Problem 2 · 1985 AJHSME
Easy
Arithmetic & Operations
arithmetic-series
90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 =
Show answer
Answer: B — 945.
Show hints
Hint 1 of 2
Don't add left to right. In an evenly-spaced run, pair the smallest with the largest: 90 + 99, 91 + 98, 92 + 97… each pair makes the same total. What is that total, and how many pairs are there?
Still stuck? Show hint 2 →
Hint 2 of 2
Sum of an evenly-spaced run = (first + last)⁄2 × count — the (first+last)⁄2 is just the average, sitting right in the middle. So a long sum becomes one multiplication.
Show solution
Approach: average × count
- These ten numbers climb by 1, so their average is the middle value: (90 + 99)⁄2 = 94.5. There are 10 of them.
- Sum = 94.5 × 10 = 945 — the average times the count.
- Sanity check: all ten numbers are in the 90s, so the total must be between 10×90 = 900 and 10×99 = 990. 945 lands right in the middle, as it should.
Another way — pair from the ends:
- Pair 90+99, 91+98, 92+97, 93+96, 94+95 — five pairs, each summing to 189.
- 5 × 189 = 945.
Mark:
· log in to save