AMC 8 · 2009 · #21

Grade 6 arithmetic
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
mean-median-mode-rangeratio-proportionset-partition identify-subproblemspattern-recognition ↑ Prerequisites: mean-median-mode-rangeratio-proportion
📏 Medium solution 💡 3 insights
Problem
A rectangular table of numbers has 40 rows and 75 columns. Andy sums each row and averages his 40 row sums to get A. Bethany sums each column and averages her 75 column sums to get B. Find AB\frac{A}{B}.

Pick an answer.

(A)
$\frac{64}{225}$
(B)
$\frac{8}{15}$
(C)
1
(D)
$\frac{15}{8}$
(E)
$\frac{225}{64}$

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Find an Invariant

The actual numbers in the array are never given, which is the tell that Tool #11 (Find an Invariant) applies: there must be a quantity that does not depend on the specific entries. That invariant is S, the grand total of all numbers in the array — counted by rows or by columns, it is the same value. Tool #4 (Introduce a Variable) lets us name S so we can write A and B in terms of it. Once both averages share S, the ratio AB\frac{A}{B} falls out as a ratio of two counts.

1STEP 1

Adding row by row or column by column visits each cell once, so both give the same grand total S.

S = sum of all entries in the 40 × 75 array
2STEP 2

Andy's 40 row sums total S and A is their average, so S = 40A.

A = S40\frac{S}{40} → S = 40A
3STEP 3

Bethany's 75 column sums total the same S and B is their average, so S = 75B.

B = S75\frac{S}{75} → S = 75B
4STEP 4

Both 40A and 75B equal that same S, so 40A = 75B and AB\frac{A}{B} = 158\frac{15}{8}.

40A = 75B → AB\frac{A}{B} = 7540\frac{75}{40} = 158\frac{15}{8} → (D)
Answer
158\frac{15}{8}
Sanity check with an all-1s array: every entry is 1, so each row sum is 75 and each column sum is 40. Then A = 75, B = 40, and AB\frac{A}{B} = 7540\frac{75}{40} = 158\frac{15}{8}. The ratio matches answer (D), independent of the specific entries — exactly the invariant we expected. Also, A ＞ B makes sense because each row has more cells (75) than each column has (40), so row sums are typically larger than column sums.
💡Key takeaway

Row sums and column sums add up to the same grand total — once you spot that invariant, this AMC 8 problem becomes a Grade 6 mean-and-ratio exercise!

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