If the numerator of a fraction is increased by 2 and the denominator is decreased by 1, then it becomes 2/3. If the numerator is increased by 1 and the denominator is increased by 2, then it becomes 1/3. Find the fraction. 

Given:

If the numerator of a fraction is increased by 2 and the denominator is decreased by 1, then it becomes 2/3.

If the numerator is increased by 1 and the denominator is increased by 2, then it becomes 1/3.

Calculation:

Let the fraction be \(\dfrac{x}{y}\)

According to the first condition: \(\dfrac{x+2}{y-1} = \dfrac{2}{3}\)

According to the second condition: \(\dfrac{x+1}{y+2} = \dfrac{1}{3}\)

First condition: \(\dfrac{x+2}{y-1} = \dfrac{2}{3}\)

⇒ 3(x + 2) = 2(y - 1)

⇒ 3x - 2y = -8    ........(1)

Second condition: \(\dfrac{x+1}{y+2} = \dfrac{1}{3}\)

⇒ 3(x + 1) = y + 2

⇒ 3x - y = -1      .........(2)

Solving the system of equations:

3x - 2y = -8

3x - y = -1

Subtracting the second equation from the first:

(3x - 2y) - (3x - y) = -8 - (-1)

⇒ -2y + y = -8 + 1

⇒ y = 7

Substitute y = 7 into 3x - y = -1:

⇒ 3x - 7 = -1

⇒ x = 2

∴ The fraction is \(\dfrac{2}{7}\).