Question

The denominator of a fraction is greater than its numerator by 7. If the numerator is increased by 1 and the denominator is increased by 4, the fraction becomes 1/3. Find the original fraction.

Answer 1

❍ Let's say, the numerator of the fraction be m and denominator of the fraction be n respectively.

Hence,

• The fraction = m/n.

As per Question, the denominator 'n' of a fraction is greater than it's numerator 'm' by 7.

Therefore,

⇥ (Denominator = Numerator + 7)

⇥ n = (m + 7)⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ —eq.( I )

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$\underline{\bigstar\:{\pmb{\frak{\pink{According \;to\; the\; Question :}}}}}$

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• If the numerator 'm' is increased by 1 and the denominator 'n' is increased by 4. The new number becomes ⅓. After converting it into fraction, we get —

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$:\implies\sf\Bigg\{\dfrac{m + 1}{m + 7 + 4}\Bigg\} = \Bigg\{\dfrac{1}{3}\Bigg\} \\\\\\:\implies\sf \Bigg\{\dfrac{m + 1}{m + 11}\Bigg\} = \Bigg\{\dfrac{1}{3}\Bigg\} \\\\\\:\implies\sf 3\Big\{m + 1\Big\} = 1\Big\{m + 11\Big\}\\\\\\:\implies\sf 3m + 3 = m + 11\\\\\\:\implies\sf 3m - m = 11 - 3\\\\\\:\implies\sf 2m = 8\\\\\\:\implies\sf m = \cancel\dfrac{8}{2}\\\\\\:\implies{\pmb{\sf{m = 4}}}$

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¤ The value of 'm' is 4. Now we'll substitute this value in the eq. ( I ) to find out the denominator of the given fraction 'n' —

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$\dashrightarrow\sf Denominator = \Big\{Numerator + 7\Big\} \\\\\\\dashrightarrow\sf n = \Big\{m + 7\Big\} \\\\\\\dashrightarrow\sf n = 4 + 7 \\\\\\\dashrightarrow{\pmb{\sf{n = 11}}}$

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$\therefore{\underline{\sf{Hence, \: the \: required \: fraction \: is \: {\pmb{\bf{\dfrac{4}{11}}}}.}}}$

$\rule{250px}{.2ex}$

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★ V E R I F I C A T I O N :

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$\dashrightarrow\sf \Bigg\{\dfrac{Numerator + 1}{Denominator + 4}\Bigg\} = \Bigg\{\dfrac{1}{3}\Bigg\} \\\\\\\dashrightarrow\sf \Bigg\{\dfrac{m + 1}{n + 4}\Bigg\} = \Bigg\{\dfrac{1}{3}\Bigg\} \\\\\\\dashrightarrow\sf \dfrac{4 + 1}{11 + 4} = \dfrac{1}{3} \\\\\\\dashrightarrow\sf \cancel\dfrac{5}{15} = \dfrac{1}{3} \\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{\dfrac{1}{3} = \dfrac{1}{3}}}}}$

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⠀⠀⠀⠀⠀⠀⠀Hence Verified!

Answer 2

Given :-

The  denominator of a fraction is greater than its numerator by 7. If the numerator is increased by 1 and the denominator is increased by 4, the fraction becomes 1/3.

To Find :-

Original fraction

Solution :-

Let the numerator  be x

Fraction = x/x + 7

Given,

Numerator increased by 1 and denominator increased by 4

x + 1/x + 7 + 4 = 1/3

x + 1/x + 11 = 1/3

3(x + 1) = 1(x + 11)

3x + 3 = x + 11

3x - x = 11 - 3

2x = 8

x = 8/2

x = 4

Fraction = x/x + 7 = 4/4 + 7 = 4/11

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