If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numer

If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numerator and denominator it become 4/5. Find the fractions.

Convert 2 to a fraction with a denominator of 10:
20/2 = 10, so we multiply 2 by 10/10:
2*10/10 = 20/10

Add 2 to the numerator and denominator:
(n + 2)/(d + 2) = 9/10

Cross multiply and simplify:
10(n + 2) = 9(d + 2)
10n + 20 = 9d + 18

Move constants to right side by subtracting 20 from each side and subtracting 9d:
10n - 9d = -2

Subtract 3 from the numerator and denominator:
(n - 3)/(d - 3) = 4/5

Cross multiply and simplify:
5(n - 3) = 4(d - 3)
5n - 15 = 4d - 12

Move constants to right side by adding 15 to each side and subtracting 4d:
5n - 4d = 3

Build our system of equations:

1. 10n - 9d = -2
2. 5n - 4d = 3

Multiply equation (2) by -2:

1. 10n - 9d = -2
2. -10n + 8d = -6

Now add equation (1) to equation (2)
(10 -10)n (-9 + 8)d = -2 - 6

The n's cancel, so we have:
-d = -8

Multiply through by -1:
d = 8

Now bring back our first equation from before, and plug in d = 8 into it to solve for n:
10n - 9d = -2
10n - 9(8) = -2
10n - 72 = -2

To solve for n, we plug this equation into our search engine and we get:
n = 7

So our fraction, n/d = 7/8