Question

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

Answer 1

Let numerator of a fraction is x and denominator is y .

A/c to question,
fraction becomes 4/5 if 1 is added to both numerator and denominator.
e.g., (x + 1)/(y + 1) = 4/5
5(x + 1) = 4(y + 1)
5x + 5 = 4y + 4
4y - 5x = 1 ......... ..(i)

again, a/c to question, if 5 is subtracted from both numerator and denominator , the fraction becomes 1/2 .
e.g., (x - 5)/(y - 5) = 1/2
2(x - 5) = 1(y - 5)
2x - 10 = y - 5
2x - y = 5 ..............(ii)

solve equations (i) and (ii),
(4y - 5x) + 4(2x - y) = 1 + 4 × 5
4y - 5x + 8x - 4y = 21
3x = 21 => x = 7

put x = 7 in equation (i),
4y - 5 × 7 = 1
4y = 36 => y = 9


hence, fraction = x/y = 7/9

Answer 2

Dear Student,

Answer: Fraction is 7/9.
➖➖➖➖➖➖➖➖

Solution:
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let the rational number is x/y.

As fraction becomes 4/5 if 1 is added to both numerator and denominator

$\frac{x + 1}{y + 1} = \frac{4}{5} \\ \\ 5x + 5 = 4y + 4 \\ \\ 5x - 4y = - 1 \: \: \: \: \: \: \: \: \: \: eq1$
As, 5 is subtracted from both numerator and denominator, the fraction becomes 1/2

$\frac{x - 5}{y - 5} = \frac{1}{2} \\ \\ 2x - 10 = y - 5 \\ \\ 2x - y = 5 \: \: \: \: \: \: \: \: eq2$

Multiply eq 2 by 4 and subtract both equations

$5x - 4y - (8x - 4y) = - 1 - 20 \\ \\ 5x - 8x - 4y + 4y = - 21 \\ \\ - 3x = - 21 \\ \\ x = 7$

substitute the value of x into eq2
$2x - y = 5 \\ \\ 2(7) - y = 5 \\ \\ - y = 5 - 14 \\ \\ y = 9$
The fraction is
$\frac{7}{9}$
Hope it helps you.