Question

if 1 is subtracted from the numerator and 2 is added to the denominator of a fraction the value obtained is 1/2. if 3 is added to its numerator and 2 is subtracted from its denominator the resulting value is 5/4. find the fraction.​

Answer 1

Answer:

Step-by-step explanation:

Let the numerator be x and denominator be y.

Fraction = $\frac{x}{y}$

According to question,

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

2(x-1) = y + 2

2x - 2 = y + 2

2x - y = 4 --- (i)

$\frac{x + 3}{y - 2} = \frac{5}{4}$

4(x + 3) = 5(y - 2)

4x + 12 = 5y - 10

4x - 5y = -10 - 12

5y - 4x = 22 -- (ii)

Multiply (i) by 2,

4x - 2y = 8

Add this to (ii)

3 y = 30

y = 10

Substitue in (i)

2x - 10 = 4

2x = 14

x = 7

Therfroe, fraction is $\frac{7}{10}$.

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Answer 2

Let the numerator be x .

Then denominator be y.

According to the first case,

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

$\rightarrow \: 2(x - 1) = 1(y + 2)$

$\rightarrow \: 2x - 2 = y + 2$

$\rightarrow \: 2x - y = 2 + 2$

$\rightarrow \: y = 2x - 4$

According to second case,

$\frac{x + 3}{y - 2} = \frac{5}{4}$

$\rightarrow \: 4(x + 3) = 5(y - 2)$

$\rightarrow \: 4x + 12 = 5y - 10$

$\rightarrow \: 4x - 5y = - 10 - 12$

$\rightarrow \: 4x - 5y = - 22$

Now,

Putting the value of y in equation ( 2 )

$4x - 5y = - 22$

$\implies \: 4x - 5(2x - 4) = - 22$

$\implies \: 4x - 10x + 20 = - 22$

$\implies \: - 6x = - 22 - 20$

$\implies \: - 6x = - 42$

$\implies \: x = 7$

Putting the value of x in equation ( 1 )

$y = 2x - 4$

$\implies \:y = 2 \times 7 - 4$

$\implies \: y = 14 - 4$

$\implies \: y = 10$

Hence, Fraction = 7/10