Question

The numerator of a fraction is 5 less than the denominator. If 3 is added to both its numerator and denominator, it becomes 1/2. Find the fraction​

Answer 1

$\large\sf\underline{Given\::}$

• Numerator of a number is 5 less than the denominator.

• When 3 is added to both the numerator and denominator it becomes $\sf\:\frac{1}{2}$ .

‎

‎

$\large\sf\underline{To\:find\::}$

• The fraction which would verify my given part.

‎

‎

$\large\sf\underline{Understanding\:the\:concept\::}$

Here in the question we are given that the numerator is 5 less than denominator. So we will assume the value of denominator and then eventually get the value of numerator . Now again we are said that if we add 3 to both numerator and denominator the fraction becomes $\sf\:\frac{1}{2}$. Therefore we will make up an equation according to the question and then solve it. Doing so we will get the value of x. Then we will substitute value of x in both numerator and denominator and we will end up getting the required fraction. So let's begin !

‎

‎

$\large\sf\underline{Solution\::}$

We are given that the numerator is 5 less than denominator so let's assume that :

• Numerator = ( x - 5 )

• Denominator = x

Now according to the question we need to add 3 to both numerator and denominator :

$\sf\to\:\frac{x-5+3}{x+3}$

After adding 3 it is said that the fraction becomes $\sf\:\frac{1}{2}$ . So now equating both we get :

$\sf\to\:\frac{x-5+3}{x+3}= \frac{1}{2}$

• Solving them

$\sf\implies\:\frac{x-2}{x+3}= \frac{1}{2}$

• Cross multiplying

$\sf\implies\:2(x-2)= 1(x+3)$

$\sf\implies\:2x-4= x+3$

• Transposing x to other side

$\sf\implies\:2x- x-4=3$

$\sf\implies\:x-4=3$

• Transposing 4 to other side

$\sf\implies\:x=3+4$

$\large{\mathfrak\red{\implies\:x\:=\:7}}$

‎

‎

Let's substitute the value of x in the fraction :

We know,

$\sf\:Fraction\:=\frac{Numerator}{Denominator}$

$\sf\implies\:Fraction\:=\frac{x-5}{x}$

$\sf\implies\:Fraction\:=\frac{7-5}{7}$

${\sf{{\orange{\implies\:Fraction\:=\frac{2}{7}}}}}$

‎ ___________________

‎ $\large\sf\underline{Verification\:of\:my\:answer\::}$

We got our fraction to be

• ${\sf{{\orange{Fraction\:=\frac{2}{7}}}}}$.

Let's verify it with given part .

It was said that numerator is 5 less than denominator let's see if it is so :

• Numerator = 2

• Denominator = 7

$\sf\:Denominator\:-Numerator\:=5$

$\sf\implies\:7\:-2\:=5$

$\sf\implies\:5\:=5$

$\dag\:\underline{\sf [\:Hence\:verified\:]}$

‎

‎

Another case was if we add 3 to both numerator and denominator we would get $\sf\:\frac{1}{2}$.

Let's verify this case also :

$\sf\:\frac{2+3}{7+3}=\frac{1}{2}$

$\sf\implies\:\frac{5}{10}=\frac{1}{2}$

$\sf\implies\:\cancel{\frac{5}{10}}=\frac{1}{2}$

$\sf\implies\:\frac{1}{2}=\frac{1}{2}$

$\dag\:\underline{\sf [\:Hence\:verified\:]}$

‎ ___________________

Hence both the given part is verified so we are confirmed that required fraction is $\small{\underline{\boxed{\mathrm\pink{\frac{2}{7}}}}}$.

‎

‎

!! Hope it helps !!