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 numerator is doubled, we get 2/5. Find the fraction.

Answer 1

Answer:

The fraction is 12/25.

Step-by-step explanation:

Given :-

• If the numerator is multiplied by 2 and the denominator is reduced by 5, we get 6/5.
• If the numerator of the given fraction is increased by 8 and the denominator is doubled, we get 2/5.

To find :-

• The fraction.

Solution :-

Let the numerator of the fraction be x and the denominator of the fraction be y.

According to the 1st condition ,

• If the numerator is multiplied by 2 and the denominator is reduced by 5, we get 6/5.

$\to\sf{\dfrac{2x}{y-5}=\dfrac{6}{5}}$

$\to\sf{10x=6y-30}$

$\to\sf{10x=6(y-5)}$

$\to\sf{5x=3y-15}$

$\to\sf{5x-3y=-15.................(i)}$

According to the 2nd condition,

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

$\to\sf{\dfrac{x+8}{2y}=\dfrac{2}{5}}$

$\to\sf{5x+40=4y}$

$\to\sf{5x-4y=-40...............(ii)}$

Now subtract eq (I) from eq (ii).

5x-4y-(5x-3y)=-40-(-15)

→ 5x-4y-5x+3y = -40+15

→ -y = -25

→ y = 25

• Denominator = 25

Now put y = 25 in eq(i) for getting the value of x.

5x-3y = -15

→ 5x -3×25 = -15

→5x = -15 + 75

→5x = 60

→ x = 12

• Numerator = 12

Therefore,

${\boxed{\sf{The\: fraction=\dfrac{12}{25}}}}$

Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• If the numerator is multiplied by 2 and the denominator is reduced by 5, the fraction becomes 6/5.

• If the numerator of the given fraction is increased by 8 and the denominator is doubled, we get ⅖.

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The original fraction.

$\bf{\underline{\underline\green{Solution:-}}}$

Let the numerator be x

The denominator be y

According to the 1st condition:-

If the numerator is multiplied by 2

The numerator = 2x

If denominator is reduced by 5

The denominator = y - 5

The fraction becomes 6/5

$\rm \implies \dfrac{2x}{y - 5} = \dfrac{6}{5}$

$\rm \implies {5(2x)} = {6(y - 5)}$

$\rm \implies {10x} = {6y -30}$

$\rm \implies {10x - 6y} = {-30}$

Dividing the whole equation by 2

$\rm \implies {5x - 3y} = {-15}....(i)$

Acccording to the 2nd condition:-

If numerator is increased by 8

The numerator = x + 8

If denominator is doubled

The denominator = 2y

The fraction becomes ⅖.

$\rm \implies\dfrac{x + 8}{2y} = \dfrac{2}{5}$

$\rm \implies{5(x + 8)} = {2(2y)}$

$\rm \implies{5x + 40} = {4y}$

$\rm \implies{5x - 4y} = { - 40} ....(ii)$

Subtracting equation (ii) from (i)

$\rm \implies{5x - 3y - (5x - 4y)} = { - 15 - ( - 40)}$

$\rm \implies{5x - 3y - 5x + 4y} = { - 15 + 40}$

$\rm \implies{y} = { 25}$

Substituting y = 25 in equation (i)

$\rm \implies5x - 3y = { - 15}$

$\rm \implies5x - 3(25)= { - 15}$

$\rm \implies5x - 75= { - 15}$

$\rm \implies5x = { - 15 + 75}$

$\rm \implies5x = { 60}$

$\rm \implies x = { 12}$

The numerator = x = 12

The denominator = y = 25

$\underline{ \boxed{ \purple{\rm \therefore \: The \: fraction = \frac{12}{25} } }}$