Question

The denominator of a fraction exceeds its numerator by 7.
If the numerator is increased by 16 and the denominator
is decreased by 1 we get 4/2 .
find the original fraction​

Answer 1

Given:

• The denominator of a fraction exceeds its numerator by 7.
• If the numerator is increased by 16 and the denominator is decreased by 1 we get 4/2.

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To find:

• Original fraction?

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Solution:

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☯ Let Numerator and denominator of a fraction be x and y respectively.

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Now,

$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

• The denominator of a fraction exceeds its numerator by 7.

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$:\implies\sf y = x + 7 \qquad\qquad\bigg\lgroup\bf \:eq\:(1)\bigg\rgroup$

Also,

• If the numerator is increased by 16 and the denominator is decreased by 1 we get 4/2.

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$:\implies\sf \dfrac{x + 16}{y - 1} = \dfrac{4}{2}\\\\\\ :\implies\sf \dfrac{x + 16}{(x + 7) - 1} = \dfrac{4}{2}\qquad\qquad\bigg\lgroup\bf From\:eq\:(1)\bigg\rgroup\\\\\\ :\implies\sf \dfrac{x + 16}{x + 6} = \cancel{\dfrac{4}{2}}\\\\\\ :\implies\sf \dfrac{x + 16}{x + 6} = \dfrac{2}{1}\\\\\\ :\implies\sf x + 16 = 2(x + 6)\\\\\\ :\implies\sf x + 16 = 2x + 12\\\\\\ :\implies\sf 2x - x = 16 - 12\\\\\\ :\implies{\underline{\boxed{\frak{\purple{x = 4}}}}}\;\bigstar$

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Now, Putting value of x in eq (1),

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$\implies\sf y = 4 + 7\\\\\\ :\implies{\underline{\boxed{\frak{\pink{y = 11}}}}}\;\bigstar$

Therefore,

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• Numerator of a fraction, x = 4
• Denominator of a fraction, y = 11

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$\therefore\;{\underline{\sf{Hence,\;the\; original\;fraction\;is\; \bf{ \dfrac{4}{11}}.}}}$

Answer 2

Answer:

Given :-

• The denominator of a fraction exceeds it's numerator by 7.
• If the denominator is increased by 16 and the denominator by 1 we get $\dfrac{4}{2}$.

To Find :-

• What is the original fraction.

Solution :-

» Let, the numerator be x

» And, the denominator be x + 7

» Then, the fraction will be $\sf\dfrac{x}{x + 7}$

According to the question,

⇒ $\sf\dfrac{x + 16}{x + 7 - 1}$ = $\dfrac{4}{2}$

⇒ $\sf\dfrac{x + 16}{x + 6}$ = $\dfrac{4}{2}$

⇒ 2(x + 16) = 4(x + 6)

⇒ 2x + 32 = 4x + 24

⇒ 2x - 4x = 24 - 32

⇒ - 2x = - 8

⇒ x = $\sf\dfrac{\cancel{- 8}}{\cancel{- 2}}$

➠ x = 4

Hence, the required original fraction will be,

↦ $\sf\dfrac{4}{x + 7}$

↦ $\sf\dfrac{4}{4 + 7}$

➥ $\sf\dfrac{4}{11}$

$\therefore$ The original fraction will be $\sf\boxed{\bold{\small{\dfrac{4}{11}}}}$.