Question

Question :

the numerator of a fraction is 7 less than it's denominator. If 4 is subtracted from the numerator and the denominator is increased by 1, the resulting fraction equals 1/3. Find the original fraction.​

Answer 1

$\large \dag$ Question :-

The numerator of a fraction is 7 less than it's denominator. If 4 is subtracted from the numerator and the denominator is increased by 1, the resulting fraction equals 1/3. Find the original fraction.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf  \green{The   \: Original \:  Fraction  \: is \:  \frac{10}{17} }} }$

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

Let Numerator and Denominator of Original Fraction be :

• Numerator = x

As Per the question numerator of the fraction is 7 less than it's denominator so,

• Denominator should be = x + 7

So ,

$\text{Original Fraction = } \frac{\text x}{\text x + 7}$

❒ When 4 is subtracted from the numerator and the denominator is increased by 1 :

$\\ \rm \text{Fraction Becomes } :  \frac{x -4}{x + 8}$

⏩ According To Question :

$\\ \large \blue \bigstar  \:   \red{ \bf  \frac{x -4}{x + 8} =  \frac{1}{3}  }$

$:\longmapsto \rm 3(x - 4) = 1(x + 8)$

$:\longmapsto \rm 3x - 12 = x + 8$

$:\longmapsto \rm 3x - x = 8 + 12$

$:\longmapsto \rm 2x = 20$

$:\longmapsto \rm x = \cancel \frac{20}{2}$

$\purple{ \large :\longmapsto  \underline {\boxed{{\bf x = 10} }}}$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Numerator  \: of \:  Original  \: Fraction = 10 }} }$

☆ As numerator of the fraction is 7 less than it's denominator

$\rm\therefore \:  Denominator  = 10   + 7$

$\blue\dashrightarrow\underline{\underline{\sf  \orange{Denominator  \: of \:  Original  \: Fraction = 17}} }$

Therefore,

$\large\underline{\pink{\underline{\frak{\pmb{ Original  \: Fraction  =  \dfrac{10}{17} }}}}}$

Answer 2

Given : The numerator of a fraction is 7 less than it's denominator. If 4 is subtracted from the numerator and the denominator is increased by 1, the resulting fraction equals 1/3. Find the original fraction.

Need to Find : The original fraction

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★ Cᴏɴᴄᴇᴘᴛ :

According to the question, here we are given that the Numerator is less than the denominator. Now, 4 is subtracted from the numerator and 1 is added to the denominator. The new fraction becomes ⅓.

★ Sᴏʟᴜᴛɪᴏɴ :

$\rm {Let \: us \: assume \: the \: numerator \: be \: \frak{a}}$

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

$\quad\quad \frak{ Original \: Fraction = \blue{ \frac{a}{a + 7}}}$

$\rm{Now, \: 4 \: is \: subtracted \: from \: the \: numerator \: and \: 1 \: is \: added \: to \: the \: denominator}$

$\quad\quad \small{ \frak{Original \: Fraction = \frac{a - 4}{a + 7 + 1} }} \\ \\ \small\quad \quad\frak{Original \: Fraction = \frac{a - 4}{a + 8} }$

$\rm{New \: Fraction \: becomes \: \frak{ \frac{1}{3} }}$

$\underline{\rm{\orange{Equation}}}: \\ \\ \hookrightarrow \small \frak{ \frac{a - 4}{a + 8} = \frac{1}{3} } \\ \\ \hookrightarrow \small \frak{ 3(a - 4) = 1(a + 8) } \\ \\ \hookrightarrow \small \frak{ 3a - 12 = a + 8 } \\ \\ \hookrightarrow \small \frak{ 3a - a = 8 + 12 } \\ \\ \hookrightarrow \small \frak{ 2a = 20 } \\ \\ \hookrightarrow \small \frak{ a = \cancel \frac{20}{2}} \\ \\ \star \quad \underline{ \boxed{ \small \green{\frak{ a = 10 }}}}$

$\rm{Now, \: finding \: original \: fraction}$

$\quad \quad\frak{Original \: Fraction = \frac{a}{a + 7}} \\ \\ \quad\quad\frak{Original \: Fraction = \frac{10}{10 + 7}} \\ \\ \qquad\star\quad \underline{\frak{Original \: Fraction = \frac{10}{17}}}$

$\small{ \mathbb{ \red{HENCE \: THE \: ORIGINAL \: FRACTION \: \: IS } }} \: \: \underline{\boxed{ \pink{\frak{ \frac{10}{17} }}}}$