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

Let the two numbers be x and y
x/y=3/4
Therefore x=3z
y=4z, where z is the GCF of x and y

(3z+k)/(4z+k)=4/5
(3z+k)*5=(4z+k)*4
15z+5k=16z+4k
k=z--->eq1
3z+k+4z+k=117
7z+2k=117--->eq2

Using value of z from eq1 in eq2

7k+2k=117
k=117/9
=13

You want the answer to all