2019 AMC 10B Problem 11

Let each jar hold $N$ marbles. In Jar $1$ the greens are $\frac{1}{10}$ of the marbles, and in Jar $2$ they are $\frac{1}{9}$. Both fractions must be whole numbers, so $N$ is a multiple of $90$; write $N = 90k$.

• Jar $1$: $9k$ green, $81k$ blue.
• Jar $2$: $10k$ green, $80k$ blue.

Total green: $9k + 10k = 19k = 95$, so $k = 5$ (each jar holds $450$ marbles).

Blue difference: $81k - 80k = k = 5$.

The answer is $\boxed{\textbf{(A)}\ 5}$.

A ratio $9:1$ says the greens are $\frac{1}{10}$ of the jar, and $8:1$ says $\frac19$; the shared jar size is the link between the two jars. Introducing one scaling variable for the common size turns both ratios into integer counts and reduces the problem to a single equation. The requested difference is just the coefficient of that variable.

Directly: $\frac{N}{10} + \frac{N}{9} = 95$ gives $\frac{19N}{90} = 95$, so $N = 450$. Blue counts are $405$ and $400$, differing by $5$.

• Applying the ratios to the total green count (95) instead of to each jar, or forgetting that both jars hold the same number of marbles.
• Answering the difference in green marbles ($10k - 9k = 5$ happens to match here) by luck, or reporting $k$'s multiple such as $50$ without checking which quantity was asked.