Question

There are 200 red and 90 blue marbles in Box A. There are 80 red and 100 blue marbles in Box B. Find the total number of red and blue marbles that must be transferred from Box A to Box B so that 80% of the marbles in Box A and 50% of the marbles in Box B are red?​

Answer 1

So we need to transfer 37 red and 73 blue marbles from Box A to Box B.

Explanation:

Let x be the number of red marbles to be transferred from Box A to Box B, and let y be the number of blue marbles to be transferred from Box A to Box B. Then the total number of marbles in each box after the transfer is:

Box A: 200 + 90 - x - y = 290 - x - y Box B: 80 + 100 + x + y = 180 + x + y

We want 80% of the marbles in Box A and 50% of the marbles in Box B to be red, so we can set up the following equations:

0.8(290 - x - y) = 232 - 0.8x - 0.8y (80% of Box A is red) 0.5(180 + x + y) = 90 + 0.5x + 0.5y (50% of Box B is red)

To solve for x and y, we can set these two expressions equal to each other:

232 - 0.8x - 0.8y = 90 + 0.5x + 0.5y

Simplifying and rearranging, we get:

1.3x + 1.3y = 142 13x + 13y = 1420 x + y = 110

So we need to transfer a total of 110 marbles from Box A to Box B. To find the number of red marbles to transfer (x), we can use the first equation:

0.8(290 - x - y) = 232 - 0.8x - 0.8y 232 - 0.8x - 0.8y = 232 - 0.8x - 58 0.8y = 58 y = 72.5

Since we can't have a half marble, we'll round up to 73 blue marbles to transfer. Therefore, the number of red marbles to transfer is:

x = 110 - 73 = 37

So we need to transfer 37 red and 73 blue marbles from Box A to Box B.