The ratio of two positive numbers is 3 to 4. If k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117. What is the value of k?

It's easy! The value of k is 13. How?

First of all, let's consider the two numbers as multiples of 3 and 4 (because their initial ratio is 3:4).

So, the numbers are 3x and 4x.

Now, k is added to both numbers and the new ratio becomes 4:5 which means,

(3x + k) : (4x + k) = 4:5

We can write ratios in fraction formats as well...

(3x + k)/(4x + k) = 4/5

Now cross multiply,

5 * (3x + k) = 4 * (4x + k)
15x + 5k = 16x + 4k
15x + 5k - 16x - 4k = 0
-x + k = 0

From here, we get x = k.

Now, we also know that the sum of new numbers is 117. 

(3x + k) + (4x + k) = 117

Substituting x=k in the equation above:

(3k + k) + (4k + k) = 117
4k + 5k = 117
9k = 117
k = 117/9
k = 13

Therefore, the value of k is 13. 

You can verify the answer by putting the value back in any of the above equations:

4k + 5k = 117

4*13 + 5*13
52 + 65
117

LHS = RHS