Problem 13 · 1992 AJHSME
Hard
Arithmetic & Operations
mean-median-mode
Five test scores have a mean of 90, a median of 91, and a mode of 94. The sum of the two lowest test scores is
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Answer: B — 171.
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Hint 1 of 3
You don't need each individual low score — only their SUM. What's the fastest route from "mean of all five" to "sum of the bottom two"?
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Hint 2 of 3
Mean unlocks the grand total (mean × count); the two lowest are then just total − (the three you can pin down). Find the known three.
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Hint 3 of 3
Line the 5 scores up in order. The median is the middle (3rd) one; the mode 94 must appear at least twice, and the only room for two 94's is the 4th and 5th slots (the 3rd is already 91).
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Approach: turn mean into the total, pin the top three, subtract
- Mean 90 over 5 scores means they total 5 × 90 = 450. Write the scores in increasing order; the median is the 3rd = 91.
- The mode is 94, so 94 must appear at least twice. The 3rd slot is already 91, so the two 94's have to be the 4th and 5th (the largest). The top three are therefore 91, 94, 94 = 279.
- The two lowest are everything else: 450 − 279 = 171.
- Why this transfers: when a question asks for a SUM of unknowns, don't solve for each one — get the grand total from the mean and subtract the part you can determine. The individual low scores stay unknown, yet their sum is forced.
- Why not ‘not determined’: the median and mode lock all three top scores exactly, so the bottom two have no wiggle room in their total.
Mark:
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