Find a fraction which reduce to 2/3 if the numerator and the denominator are each increased by 1 and reduce to 3/5 if the numerator and the denominator are each reduced by 2

original fraction --- x/y

first new fraction: (x+1)/(y+1) = 2/3
3x + 3 = 2y + 2
3x - 2y = -1 **

second new fraction: (x-2)/(y-2) = 3/5
5x - 10 = 3y - 6
5x - 3y = 4 ***

solve the two equations ** and ***

To solve this problem, let's start by assuming the fraction is written as x/y, where x is the numerator and y is the denominator.

According to the given conditions, if the numerator and denominator are each increased by 1, the fraction reduces to 2/3. This can be expressed as (x+1)/(y+1) = 2/3.

Similarly, if the numerator and denominator are each reduced by 2, the fraction reduces to 3/5. This can be expressed as (x-2)/(y-2) = 3/5.

To find the value of x and y, we can solve these two equations simultaneously.

First, let's solve the equation (x+1)/(y+1) = 2/3 for x:

Cross multiplying, we have: 3(x+1) = 2(y+1)
Expanding, we get: 3x + 3 = 2y + 2
Rearranging, we get: 3x - 2y = -1 .............(Equation 1)

Now, let's solve the equation (x-2)/(y-2) = 3/5 for x:

Cross multiplying, we have: 5(x-2) = 3(y-2)
Expanding, we get: 5x - 10 = 3y - 6
Rearranging, we get: 5x - 3y = 4 .............(Equation 2)

We now have a system of linear equations with two variables (x and y). We can use any method to solve it, such as substitution or elimination.

Let's solve it using the elimination method:

Multiplying Equation 1 by 3 and Equation 2 by 2, we get:

(Equation 1): 9x - 6y = -3
(Equation 2): 10x - 6y = 8

Now, subtracting Equation 1 from Equation 2, we eliminate the "y" term:

(10x - 6y) - (9x - 6y) = 8 - (-3)
Simplifying, we get: x = 11

Now substituting the value of x = 11 into Equation 1, we can solve for y:

3(11) - 2y = -1
33 - 2y = -1
-2y = -34
y = 17

Therefore, the fraction is 11/17, which satisfies both conditions.