A, B, C.
- Those three are 6/7, 6/8, 6/6. Same top of 6, so the biggest bottom makes the smallest fraction: that's 6/8, which is 6/(x+1).
- Why this transfers: when fractions share a numerator, more in the denominator means a smaller value — you split the same 6 among more pieces. You'll reuse this to compare fractions on sight without finding a common denominator.
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