2019 AMC 10B Problems/Problem 11

Problem

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$ , and the ratio of blue to green marbles in Jar $2$ is $8:1$ . There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$ ?

$\textbf{(A) } 5\qquad\textbf{(B) } 10 \qquad\textbf{(C) }25 \qquad\textbf{(D) } 45 \qquad \textbf{(E) } 50$

Solution

Call the number of marbles in each jar $x$ (because the problem specifies that they each contain the same number). Thus, $\frac{x}{10}$ is the number of green marbles in Jar $1$ , and $\frac{x}{9}$ is the number of green marbles in Jar $2$ . Since $\frac{x}{9}+\frac{x}{10}=\frac{19x}{90}$ , we have $\frac{19x}{90}=95$ , so there are $x=450$ marbles in each jar.

Because $\frac{9x}{10}$ is the number of blue marbles in Jar $1$ , and $\frac{8x}{9}$ is the number of blue marbles in Jar $2$ , there are $\frac{9x}{10}-\frac{8x}{9}=\frac{x}{90} = 5$ more marbles in Jar $1$ than Jar $2$ . This means the answer is $\boxed{\textbf{(A) } 5}$ .

2019 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2019 AMC 10B Problems/Problem 11

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