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2005 AMC 10A Problems/Problem 6

Problem

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?

$\mathrm{(A) \ } 23\qquad \mathrm{(B) \ } 24\qquad \mathrm{(C) \ } 25\qquad \mathrm{(D) \ } 26\qquad \mathrm{(E) \ } 27$

Solution

Since the average of the first $20$ numbers is $30$, their sum is $20\cdot30=600$.

Since the average of $30$ other numbers is $20$, their sum is $30\cdot20=600$.

So the sum of all $50$ numbers is $600+600=1200$

Therefore, the average of all $50$ numbers is $\frac{1200}{50}=24 \Longrightarrow \mathrm{(B)}$

See Also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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