Find a fraction which reduces to 3/4 when the numerator and denominator are each decreased by 1, and which reduces to 4/5 when the numerator and denominator are each increased by 1.

To find the desired fraction, let's start by assigning variables to the numerator and denominator of the fraction. Let's call the numerator N, and the denominator D.

According to the problem statement, when both the numerator and denominator are decreased by 1, the fraction reduces to 3/4. This can be expressed as:

(N - 1)/(D - 1) = 3/4 ---(1)

Similarly, when both the numerator and denominator are increased by 1, the fraction reduces to 4/5:

(N + 1)/(D + 1) = 4/5 ---(2)

To solve this system of equations, we will use the method of substitution. Solve equation (1) for N, and equation (2) for D:

From equation (1): N = (3/4)(D - 1) + 1 ---(3)

From equation (2): D = (5/4)(N + 1) - 1 ---(4)

Now, substitute equation (3) into equation (4) as follows:

D = (5/4)[(3/4)(D - 1) +1 +1] - 1

Simplify equation (4):

D = (5/4)(3D/4 + 2) - 1
D = (5/4)(3D/4) + (5/4)(2) - 1
D = (15D/16) + 10/4 - 4/4
D = (15D/16) + 6/4
D = (15D + 24)/16

Now, let's solve for D:

16D = 15D + 24
16D - 15D = 24
D = 24

Substituting D = 24 back into equation (3) to solve for N:

N = (3/4)(24 - 1) + 1
N = (3/4)(23) + 1
N = 69/4 + 1
N = 69/4 + 4/4
N = 73/4

Therefore, the fraction we are looking for is 73/24.