Question

$\huge\boxed{\tt\purple{✠Question}}$
a fraction becomes 1/3 if 1 is subtracted from both its numerator and dinominator if 1 is added to both numerator and dinominator, it becomes 1/2 find the fraction​.

$\pink{\large \qquad \boxed{\boxed{\begin{array}{cc} \maltese \tt Don't \: Spam \\ \\ \maltese \tt LaTex \: Answer \: is \: Required \\ \\ \maltese \: \: \tt All \: The \: Best\end{array}}}}$
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Answer 1

$\large \dag$ Question :-

A Fraction becomes 1/3, If 1 is subtracted from both numerator and denominator, If 1 is added to both numerator and denominator, it becomes 1/2 find the fraction.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\pink{\underline{\frak{\pmb{\text The \: Original \: Fraction \: is \: \dfrac{3}{7}}}}}}$

$\large \dag$ Step by step Explanation :-

Let Numerator and Denominator of Original Fraction be :

• Numerator = $\large \rm x$

• Denominator = $\large \rm y$

So,

$\text{Original Fraction = }\rm \frac{x}{y}$

❒ When 1 is subtracted from both numerator and denominator :-

$\text{Fraction Becomes : } \rm \frac{\text x - 1}{\text y - 1}$

$\large \maltese\:$ According To Question :

$\large \blue \bigstar \: \: \red{ \bf \frac{x - 1}{y - 1} = \frac{1}{3} }$

⏩ On Cross Multiplying :

$:\longmapsto \rm 3(x - 1) = y - 1$

$:\longmapsto \rm 3x - 3 = y - 1$

$:\longmapsto \rm 3x - y = - 1 + 3$

$\large:\longmapsto \underline \green{{ \underline{\rm 3x - y = 2}}} \: \: - - - (1)$

❒ When 1 is added to both numerator and dinominator :-

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

$\large \maltese\:$ According To Question :

$\large \blue \bigstar \: \: \red{ \bf \frac{x + 1}{y + 1} = \frac{1}{2} }$

⏩ On Cross Multiplying :

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

$:\longmapsto \rm 2x + 2 = y + 1$

$:\longmapsto \rm 2x - y = 1 - 2$

$\large:\longmapsto \underline \green{{ \underline{\rm 2x - y = -1}}} \: \: - - - (2)$

✧ Subtracting (2) from (1) :

$:\longmapsto \rm 3x - y - (2x - y) = 2 - ( - 1)$

$:\longmapsto \rm 3x - \cancel y - 2x + \cancel y = 2 + 1$

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

✧ Putting Value of x in (1) :

$:\longmapsto \rm 3 \times 3 - y = 2$

$:\longmapsto \rm 9 - y = 2$

$:\longmapsto \rm y = 9 - 2$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf y = 7} }}}$

∴ Numerator of Fraction = 3

And Denominator of Fraction = 7

Therefore,

$\underline{\orange{\underline{\frak{\pmb{\text The \: Original \: Fraction \: is \: \dfrac{3}{7}}}}}}$

Answer 2

Answer:

‣ Required fraction $\mapsto\:{\boxed{\tt{\dfrac{3}{7}}}}$

Explanation:

Given information,

A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator, if 1 is added to both numerator and denominator, it becomes 1/2. Find the fraction.

Let,

• $\tt Fraction = \dfrac{Numerator}{Denominator} = \dfrac{a}{b}$

Case – ❶

⚘ If 1 is subtracted from both its numerator and denominator, fraction becomes 1/3.

Therefore,

$\\ \longrightarrow\:\tt \dfrac{a - 1}{b - 1} = \dfrac{1}{3}$

⚘ By doing cross multiplication we get,

$\\ \longrightarrow\:\tt 3(a - 1) = 1(b - 1)$

$\\ \longrightarrow\:\tt 3a - 3 = b - 1$

$\\ \longrightarrow\:\tt 3a - 3 + 1 = b$

$\\ \longrightarrow\:\tt \Big| b = 3a - 2 \Big|\qquad- Eq^{n}\:(1)$

Case – ❷

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

Therefore,

$\\ \longrightarrow\:\tt \dfrac{a + 1}{b + 1} = \dfrac{1}{2}$

⚘ By doing cross multiplication we get,

$\\ \longrightarrow\:\tt 2(a + 1) = 1(b + 1)$

$\\ \longrightarrow\:\tt 2a + 2 = b + 1$

$\\ \longrightarrow\:\tt 2a + 2 - 1 = b$

$\\ \longrightarrow\:\tt \Big| b = 2a + 1 \Big|\qquad- Eq^{n}\:(2)$

⚘ From Eqⁿ (1) and Eqⁿ (2) we get,

$\\ \longrightarrow\:\tt 3a - 2 = 2a + 1$

$\\ \longrightarrow\:\tt 3a - 2a = 1 + 2$

$\\ \longrightarrow\:\tt 1a = 3$

$\\ \longrightarrow\:{\underline{\boxed{\tt{a = 3}}}}\:\bigstar$

⚘ Putting value of 'a' in Eqⁿ (2),

$\\ \longrightarrow\:\tt b = 2(3) + 1$

$\\ \longrightarrow\:\tt b = (2\:\times\:3) + 1$

$\\ \longrightarrow\:\tt b = 6 + 1$

$\\ \longrightarrow\:{\underline{\boxed{\tt{b = 7}}}}\:\bigstar$

Hence,

‣ $\sf Fraction = \dfrac{Numerator}{Denominator} = \dfrac{a}{b} = {\boxed{\bf{\dfrac{3}{7}}}}$

Verification:

We know that,

$\\ \longrightarrow\:\tt \dfrac{a - 1}{b - 1} = \dfrac{1}{3}$

⚘ Put a = 3 and b = 7 in above equation we get,

$\\ \longrightarrow\:\tt \dfrac{3 - 1}{7 - 1} = \dfrac{1}{3}$

$\\ \longrightarrow\:\tt {\cancel{\dfrac{2}{6}}} = \dfrac{1}{3}$

$\\ \longrightarrow\:\tt \dfrac{1}{3} = \dfrac{1}{3}$

$\\ \longrightarrow\:{\underline{\boxed{\tt{LHS = RHS}}}}\:\bigstar$

Also,

$\\ \longrightarrow\:\tt \dfrac{a + 1}{b + 1} = \dfrac{1}{2}$

⚘ Put a = 3 and b = 7 in above equation we get,

$\\ \longrightarrow\:\tt \dfrac{3 + 1}{7 + 1} = \dfrac{1}{2}$

$\\ \longrightarrow\:\tt {\cancel{\dfrac{4}{8}}} = \dfrac{1}{2}$

$\\ \longrightarrow\:\tt \dfrac{1}{2} = \dfrac{1}{2}$

$\\ \longrightarrow\:{\underline{\boxed{\tt{LHS = RHS}}}}\:\bigstar$

Hence, Verified ✔

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