Problem 1 · 1989 AJHSME
Easy
Arithmetic & Operations
pair-terms
(1 + 11 + 21 + 31 + 41) + (9 + 19 + 29 + 39 + 49) =
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Answer: E — 250.
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Hint 1 of 3
Don't add straight down each group — look at how the two groups line up term-by-term. What do 1 and 9 have in common? 11 and 19?
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Hint 2 of 3
Re-grouping a sum into friendly pairs that each make a round number is far easier than adding ten messy numbers in a row.
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Hint 3 of 3
Each matched pair (1+9, 11+19, …) ends in 0, so you only ever add tens.
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Approach: re-pair into round tens
- Notice the two groups are built from the same ten-step pattern, just shifted: 1 pairs with 9, 11 with 19, 21 with 29, and so on. Each pair adds to a number ending in 0 — that's the whole point of pairing instead of adding straight down.
- The five pairs give 10 + 30 + 50 + 70 + 90 = 250.
- Why this transfers: whenever a sum can be reshuffled so terms combine into round numbers, do the reshuffling first — addition is allowed in any order, and round numbers carry almost no chance of a slip.
- Sanity check: ten numbers averaging about 25 should total roughly 250, which matches.
Another way — sum each group separately:
- First group: 1+11+21+31+41 = 105 (five terms averaging 21). Second group: 9+19+29+39+49 = 145 (five terms averaging 29).
- 105 + 145 = 250. The pairing trick just avoids these two intermediate totals.
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