Question

The numerator of a fraction is 5 less than its denominator, if 2 is subtracted from the numerator and 2 is added to the denominator the fraction becomes .Find the original fraction​

Answer 1

Answer:

Appropriate Question :-

• The numerator of a fraction is 5 less than its denominator, if 2 is subtracted from the numerator and 2 is added to the denominator, the fraction becomes 2/5. Find the original fraction.

Given :-

• The numerator of a fraction is 5 less than its denominator, if 2 is subtracted from the numerator and 2 is added to the denominator, the fraction becomes 2/5.

To Find :-

• What is the original fraction.

Solution :-

Let,

$\mapsto \bf Numerator =\: x$

$\mapsto \bf Denominator =\: x + 5$

Hence, the required original fraction is :

$\leadsto \sf Original\: Fraction =\: \dfrac{Numerator}{Denominator}$

$\leadsto \sf\bold{\pink{Original\: Fraction =\: \dfrac{x}{x + 5}}}$

According to the question,

$\bigstar$ 2 is subtracted from the numerator and 2 is added to the denominator, the fraction becomes 2/5.

$\implies \bf \bigg\{\dfrac{Numerator - 2}{Denominator + 2}\bigg\} =\: \bigg\{\dfrac{2}{5}\bigg\}$

$\implies \sf \dfrac{x - 2}{x + 5 + 2} =\: \dfrac{2}{5}$

$\implies \sf \dfrac{x - 2}{x + 7} =\: \dfrac{2}{5}$

By doing cross multiplication we get,

$\implies \sf 5(x - 2) =\: 2(x + 7)$

$\implies \sf 5x - 10 =\: 2x + 14$

$\implies \sf 5x - 2x =\: 14 + 10$

$\implies \sf 3x =\: 24$

$\implies \sf x =\: \dfrac{24}{3}$

$\implies \sf\bold{\purple{x =\: 8}}$

Hence, the required original fraction will be :

$\longrightarrow \sf Original\: Fraction =\: \dfrac{x}{x + 5}$

$\longrightarrow \sf Original\: Fraction =\: \dfrac{8}{8 + 5}$

$\longrightarrow \sf\bold{\red{Original\: Fraction =\: \dfrac{8}{13}}}$

${\small{\bold{\underline{\therefore\: The\: original\: fraction\: is\: \dfrac{8}{13}\: .}}}}$

Answer 2

Given :

The numerator of a fraction is 5 less than its denominator, if 2 is subtracted from the numerator and 2 is added to the denominator the fraction becomes 2/5. Find the original fraction

How To Solve :

• Here we are provided that the numerator of a fraction is 5 less than its denominator which means 5 will be added to the variable in the denominator. After that the second clue is that 2 is substracted from the numerator and 2 will is added in the denominator with the previous ones. After which the new fraction becomes 2/5.

Solution :

Let us assume :

The numerator be a

Now,

The numerator of a fraction is 5 less than its denominator which means 5 will be added to the denominator. Hence, the denominator is a + 5

We know that original fraction should be :

$\boxed{\frak{ \frac{ \green{Numerator}}{ \blue{Denominator}}}}$

Hence, putting the values we get :

$\twoheadrightarrow \red{\frak{ \frac{a}{a + 5} }}$

After that the next part which states that 2 is subtracted from the numerator and 2 is added to the denominator

$\twoheadrightarrow \pink{ \frak{\frac{a - 2}{a + 5 + 2} }}$

$\twoheadrightarrow \pink{ \frak{\frac{a - 2}{a + 7} }}$

The new fraction becomes 2/5

Hence, the equation is :

$\twoheadrightarrow \purple{ \frak{\frac{a - 2}{a + 7} = \frac{2}{5} }}$

Cross multiplying the equation we get

$\twoheadrightarrow \purple{ \frak{5(a - 2) = 2(a + 7)}}$

$\twoheadrightarrow \purple{ \frak{5a - 10 = 2a + 14}}$

Transposing 2a and -10 we get -2a and 10

$\twoheadrightarrow \purple{ \frak{5a -2a=14+10}}$

$\twoheadrightarrow \purple{ \frak{3a=24}}$

$\twoheadrightarrow \purple{ \frak{a=\frac{24}{3}}}$

$\twoheadrightarrow \purple{ \frak{a=\cancel{\frac{24}{3}}}}$

$\:\:\:\:\:\:\:\:\star\:\:\:\underline{\boxed{ \purple{\frak{a=8}}}}$

_____________________________________

Now, getting the original fraction :

The original fraction was :

$\leadsto \green {\frak{ \frac{a}{a + 5}}}$

Putting a = 8 in the original fraction we get :

$\leadsto \green{ \frak{ \frac{8}{8 + 5} }}$

$\leadsto \green{ \frak{ \frac{8}{13} }}$

Henceforth, the original fraction is 8/13