AMC8 2009

AMC8 2009 · Q21

AMC8 2009 · Q21. It mainly tests Averages (mean), Arithmetic misc.

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 $\frac{A}{B}$?

Andy 和 Bethany 有一个有 40 行和 75 列的矩形数字阵列。Andy 计算每行的数字之和。他的 40 个和的平均值为 $A$。Bethany 计算每列的数字之和。她的 75 个和的平均值为 $B$。$rac{A}{B}$ 的值为多少？

(A) \frac{64}{225} \frac{64}{225}

(B) \frac{8}{15} \frac{8}{15}

(D) \frac{15}{8} \frac{15}{8}

(E) \frac{225}{64} \frac{225}{64}

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): There are 40 rows, so the sum of the 40 row sums is $40A$. This number is also the sum of all of the numbers in the array because each number in the array is added to obtain one of the row sums. Similarly, there are 75 columns, so the sum of the 75 column sums is $75B$, and this, too, is the sum of all of the numbers in the array. So $40A = 75B$, and $\frac{A}{B} = \frac{75}{40} = \frac{15}{8}$.

答案（D）：有40行，所以40个行和的总和是$40A$。这个数也是数组中所有数字的总和，因为数组中的每个数字都会被加到某一个行和中。同样地，有75列，所以75个列和的总和是$75B$，而这也等于数组中所有数字的总和。因此$40A = 75B$，并且$\frac{A}{B} = \frac{75}{40} = \frac{15}{8}$。

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