Question
The sum of the numerator and denominator of a positive fraction is 11. If 2 is added to both numerator and denominator, the fraction is increased by $1/24$. The difference of numerator and denominator of the fraction is
Answer: Option B
Answer: (b)Let numerator be $x$, then denominator = 11 – $x$. Fraction =$x/{11 - x}$Again, ${x + 2}/{11 – x + 2} = x/{11 - x} + 1/24$⇒ ${x + 2}/{13 - x} – x/{11 - x} = 1/24$⇒ ${11x – x^2 + 22 – 2x – 13x +x^2}/{(13 - x)(11 - x)} = 1/24$⇒ ${22 – 4x}/{(13 - x)(11 - x)} = 1/24$⇒ $528 – 96x = 143 – 24x + x^2$⇒ $x^2 + 72x – 385$ = 0⇒ $x^2 + 77x – 5x – 385$ = 0 ⇒ $x (x + 77) - 5 (x + 77)$ = 0⇒ (x - 5) (x + 77) = 0 ⇒ x = 5 Denominator = 11 - 5 = 6 Difference = 6 - 5 = 1
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Answer: (b)Let numerator be $x$, then denominator = 11 – $x$. Fraction =$x/{11 - x}$Again, ${x + 2}/{11 – x + 2} = x/{11 - x} + 1/24$⇒ ${x + 2}/{13 - x} – x/{11 - x} = 1/24$⇒ ${11x – x^2 + 22 – 2x – 13x +x^2}/{(13 - x)(11 - x)} = 1/24$⇒ ${22 – 4x}/{(13 - x)(11 - x)} = 1/24$⇒ $528 – 96x = 143 – 24x + x^2$⇒ $x^2 + 72x – 385$ = 0⇒ $x^2 + 77x – 5x – 385$ = 0 ⇒ $x (x + 77) - 5 (x + 77)$ = 0⇒ (x - 5) (x + 77) = 0 ⇒ x = 5 Denominator = 11 - 5 = 6 Difference = 6 - 5 = 1
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