Question

If the numerator of a fraction is increased by 1, its value becomes 3/4. If its denominator is increased by 2, its value becomes 1/2. Then (i) Write the equations for the given conditions , (ii) Find the fraction.

Answer 1

Answer:

So the fraction is 5/8

Step-by-step explanation:

Let the fraction is

$\frac{x}{y}$

Now,

When the numerator is increased by 1 then the equation is,

$\frac{x + 1}{y} = \frac{3}{4} \\ 4x + 4 = 3y \\ 4x - 3y = - 4$

and,

when the denominator is increased by 2 the. the equation is,

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

$do \: (1) - (2) \times 3 \\ we \ \: get \\ 4x - 3y - 6x + 3y = - 4 - 6 \\ - 2x = - 10 \\ x = 5$

put the value of x into equation (2) we get,

10 - y = 2

so, y = 8

Then the fraction is,

$\frac{5}{8}$

Answer 2

Given :

• If the numerator of a fraction is increased by 1, it's value becomes 3 / 4.
• If the denominator is increased by 2, it's value becomes 1 / 2.

To find :

• The equations for the given conditions.
• The Fraction.

According to the question :

Let numerator be 'n' and denominator be 'd'.

• ∴ The fraction will be n / d

It is given that,

• When numerator is increased by 1, it's value becomes 3 / 4,

$⟶\sf\dfrac{n\:+\:1}{d}\:=\:\dfrac{3}{4}$

$⟶\sf{4\:(n\:+\:1)\:=\:3d}$

$⟶\sf{4n\:+\:4\:=\:3d}$

$⟶\sf{4n\:-\:3d\:=\:4\:-------(1)}$

• When denominator is increased by 2, it's original value becomes 1 / 2,

$⟶\sf\dfrac{n}{d\:+\:2}\:=\:\dfrac{1}{2}$

$⟶\sf{2\:n\:=\:1\:(d\:+\:2)}$

$⟶\sf{2\:n\:=\:d\:+\:2\:-------(2)}$

To find any one of the value, let's cancel 'n',

And to make both the value of 'n' same, let's multiply equation (2) by 2.

∴ It will be, 4n = 2d + 4

Subtracting (1) from (2):

$⟶\sf{4n\:=\:2d\:+\:4}$

$⟶\sf{4n\:-\:3d\:=\:4}$

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$⟶\sf{d\:=\:8}$

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[ Both 4n s are positive. To change the sign of any one, we have to subtract. so (+4n) and (-4n) got cancelled ]

Substituting 'd' value in (2) :

$⟶\sf{2n\:=\:d\:+\:2}$

$⟶\sf{2n\:=\:8\:+\:2}$

$⟶\sf{2n\:=\:10}$

$⟶\sf{n\:=\:\dfrac{10}{2}}$

$⟶\sf{n\:=\:5}$

• ∴ The fraction is $\sf{\boxed{\dfrac{5}{8}}}$