In a fraction, if 1 is added to both the numerator at the denominator, the fraction becomes 1/2. If numerator is subtracted from the denominator, the fraction becomes 3/4. Find the fraction.Correct answer is '3/7'. Can you explain this answer?

Question

Answer

Given:
- When 1 is added to both the numerator and denominator, the fraction becomes 1/2.
- When the numerator is subtracted from the denominator, the fraction becomes 3/4.

To find: The original fraction.

Solution:
Let the original fraction be x/y.

Condition 1: When 1 is added to both the numerator and denominator, the fraction becomes 1/2.

Adding 1 to the numerator and denominator, the new fraction becomes (x+1)/(y+1).
According to the given condition, (x+1)/(y+1) = 1/2.

Cross-multiplying and simplifying, we get:
2x + 2 = y + 1
2x - y = -1 ---(Equation 1)

Condition 2: When the numerator is subtracted from the denominator, the fraction becomes 3/4.

Subtracting numerator from the denominator, the new fraction becomes (y-x)/y.
According to the given condition, (y-x)/y = 3/4.

Cross-multiplying and simplifying, we get:
4y - 4x = 3y
y = 4x/(-1) ---(Equation 2)

Substituting Equation 2 in Equation 1, we get:
2x - 4x/(-1) = -1
2x + 4x = 1
6x = 1
x = 1/6

Substituting x in Equation 2, we get:
y = 4x/(-1) = -4/6 = -2/3

Since the fraction x/y must be positive, we take the absolute values of x and y. Therefore, the original fraction is |x/y| = |-1/2| = 1/2.

Hence, the solution is 1/2.

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