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?

Question

Sanjana Patil · Accepted Answer

To solve the problem, let's denote the two numbers as 3x and 4x, since their ratio is 3 to 4.
Given that if k is added to each number, the new ratio will be 4 to 5:(3x + k) / (4x + k) = 4 / 5
Cross-multiplying, we get:5(3x + k) = 4(4x + k)15x + 5k = 16x + 4k15x + 5k - 16x - 4k = 0-x + k = 0k = x
We also know that the sum of the numbers is 117:3x + 4x = 1177x = 117x = 117 / 7x = 17
Therefore, k = x = 17
Thus, the value of k is 17.
I hope you'll find this helpful !

A.D Singh · Answer

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 + 4k15x + 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) = 1174k + 5k = 1179k = 117k = 117/9k = 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*1352 + 65117
LHS = RHS