Question

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

Answer 1

Answer:

Step-by-step explanation:

Given,

• A fraction becomes equal to 4/5 if 1 is added both numerator and denominator.
• If however 5 is substracted from both numerator and denominator the fraction becomes equal to 1/2.

To Find,

• The Fraction.

Solution,

Let the fraction be x/y.

Then,

According to the question,

x + 1/y + 1 = 4/5

And, x - 5/y - 5 = 1/2

⇒ 5x + 5 = 4y and 2x - 10 = y - 5

⇒ 5x - 4y + 1 = 0 and 2x - y - 5 = 0

By using cross-multiplication, we have

⇒ x/- 4 × (- 5) - (- 1) × 1 = - y/5 × (- 5) - 2 × 1 = 1/5 × (- 1) - 2 × (- 4)

⇒ x/20 + 1 = y/25 + 2 = 1/- 5 + 8

⇒ x/21 = y/27 = 1/3

⇒ x = 21/3 = 7

⇒ y = 27/3 = 9

Hence, the required fraction is 7/9.

Answer 2

★ Solution :-

Let the numerator be x.

Let the denominator be y.

So, the fraction is

$\sf \leadsto \dfrac{x}{y}$

In the first case,

$\sf \leadsto \dfrac{x + 1}{y + 1} = \dfrac{4}{5}$

$\sf \leadsto 5(x + 1) = 4(y + 1)$

$\sf \leadsto 5x + 5 = 4y + 4$

$\sf \leadsto 5x - 4y = 4 - 5$

$\sf \leadsto 5x - 4y = -1 \: \: --- (i)$

In the second case,

$\sf \leadsto \dfrac{x - 5}{y - 5} = \dfrac{1}{2}$

$\sf \leadsto 2(x - 5) = 1(y - 5)$

$\sf \leadsto 2x - 10 = 1y - 5$

$\sf \leadsto 2x - 1y = -5 + 10$

$\sf \leadsto 2x - 1y = 5 \: \: --- (ii)$

By first equation,

$\sf \leadsto 5x - 4y = -1$

$\sf \leadsto 5x = -1 + 4y$

$\sf \leadsto x = \dfrac{-1 + 4y}{5}$

Now, we should find the value of y by second equation.

$\sf \leadsto 2x - 1y = 5$

$\sf \leadsto 2 \bigg( \dfrac{-1 + 4y}{5} \bigg) - 1y = 5$

$\sf \leadsto \dfrac{-2 + 8y}{5} - 1y = 5$

$\sf \leadsto \dfrac{-2 + 8y - 5y}{5} = 5$

$\sf \leadsto \dfrac{-2 + 3y}{5} = 5$

$\sf \leadsto -2 + 3y = 5 \times 5$

$\sf \leadsto -2 + 3y = 25$

$\sf \leadsto 3y = 25 + 2$

$\sf \leadsto 3y = 27$

$\sf \leadsto y = \dfrac{27}{3}$

$\sf \leadsto y = 9$

Now, we can find the value of x by first equation.

$\sf \leadsto 5x - 4y = -1$

$\sf \leadsto 5x - 4(9) = -1$

$\sf \leadsto 5x - 36 = -1$

$\sf \leadsto 5x = -1 + 36$

$\sf \leadsto 5x = 35$

$\sf \leadsto x = \dfrac{35}{5}$

$\sf \leadsto x = 7$

We know that,

Value of x :- 7

Value of y :- 9

So, the fraction is $\sf \dfrac{7}{9}$