Problem 14 · 2026 AMC 8
Medium
Algebra & Patterns
arithmetic-sequence
Jami picked three equally spaced integers on the number line. The sum of the first and second is 40, and the sum of the second and third is 60. What is the sum of all three numbers?
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Answer: B — 75.
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Hint 1 of 2
‘Equally spaced’ is the magic phrase: the middle number is exactly the average of the outer two. So how does the middle number relate to the whole sum of three?
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Hint 2 of 2
Add the two given sums (40 + 60). The middle number gets counted twice and the outer two once each — turn that into the value of the middle, then triple it.
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Approach: the middle number is the average — and the whole sum is just 3 times it
- For three equally spaced numbers, the middle one is the average of all three, so the total is simply 3 × middle. Find the middle.
- Add the two given sums: (first + second) + (second + third) = 40 + 60 = 100. The middle got counted twice, the outer two once each, so 100 = (first + second + third) + second. Also first + third = 2·second, giving 100 = 4·second, so the middle is 25.
- Total = 3 × 25 = 75.
- Why this transfers: in any evenly-spaced list, the middle term is the mean, so sum = (count) × (middle). Spotting ‘equally spaced’ lets you skip solving for the individual numbers.
Another way — name the spacing and watch it cancel:
- Call the numbers m−d, m, m+d. The first sum is 2m−d = 40 and the second is 2m+d = 60.
- Add them: 4m = 100, so m = 25. The total is exactly 3m = 75 — the spacing d never matters.
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