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Question

A fraction becomes $$\dfrac{4}{5}$$ if $$1$$ is added to both numerator and denominator. If, however, $$5$$ is subtracted from both numerator and denominator, the fraction becomes $$\dfrac{1}{2}$$. What is the fraction?


Solution

Let the fraction be $$\dfrac xy$$
$$\Rightarrow \dfrac { x+1 }{ y+1 } =\dfrac { 4 }{ 5 } $$
$$\Rightarrow   5x+5 = 4y+4$$
$$\Rightarrow   y = \dfrac{5x+1}4 $$

$$\Rightarrow \dfrac { x-5 }{ y-5 } =\dfrac { 1 }{ 2 } $$
$$\Rightarrow \cfrac { x-5 }{ \dfrac {5x+1} {4}-5 } =\cfrac { 1 }{ 2 } $$
$$\Rightarrow \cfrac { 4(x-5) }{ 5x-19 } =\cfrac12\\ \Rightarrow 8x-40=5x-19\\ \Rightarrow x=7\\ \Rightarrow y=\dfrac { 5\times7+1 }{ 4 } =9$$

Fraction is $$\dfrac{ 7 }{ 9 }$$

Maths

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