If the numerator of a fraction is increased b | vedclass

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(A) Let the fraction be $\frac{x}{y}$.According to the first condition,$\frac{x+2}{y+3} = \frac{7}{9}$.Cross-multiplying gives $9(x+2) = 7(y+3)$,which

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$\frac{5}{6}$

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(A) Let the fraction be $\frac{x}{y}$.According to the first condition,$\frac{x+2}{y+3} = \frac{7}{9}$.Cross-multiplying gives $9(x+2) = 7(y+3)$,which simplifies to $9x + 18 = 7y + 21$,or $9x - 7y = 3$ ..........$(1)$According to the second condition,$\frac{x-1}{y-1} = \frac{4}{5}$.Cross-multiplying gives $5(x-1) = 4(y-1)$,which simplifies to $5x - 5 = 4y - 4$,or $5x - 4y = 1$ ..........$(2)$To solve the system of equations,multiply $(1)$ by $4$ and $(2)$ by $7$:$36x - 28y = 12$ ..........$(3)$$35x - 28y = 7$ ..........$(4)$Subtracting $(4)$ from $(3)$ gives $x = 5$.Substituting $x = 5$ into $(2)$:$5(5) - 4y = 1 \Rightarrow 25 - 4y = 1 \Rightarrow 4y = 24 \Rightarrow y = 6$.Thus,the original fraction is $\frac{5}{6}$.

If the numerator of a fraction is increased by $2$ and the denominator is increased by $3,$ the fraction becomes $\frac{7}{9},$ and if both the numerator and the denominator are decreased by $1,$ the fraction becomes $\frac{4}{5}.$ What is the original fraction?

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