Fremontian 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
Just curious - Is this solvable in < 2mins?
We have been given the following information :
In room A, ratio of juniors to seniors = 4 : 5
In room B, ratio of juniors to seniors = 3 : 17
Total ratio of juniors to seniors = 5 : 7
Now, from this information we can infer the following :
In room A, the if the number of juniors are '4x' then the number of seniors will be '5x'. Also, the total number of students in room A will be 4x + 5x = 9x.
Similarly, in room B, if the number of juniors are '3y' then the number of seniors will be '17y' and the total number of students in room B will be 3y + 17y = 20y.
And finally, if the total number of juniors are '5z' then the total number of seniors will be '7z'
Note : We use different variables (x, y and z) in each case because the value of the multiplying factor can be different in each case.
Now, with this information, we can form the following equations :
Equation 1 : 4x + 3y = 5z
That is, the number of juniors in room A plus the number of juniors in room B = Total number of juniors.
Equation 2 : 5x + 17y = 7z
That is, the number of seniors in room A plus the number of seniors in room B = Total number of seniors.
Now, we can solve these two equations and get the values of 'x' and 'y' in terms of 'z'.
Multiplying Eq.(1) by 5 and Eq.(2) by 4 and then subtracting Eq.(1) from Eq.(2), we get :
\(68y - 15y = 28z - 25z\) OR \(y = \frac{3z}{53}\)
Substituting this value of 'y' in Eq.(1) we get : \(x = \frac{64z}{53}\)
Now, the ratio of students in room A to students in room B = \(\frac{9x}{20y}\)
Substituting values of 'x' and 'y' in terms of 'z' we get : Ratio of students = \(\frac{9*\frac{64z}{53}}{20*\frac{3z}{53}\) = \(9*\frac{64z}{53}*\frac{53}{3z}*\frac{1}{20}\) = \(\frac{48}{5}\)
Answer : D
Ps. The amount of time it takes to solve this problem depends on how long it takes you to figure out the approach. Once you know how to go about it, the calculations should be manageable in about 2 minutes.
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