Problem 6 · 2022 AMC 8
Easy
Algebra & Patterns
substitutionarithmetic-sequence
Three positive integers are equally spaced on a number line. The middle number is 15 and the largest number is 4 times the smallest number. What is the smallest of these three numbers?
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Answer: C — 6.
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Hint 1 of 2
“Equally spaced” is the key word: the middle number sits exactly halfway between the other two, so it's their average.
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Hint 2 of 2
So smallest + largest = 2 × 15 = 30. And largest = 4 × smallest, so smallest + 4×smallest = 30.
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Approach: equally spaced ⇒ middle is the average of the outer two
- Insight: “equally spaced” means the middle is the average of the outer two, so the two outer numbers add to 2 × 15 = 30 — no spacing variable needed.
- The two outer numbers are the smallest and 4 times the smallest, which together make 5 of the smallest. So 5 × smallest = 30 ⇒ smallest = 6.
- Sanity check: the three numbers are 6, 15, 24 — gaps 9 and 9, equally spaced, and 24 = 4 × 6. ✓
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