Let the numerator be "x"...... and denominator be "y"...... A/Q x+y=18-----------(1) Nd x/(y+2)=1/3.... =>3x-y=2--------(2) Now, add eq (1) and (2).... 3x-y=2 x+y=18 => 4x= 20.. =>x=5..... now, put the value of x in eq (2).. y=18-5 =>. y=13..... x=5. ,y=13.... Therefore, the fraction is 5 /13.....

The Problem:
The sum of the numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. We need to find the fraction.

Step 1: Understanding the Problem
Let's denote the numerator of the fraction as 'n' and the denominator as 'd'. According to the given information, we have two conditions:
1. The sum of the numerator and denominator is 18: n + d = 18
2. If the denominator is increased by 2, the fraction reduces to 1/3: n / (d + 2) = 1/3

Step 2: Solving the Equations
We can use the first equation to express the numerator in terms of the denominator:
n = 18 - d

Substituting this value of 'n' in the second equation:
(18 - d) / (d + 2) = 1/3

Step 3: Simplifying the Equation
To simplify the equation, we can cross-multiply:
3(18 - d) = (d + 2)

Expanding and simplifying:
54 - 3d = d + 2

Rearranging the equation:
3d + d = 54 - 2

Combining like terms:
4d = 52

Dividing both sides by 4:
d = 13

Step 4: Finding the Numerator
Using the value of 'd' obtained, we can substitute it back into the first equation to find the numerator:
n + 13 = 18
n = 18 - 13
n = 5

Step 5: Writing the Fraction
Now that we have the values of both the numerator and denominator, we can write the fraction:
The fraction is 5/13.

Step 6: Checking the Solution
To verify our solution, let's substitute the values back into the second condition:
n / (d + 2) = 1/3
5 / (13 + 2) = 1/3
5 / 15 = 1/3
1/3 = 1/3

The equation holds true, confirming that our solution is correct.

Conclusion:
The fraction that satisfies the given conditions is 5/13.