Problem 21 · 2009 AMC 8
Medium
Algebra & Patterns
invariant-total
Andy and Bethany have a rectangular array of numbers with 40 rows and 75 columns. Andy adds the numbers in each row. The average of his 40 sums is A. Bethany adds the numbers in each column. The average of her 75 sums is B. What is the value of AB?
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Answer: D — 15/8.
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Hint 1 of 2
The actual numbers never matter. Andy's row-sums and Bethany's column-sums both add up EVERY entry of the array exactly once — so the two grand totals are identical.
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Hint 2 of 2
Average = total ÷ (how many you averaged). Andy divided the grand total by 40, Bethany by 75. Write both totals as one shared value S.
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Approach: both grand totals are the same number
- Let S = sum of all entries. Adding by rows or by columns both reach S, so 40A = S and 75B = S. Hence 40A = 75B.
- Rearrange: A/B = 75/40 = 15/8.
- Sanity check: fewer rows (40) than columns (75) means each row-sum is bigger on average, so A > B — and 15/8 > 1. Good. This transfers as a counting invariant: when one total can be tallied two ways, set the two tallies equal.
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