When positive integer x is divided by positive integer y…
This problem will be best solved using the remainder formula. Let’s first state the remainder formula:
When positive integer x is divided by positive integer y, if integer Q is the quotient and r is the remainder, then x/y = Q + r/y.
In this problem we are given that when positive integer x is divided by positive integer y, the remainder is 9. So we can say:
x/y = Q + 9/y
We also are given that x/y = 96.12. Using the remainder formula we can say:
x/y = 96.12
x/y = 96 + 0.12
x/y = 96 + 12/100
Because Q is always an integer, we see that Q must be 96, and thus the remainder 9/y must be 12/100. We can now equate 9/y to 12/100 and determine the value of y.
9/y = 12/100
12y = 9 x 100
y = 900/12 = 75
Note: Had we simplified 12/100 to 3/25 first, we would have also obtained the same answer. See below.
9/y = 3/25
3y = 9 x 25
y = 3 x 25 = 75