# If we add 1 to the numerator and subtract 2 from the denominator. A fraction reduces to 1. It becomes $\dfrac{1}{2}$. If we only add 1 the denominator. Find the fraction by the method of elimination?

Last updated date: 28th Mar 2023

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Answer

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Hint: We have to assume the numerator and denominator as x and y. Then, we can use the cross-multiplication and elimination method to solve for x and y and hence obtain the fraction.

Complete step-by-step Solution:

Before proceeding we should know that cross multiplication method is the method where the numerator from the left-hand side is multiplied with the denominator from the right-hand side and numerator from the right-hand side is multiplied with the denominator from the left-hand side.

Let, the numerator and denominator be $x$ and $y$.

Therefore, the fraction is $\dfrac{x}{y}$.

The question we have,

$\dfrac{x+1}{y-1}=1......(i)$

$\dfrac{x}{y+1}=\dfrac{1}{2}......(ii)$

Cross multiplying equation (i) and (ii) we get,

$x+1=y-1......(iii)$

$2x=y+1.....(iv)$

Multiplying equation (iii) on both sides by 2 we get,

$\Rightarrow 2x+2=2y-2.....(v)$

Subtracting equation (iv) from equation (v) we get,

$\Rightarrow 2x+2-2x=2y-2-y-1$

$\Rightarrow 2=y-3$

$\Rightarrow y=5$

Now, substitute the value of$y=5$ in equation (iv) we get,

$\Rightarrow 2x=5+1$

$\therefore x=3$

Hence, the required fraction is $\dfrac{3}{5}$ .

Note: The alternate method to solve this question, if we are provided with options in the question then without solving we can check the answers through options, from options we just substitute the values of numerator and denominator and the option which satisfies the question is the answer. Do not mess up with the question frame your equation accordingly as mentioned in the question. Try to cancel the terms which need to be cancelled otherwise it seems hectic as we proceed further. This question can also be cross-checked after getting the answer by just substituting the value of numerator and denominator in the question.

Complete step-by-step Solution:

Before proceeding we should know that cross multiplication method is the method where the numerator from the left-hand side is multiplied with the denominator from the right-hand side and numerator from the right-hand side is multiplied with the denominator from the left-hand side.

Let, the numerator and denominator be $x$ and $y$.

Therefore, the fraction is $\dfrac{x}{y}$.

The question we have,

$\dfrac{x+1}{y-1}=1......(i)$

$\dfrac{x}{y+1}=\dfrac{1}{2}......(ii)$

Cross multiplying equation (i) and (ii) we get,

$x+1=y-1......(iii)$

$2x=y+1.....(iv)$

Multiplying equation (iii) on both sides by 2 we get,

$\Rightarrow 2x+2=2y-2.....(v)$

Subtracting equation (iv) from equation (v) we get,

$\Rightarrow 2x+2-2x=2y-2-y-1$

$\Rightarrow 2=y-3$

$\Rightarrow y=5$

Now, substitute the value of$y=5$ in equation (iv) we get,

$\Rightarrow 2x=5+1$

$\therefore x=3$

Hence, the required fraction is $\dfrac{3}{5}$ .

Note: The alternate method to solve this question, if we are provided with options in the question then without solving we can check the answers through options, from options we just substitute the values of numerator and denominator and the option which satisfies the question is the answer. Do not mess up with the question frame your equation accordingly as mentioned in the question. Try to cancel the terms which need to be cancelled otherwise it seems hectic as we proceed further. This question can also be cross-checked after getting the answer by just substituting the value of numerator and denominator in the question.

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