Question

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of new fraction and the original fraction is \( \frac{29}{20} \). Find the original fraction.

Hint:

When solving problems involving fractions, ensure to combine the fractions by finding a common denominator and then solve the resulting equation.

Answer 1

Let the original fraction be \( \frac{x}{x+3} \), where \( x \) is the numerator and \( x+3 \) is the denominator.
Step 1: The new fraction formed by adding 2 to both the numerator and denominator is \( \frac{x+2}{x+5} \).
Step 2: We are told that the sum of the original fraction and the new fraction is \( \frac{29}{20} \). Therefore, we have the equation: \[ \frac{x}{x+3} + \frac{x+2}{x+5} = \frac{29}{20}. \]
Step 3: To add the two fractions on the left-hand side, first find the common denominator: \[ \frac{x}{x+3} + \frac{x+2}{x+5} = \frac{x(x+5) + (x+2)(x+3)}{(x+3)(x+5)}. \] Simplify the numerator: \[ x(x+5) + (x+2)(x+3) = x^2 + 5x + x^2 + 5x + 6 = 2x^2 + 10x + 6. \] So, we have: \[ \frac{2x^2 + 10x + 6}{(x+3)(x+5)} = \frac{29}{20}. \]
Step 4: Now, cross multiply to solve for \( x \): \[ 20(2x^2 + 10x + 6) = 29(x+3)(x+5). \] Simplify both sides: \[ 40x^2 + 200x + 120 = 29(x^2 + 8x + 15). \] Expand the right-hand side: \[ 40x^2 + 200x + 120 = 29x^2 + 232x + 435. \]
Step 5: Now, move all terms to one side: \[ 40x^2 + 200x + 120 - 29x^2 - 232x - 435 = 0, \] \[ 11x^2 - 32x - 315 = 0. \]
Step 6: Solve the quadratic equation using the quadratic formula: \[ x = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(11)(-315)}}{2(11)}. \] Simplify: \[ x = \frac{32 \pm \sqrt{1024 + 13860}}{22} = \frac{32 \pm \sqrt{14884}}{22} = \frac{32 \pm 122}{22}. \]
Step 7: Thus, \( x = \frac{32 + 122}{22} = \frac{154}{22} = 7 \) or \( x = \frac{32 - 122}{22} = \frac{-90}{22} \) (which is not valid since \( x \) must be positive).
Step 8: Therefore, the numerator of the original fraction is \( x = 7 \), and the denominator is \( x + 3 = 10 \). Thus, the original fraction is \( \frac{7}{10} \).

Conclusion: The original fraction is \( \frac{7}{10} \).