AMC12 2005 A

AMC12 2005 A · Q5

AMC12 2005 A · Q5. It mainly tests Averages (mean), Arithmetic misc.

The average (mean) of $20$ numbers is $30$, and the average of $30$ other numbers is $20$. What is the average of all $50$ numbers?

$20$ 个数的平均数（均值）是 $30$，另外 $30$ 个数的平均数是 $20$。这 $50$ 个数的平均数是多少？

(A) 23 23

(B) 24 24

(D) 26 26

(E) 27 27

Answer

Correct choice: (B)

正确答案：（B）

Solution

The sum of the first $20$ numbers is $20 \cdot 30$ and the sum of the other $30$ numbers is $30\cdot 20$. Hence the overall average is $\frac{20 \cdot 30 + 30 \cdot 20}{50} = 24 \ \mathrm{(B)}$. This is just the harmonic mean, so the answer is $\frac{2 \cdot 20 \cdot 30}{20+30}=24 \ \mathrm{(B)}$. Solution by franzliszt

前 $20$ 个数的和是 $20 \cdot 30$，另外 $30$ 个数的和是 $30\cdot 20$。因此总体平均数是 $\frac{20 \cdot 30 + 30 \cdot 20}{50} = 24 \ \mathrm{(B)}$。这就是调和平均数，所以答案是 $\frac{2 \cdot 20 \cdot 30}{20+30}=24 \ \mathrm{(B)}$。 Solution by franzliszt

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