Question

If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes 1/2 if we only add 1 to the denominator. What is the fraction?

➡Class 10th

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Answer 1

HEYA !

Let the numerator be x and the denominator be y

Then the fraction is x / y

Now,

According to the question,

$\frac{x + 1}{y - 1} = 1 \\ \\ \Rightarrow \: \: \: x + 1 = y - 1 \\ \\ \Rightarrow \: \: \: x = y - 1 - 1 \\ \\ \Rightarrow \: \: \: x = y - 2 \: \: \: \: \: \: \: \: \: .....(1) \\ \\ \text{And} \\ \\ \frac{x}{y + 1} = \frac{1}{2} \\ \\ \Rightarrow \: \: \:2x = y + 1 \\ \\ \Rightarrow \: \: \:2(y - 2) = y + 1 \: \: \: \: \: \: \: \{\text{from (1)} \} \\ \\ \Rightarrow \: \: \:2y - 4 = y + 1 \\ \\ \Rightarrow \: \: \:2y - y = 1 + 4 \\ \\ \Rightarrow \: \: \: \boxed{ \: y = 5 \: } \\ \\ \text{putting y = 5 in (1) we get} \\ \\ x = 5 - 2 \\ \\ \Rightarrow \: \: \: \boxed{ \: x = 3\: }$
Fraction = x / y = 3/5

Hence,

The required fraction is 3/5

Answer 2

Heya mate,

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Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is x/y

If 1 is added to the numerator and 1 is subtracted from the denominator, the fraction becomes 1 . Thus, we have

$\frac{x + 1}{y - 1} = 1 \\ = > x + 1 = y - 1 \\ = > x + 1 - y + 1 = 0 \\ = > x - y + 2 = 0$


If 1 is added to the denominator, the fraction becomes 1/2. Thus, we have

$\frac{x}{y + 1} = \frac{1}{2} \\ = > 2x = y + 1 \\ = > 2x - y - 1 = 0$


So, we have two equations

$x - y + 2 = 0 \\ \\ 2x - y - 1 = 0$


Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{( - 1) \times ( - 1) - ( - 1) \times 2} = \frac{ - y}{1 \times ( - 1) - 2 \times ( 2)} = \frac{1}{1 \times ( - 1) - 2 \times ( - 1)} \\ \\ = > \frac{x}{1 + 2} = \frac{ - y}{ - 1 - 4} = \frac{1}{ - 1 + 2} \\ \\ = > \frac{x}{3} = \frac{ - y}{ - 5} = \frac{1}{1} \\ \\ = > \frac{x}{3} = \frac{y}{5} = 1 \\ \\ = > x = 3 \: . \: \: \: y = 5$




Hence, the fraction is 3/5 .

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