Question

the numerator of a fraction is 4 less than the denominator. if 1 is added to both the numerator and the denominator, the fraction becomes 1/2.​

Answer 1

⇝ Appropriate Question :-

The numerator of a fraction is 4 less than the denominator. If 1 is added to both the numerator and the denominator, the fraction becomes 1/2. Then Find the Original Fraction.

⇝ Given :-

• The numerator of a fraction is 4 less than the denominator.

• If 1 is added to both the numerator and the denominator, the fraction becomes 1/2.

⇝ To Find :-

• The Original Fraction.

⇝ Solution :-

For Original Fraction Let,

• Numerator = $\large \rm x$

As The numerator of a fraction is 4 less than the denominator.

Therefore,

• Denominator = $\large \rm (x+4)$

So ,

$\text{Original Fraction = } \frac{\text x}{\text x + 4}$

❒ When 1 is added to both the numerator and the denominator :

$\text{Fraction Becomes : } \frac{\text x + 1}{\text x + 5}$

⏩ According To Question :

$\bf \large\red{ \frac{ x + 1}{x + 5}  =  \frac{1}{2}}$

✧ On Cross Multiplying ;

$:\longmapsto \rm 2(x + 1) = x + 5$

$:\longmapsto \rm 2x + 2 = x + 5$

$:\longmapsto \rm 2x - x = 5 - 2$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf x = 3} }}}$

∴ Numerator of Fraction is 3 .

❒ As The numerator of a fraction is 4 less than the denominator.

$\therefore \rm Denominator = Numerator + 4$

$:\longmapsto \rm Denominator = 3 + 4$

∴ Denominator of Fraction is 7 .

Therefore,

$\large\underline{\pink{\underline{\frak{\pmb{ Original  \: Fraction  =  \dfrac{3}{7} }}}}}$

Answer 2

G I V E N :

The numerator of a fraction is 4 less than the denominator. If 1 is added to both the numerator and the denominator, the fraction becomes 1/2. Find the original fraction

S O L U T I O N :

Let us assume the numerator be α

Now, according to the question the numerator is 4 less then the denominator

So the denominator should be α + 4

As we know that

$\hookrightarrow \frak{Original \: Fraction = \frac{Numerator}{Denominator}}$

Now,

$\hookrightarrow \frak{Original \: Fraction = \frac{ \alpha }{ \alpha + 4} }$

Now, 1 is added to both numerator and denominator

$\hookrightarrow \frak{ \frac{ \alpha + 1}{ \alpha + 4 + 1} }$

$\hookrightarrow \frak{ \frac{ \alpha + 1}{ \alpha + 5}}$

The new fraction becomes ½

$\leadsto \red{\frak{Original \: Fraction = New \: Fraction }}$

$\leadsto \frak{ \frac{ \alpha + 1}{ \alpha + 5} = \frac{1}{2} }$

Cross multiplying we get

$\leadsto \frak{2( \alpha + 1) = 1( \alpha + 5)}$

$\leadsto\frak{2 \alpha + 2 = \alpha + 5}$

$\leadsto \frak{2 \alpha - \alpha = 5 - 2}$

$\star \: \: \red{\underline{ \pink{\boxed{ \frak{ \purple{\alpha = 3}}}}}}$

____________________________

Finding the original fraction

$\implies \frak{Original \: Fraction = \frac{ \alpha}{ \alpha + 4} }$

Putting α = 3 we get

$\implies \frak{Original \: Fraction = \frac{ 3}{ 3 + 4} }$

$\implies \frak{Original \: Fraction = \frac{3}{7} }$

• Hence, the original fraction is 3/7