sameershaik wrote:The only people in each of rooms A and B are students, and each student in each of rooms A and B is either a junior or a senior. The ratio of the number of juniors to the number of seniors in room A is 4 to 5, the ratio of the number of juniors to the number of seniors in room B is 3 to 17, and the ratio of the total number of juniors in both rooms A and B to the total number of seniors in both rooms A and B is 5 to 7. What is the ratio of the total number of students in room A to the total number of students in room B ?
A) 29/12
B) 59/10
C) 65/8
D) 48/5
E) 29/3

For this question, we want to avoid using too many variables.
Let JA = number of juniors in room A
Let SA = number of juniors in room A
Let JB = number of juniors in room B
Let SB = number of juniors in room B

From the first part of the question, we see that JA/SA = 4/5
So, we can say that JA=4x and SA=5x for some value of x.

Similary, the next part of the question tells us that JB/SB = 3/17
So, we can say that JB=3y and SB=17y for some value of y.

I'll now redefine my variable as follows:

Let JA = number of juniors in room A = 4x
Let SA = number of juniors in room A = 5x
Let JB = number of juniors in room B = 3y
Let SB = number of juniors in room B = 17y

The last part of the question tells us that (JA+JB)/(SA+SB) = 5/7
We can plug in our alternative variables to get (4x+3y)/(5x+17y) = 5/7
Cross multiply this equation to get 28x + 21y = 25x + 85y
Simplify to get 3x = 64y
And solve for x/y to get x/y = 64/3 (you'll see why I solved for x/y shortly)

We are asked to find the ratio (JA+SA)/(JB+SB) which is the same as (4x+5x)/(3y+17y) which is the same as 9x/20y

So, 9x/20y = (9/20)(x/y) = (9/20)(64/3) =48/5 (answer choice D)