Question

The numerator of a fraction is 3 less than the denominator. If both the numerator and denominator are increased by 2. the new fraction becomes 6/7 .
Find the original fraction.​

Answer 1

Given :

• ➻ The numerator of a fraction is 3 less than the denominator. If both the numerator and denominator are increased by 2 the fraction becomes 6/7 .

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To Find :

• ➻ Find the original fraction.

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Solution :

~ According to Question :

➢ The numerator of the fraction is 3 less than denominator merator. Hence,

$\large{\qquad{\color{gray}{\bigstar{\color{red}{\sf{ \: \: \dfrac{Numerator}{Denominator} = \dfrac{x - 3}{x}}}}}}}$

➢ If both the numerator and denominator are increased by 2 the fraction becomes 6/7 . Hence,

$\large{\qquad{\color{gray}{\bigstar{\color{red}{\sf{ \: \: \dfrac{x - 3 + 2}{x + 2} = \dfrac{6}{7}}}}}}}$

$\qquad{━━━━━━━━━━━━━━━━━━━}$

~ Doing Cross - Multiplication :

${\longmapsto{\qquad{\sf{ 7(x - 1) = 6(x + 2)}}}} \\ \\ \ {\longmapsto{\qquad{\sf{ 7x - 7 = 6x + 12 }}}} \\ \\ \ {\longmapsto{\qquad{\sf{ 7x - 6x = 12 + 7}}}} \\ \\ \ {\qquad{\sf{ x \: = {\color{green}{\sf{ 19 }}}}}}$

$\qquad{━━━━━━━━━━━━━━━━━━━}$

~ Now, Numerator and Denominator :

◐ Numerator :

${\qquad{\pink{\leadsto{\underline{\underbrace{\underline{\color{darkblue}{\sf{ Numerator = x - 3 = 19 - 3 = 16 }}}}}}}}}$

◐ Denominator :

${\qquad{\pink{\leadsto{\underline{\underbrace{\underline{\color{darkblue}{\sf{ Denominator = x = 19 }}}}}}}}}$

◐ Original Fraction :

$\large{\pink{\underline{\orange{\underline{\pmb{\color{purple}{\frak{ Original \: Fraction \: = \dfrac{16}{19}}}}}}}}}$

$\qquad{━━━━━━━━━━━━━━━━━━━}$

Therefore :

❝ Original fraction is 16/19 .❞

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${\red{:}{\pink{)}}}$

Answer 2

Given : The numerator of a fraction is 3 less than the denominator. If both the numerator and denominator are increased by 2. The new fraction becomes 6/7. Find the original fraction.

★ ᴄᴏɴᴄᴇᴘᴛ :

According to the question, here we are given that the numerator of a fraction is 3 less than the denominator and then the numerator as well as the denominator are both increased by 2. After which we get the new fraction as 6/7. So, here first 3 is to be added to the denominator and then both the num and den to be added by 2, after which the simplifying the equation we will get the deserved result

★ sᴏʟᴜᴛɪᴏɴ :

$\text{Let us assume the numerator be} \: \bold{ \pink{\beta}}$

$\text{Now, as we know that original fraction is} \: \frak{\frac{ \red{Numerator}}{ \purple{Denominator}}}$

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

Now, both numerator and denominator is increased by 2

$\small\frak{Original \: Fraction = \frac{\beta + 2}{ \beta + 3 + 2}} \\ \\ \small \underline{ \green{\frak{Original \: Fraction = \frac{\beta + 2 }{ \beta + 5}}}}$

Now, the new fraction becomes 6/7

Getting the equation

$\frak{Original \: Fraction = New \: Fraction} \\ \\ \hookrightarrow \small\frak{ \frac{ \beta + 2}{ \beta + 5} = \frac{6}{7}} \\ \\ \hookrightarrow \small\frak{7( \beta + 2) = 6( \beta + 5)} \\ \\ \hookrightarrow \small \frak{7 \beta + 14 = 6 \beta + 30} \\ \\ \hookrightarrow \small \frak{7 \beta - 6 \beta = 30 - 14} \\ \\ \star \quad \small \underline{ \green{\frak{\beta = 16}}}$

★ ғɪɴᴅɪɴɢ ᴏʀɪɢɪɴᴀʟ ғʀᴀᴄᴛɪᴏɴ :

$\dashrightarrow \small \frak{Original \: Fraction = \frac{\beta}{ \beta + 3}} \\ \\ \dashrightarrow \small \frak{Original \: Fraction = \frac{16}{16 + 3}} \\ \\ \star \quad \underline{ \boxed{ \blue{\small \frak{Original \: Fraction = \frac{16}{19}} }}}$