Problem 12 · 1991 AJHSME
Medium
Arithmetic & Operations
average
If 2 + 3 + 43 = 1990 + 1991 + 1992N, then N =
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Answer: D — 1991.
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Hint 1 of 3
Each side is "(sum of three numbers) ÷ something." The left side divides by 3 and equals the AVERAGE of 2, 3, 4. What's that average — and what does it force the right side to equal?
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Hint 2 of 3
Three consecutive numbers average to their middle one, so the left side is just 3. The right side must also equal 3, which pins down what N has to be.
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Hint 3 of 3
You can sidestep the big sum: notice the right side looks just like the left — three consecutive numbers over a divisor. For its value to be the middle number, the divisor must match the middle number.
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Approach: read both sides as averages of consecutive numbers
- Left side: 2, 3, 4 are consecutive, so their average is the middle value, 3. (No need to add and divide.)
- So the equation becomes (1990 + 1991 + 1992) ÷ N = 3. The three numbers on top are also consecutive with middle 1991, so their sum is 3 × 1991 (three times the middle).
- Now 3 × 1991 ÷ N = 3, which forces N = 1991.
- Why this transfers: any run of consecutive numbers (or any evenly-spaced run) sums to (count) × (middle term), and averages to the middle term. Spotting that turns "add three four-digit numbers" into a one-line observation.
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