Problem 5 · 1995 AJHSME
Medium
Fractions, Decimals & Percents
estimate-fractions
Find the smallest whole number that is larger than the sum 212 + 313 + 414 + 515.
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Answer: C — 16.
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Hint 1 of 2
You're not asked for the exact sum — only the next whole number above it. That means you can ESTIMATE instead of finding common denominators.
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Hint 2 of 2
Split each mixed number into its whole part and its fraction part. The whole parts are easy; then just decide how big the leftover fractions add up to.
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Approach: estimate — split off the whole parts, then bound the fractions
- The key realization: the question wants the smallest whole number ABOVE the sum, so we never need the exact value — a good estimate is enough.
- Whole parts: 2 + 3 + 4 + 5 = 14. The four fraction parts are ½, ⅓, ¼, ⅕ — each less than ½, and together a little more than 1 (½ + ⅓ ≈ 0.83 already, plus ¼ + ⅕ ≈ 0.45, so ≈ 1.28).
- So the total is about 15.28 — comfortably between 15 and 16. The smallest whole number larger than it is 16.
- Sanity check: the fractions can't reach 2 (the biggest, ½, isn't even close to making them sum that high), so the answer can't be 17. And they clearly exceed 1, so it isn't 15. 16 is the only fit.
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