GMAT CLUB OLYMPICS: The ratio of the number of marbles in boxes X and

We can see that the initial ratio was 5/8 and when some of the marbles were transferred, the ratio became 1/6.
Note that both ratios are divisible by 13. Actually in both cases the sum of the marbles will be 13*x

Initially: 5x and 8x
After: 7x and 6x

We can see that neither statement is sufficient by itself. Let’s analyze choice D.
1st statement says that 45 < 8x < 100
2nd statement says that 45 < 7x < 100

There are many numbers that satisfy this requirement.

So E

Answer: E

The ratio of the number of marbles in boxes X and Y was 5 to 8. After some number of marbles were transferred from box Y to box X, the ratio of the number of marbles in boxes X and Y became 7 to 6. What is the total number of marbles in the two boxes?

Let the number of marbles in box X be 5A and box Y be 8A.
Let the number of marbles transferred from box Y to box X be T.

A and T will be integers because number of marbles will be integer values

Therefore, (5A+T) / (8A-T) = 7/6
This will give us T = 2A.

We need to find total number of marbles in two boxes i.e. 5A+8A = 13A = 13T/2

Therefore, we also need to make sure that T needs to be even integer.

Statement 1: Initially the number of marbles in box Y was between 45 and 100.

This means 45 < 8A < 100.
i.e. 8A can be multiple of 8 between 45 and 100
i.e. 8A = 48, 56, 64, 72, 80, 88, or 96
i.e. A = 6, 7, 8, 9, 10, 11, 12
Since A has multiple values, therefore, we will have multiple values 13A i.e. total number of marble in two boxes.

Hence insufficient.

Statement 2: After the transfer, the number of marbles in box X became between 45 and 100.

This means 45 < 5A+T < 100
i.e. 45 < 7A < 100
i.e. 7A = 49, 56, 63, 70, 77, 84, 91, or 98
i.e. A = 7, 8, 9, 10, 11, 12, 13, or 14
Since A has multiple values, therefore, we will have multiple values 13A i.e. total number of marble in two boxes.

Hence insufficient.

Both statements together:

We still end up with A = 7, 8, 9, 10, 11 or 12
Since A has multiple values, therefore, we will have multiple values 13A i.e. total number of marble in two boxes.

[color=#00a651]Hence both statements together as well are insufficient.[/color]

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