sayanpaul wrote:The ratio between the ages of a father and a son at present is 5:2 respectively. Four years hence the ratio of the ages of the son and his mother will be 1:2 respectively. What is the ratio of the present ages of the father and the mother?

A. 3:4
B. 5:4
c. 4:3
D. 5:3
E. none of the above

Let C = the child's present cage.
Let F = the father's present cage.
The ratio between the ages of a father and a son at present is 5:2
In other words, F/C = 5/2
Solve for F first cross multiplying to get: 2F = 5C
Solve to get: F = 5C/2 = 2.5C
So, the father's present age is 2.5C

Let M = the mothers's present cage.
Four years hence the ratio of the ages of the son and his mother will be 1:2 respectively
In other words (C+4)/(M+4) = 1/2
Solve for M by first cross multiplying to get: 1(M+4) = 2(C+4)
Expand: M + 4 = 2C + 8
Solve: M = 2C + 4

What is the ratio of the present ages of the father and the mother?
We want F/M
Substitute (from above) to get: 2.5C/(2C + 4)
Multiply top and bottom by 2 to get: 5C/(4C + 8)

IMPORTANT: the ratio 5C/(4C + 8) can have several values, depending on the value of C
For example, if C = 1, then the ratio is 5/12
If C = 2, then the ratio is 10/16 (or 5/8)
If C = 3, then the ratio is 15/20 (or 2/3)
etc . . .


So, one might conclude that the correct answer is E. However, E doesn't say "cannot be determined." It says "none of the above"
At this point, I'm not too crazy about the question because it's unclear what we're supposed to do.

I supposed, we could set the ratio 5C/(4C + 8) to see which one(s) yielded a value of C that creates the given ratio.

For example, take answer choice A.
Is there a value of C such that 5C/(4C + 8) = 3/4?
Sure, just solve the equation for C to get C = 3
So when the child is presently 3 years old, the father to mother ratio is 3/4
So, answer choice A works.

What about answer choice B?
Is there a value of C such that 5C/(4C + 8) = 5/4?
No.
When we solve this equation for C, we get no solution.
So, answer choice B doesn't work.

What about answer choice C?
Is there a value of C such that 5C/(4C + 8) = 4/3?
No . . . not really.
When we solve this equation for C, we get C = -30, and this doesn't work in the real world.
So, answer choice C doesn't work.

What about answer choice D?
Is there a value of C such that 5C/(4C + 8) = 5/3?
No . . . not really.
When we solve this equation for C, we get C = -8, and this doesn't work in the real world.
So, answer choice D doesn't work.

So, one answer could be "there is no one value for the ratio 5C/(4C + 8)"
Or we could say, "among the answer choices, the only ratio that could equal 5C/(4C + 8) is answer choice A"

Cheers,
Brent