Ratio problem

kapil2012 wrote:What number must be added to two numbers which are in the ration 6:7 so that the ratio becomes 8:9 and sum of the resulting numbers is 289.

Options are 34/17/51/68/Can't be determined.

Please help me with the above question. According to me, it can't be determined however I am little confused.

Here's an approach that minimizes the number of variables.
If the two original numbers have a ratio of 6:7, we can say that those two numbers are 6x and 7x (for some value of x other than 0)

Now we want to add the same number to 6x and 7x so that the ratio of the resulting values becomes 8:9.
Well, let's add k to 6x and 7x, to get 6x+k and 7x+k

We want: (6x+k)/(7x+k) = 8/9
When we cross-multiply, we get 54x+9k = 56x+8k
This simplifies to be: 2x-k=0
This is one equation.

We're also told that the sum of the resulting numbers is 289.
In other words, (6x+k)+(7x+k) = 289
When we simplify this, we get 13x+2k = 289
This is another equation.

We can now solve following system for k
2x-k=0
13x+2k = 289
When we do so, we get k=34

So, we must add 34 to the original numbers. So, the answer is 34.

Aside: the question doesn't ask us to calculate the original numbers. But, if we solve for x, we get x=17. So, the two original numbers are 102 and 119.

Cheers,
Brent