Question

In a given fraction if the numerator is multiplied by 2 and the denominator is reduced by 5 , we get 6 /5 . But if the numerator of the given fraction is increased by 8 and the denominator is doubled we get 2/ 5 . Find the fraction.

Answer 1

Answer:

x = 12, y = 25

Step-by-step explanation:

let the numerator be x and denominator be y

according to the question:

x*2/y-5 = 6/5

so,

2x/y-5 = 6/5

by cross multiplication

5*2x = 6(y-5)

10x = 6y - 30

10x - 6y + 30 = 0. (first eq)

x + 8/ 2y = 2/5

again by cross multiplication

5(x + 8) = 2*2y

5x + 40 = 4y

5x - 4y + 40 = 0. (second eq)

[10x - 6y + 30 = 0]*1

[5x - 4y + 40 = 0]*2

10x - 6y + 30 = 0

10x - 8y + 80 = 0

- +. -

2y. - 50 = 0

2y = 50

y = 25

putting the value of y in the eq

5x - 4y + 40 = 0

5x - 4*25 + 40 = 0

5x - 100 + 40 = 0

5x = 100 - 40

5x = 60

x = 12

Answer 2

Given :-

• The numerator is multiplied by 2 and denominator is reduced by 5,we get 6/5

• The numerator of the given fraction is increased by 8 and the denominator is doubled we get 2/5

To Find :-

• The required fraction =?

Solution :-

Let the numerator be x and denominator be y

$\rm\therefore\:Fraction=\dfrac{x}{y}$

then,

According to the question:-

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\green{\bigstar}{\underline{\boxed{\bf\red{\dfrac{2x}{y-5}=\dfrac{6}{5}}}}}\green{\bigstar}$

$\rm\longrightarrow\:2x\times5=6\times(y-5)$

$\rm\longrightarrow\:10x=6y-30$

$\rm\longrightarrow\:10x-6y=-30$

$\rm\longrightarrow\:5x-3y=-15\:..........(i)$

Now,

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\purple{\bigstar}{\underline{\boxed{\bf\pink{\dfrac{x+8}{2y}=\dfrac{2}{5}}}}}\purple{\bigstar}$

$\rm\longrightarrow\:5\times\:(x+8)=2\times\:2y$

$\rm\longrightarrow\:5(x+8)=4y$

$\rm\longrightarrow\:5x+40=4y$

$\rm\longrightarrow\:5x-4y=-40\:.........(ii)$

On subtracting eq (ii) from eq (i) ,we get

$\rm\longrightarrow\:y=25$

Now,

putting the value of y in equation (i)

$\rm\:5x-(3\times\:25)=-15$

$\rm\longrightarrow\:5x-75=-15$

$\rm\longrightarrow\:5x=-15+75$

$\rm\longrightarrow\:5x=60$

$\rm\longrightarrow\:x=\cancel\dfrac{60}{5}$

$\rm\longrightarrow\:x=12$

$\rm\:Hence,the\:required\:fraction\:is\:\dfrac{12}{25}$