Question

if 1 is added to the numerator of a fraction it becomes 1/5 if 1 is taken from yhe denominator it becomes 1/7 find the fraction​

Answer 1

Appropriate Question:

• If 1 is added to the numerator of a fraction it becomes 1/5. If 1 is taken from the denominator it becomes 1/7. Find the fraction.

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☯ Let's consider that the numerator and the denominator be x and y.⠀

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Given that,

• If 1 is added to the numerator of a fraction it becomes 1/5.

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$:\implies\sf\dfrac{x + 1}{y} = \dfrac{1}{5} \\\\\\:\implies\sf 5(x + 1) = y \\\\\\:\implies\sf 5x + 5 = y \\\\\\:\implies\sf 5x - y = - 5 \qquad\qquad\bigg\lgroup\bf eq\: (1)\bigg\rgroup$⠀⠀

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

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• If 1 is taken from the denominator, It becomes 1/7.

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$:\implies\sf\dfrac{x}{y - 1} = \dfrac{1}{7} \\\\\\:\implies\sf 7x = y - 1 \\\\\\:\implies\sf 7x - y = - 1 \qquad\qquad\bigg\lgroup\bf eq\: (2)\bigg\rgroup$

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• Subtracting equation (II), from equation (I).⠀

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$:\implies\sf 7x - y = -1 \\\\\\:\implies\sf 5x - y = - 5 \\\\\\:\implies\sf 2x = 4 \\\\\\:\implies\sf x = \cancel\dfrac{4}{2} \\\\\\:\implies{\underline{\boxed{\frak{\purple{x = 2}}}}}\:\bigstar$

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• Substituting value of x, in equation (I)

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$:\implies\sf 5x - y = - 5 \\\\\\:\implies\sf 5(2) - y = - 5 \\\\\\:\implies\sf 10 - y = - 5 \\\\\\:\implies\sf y = 10 + 5 \\\\\\:\implies{\underline{\boxed{\frak{\pink{y = 15}}}}}\:\bigstar$

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Hence,⠀⠀⠀

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• Denominator, x = 15
• Numerator, y = 2

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$\therefore\:{\underline{\sf{The\: required \: fraction \ is\: \bf{\dfrac{2}{15}}.}}}$

Answer 2

$\Large\bf\pink{Let,}$

• Numerator of a fraction is x.

• Denominator of the fraction y.

$\bf\red{Thus,}$

$\longmapsto\:\:\bf\orange{Fraction\:=\:\dfrac{x}{y}\:}$

$\bf\blue{According\:to\:the\:question,}$

ƇƛƧЄ - 1 ;-

• If 1 is added to the numerator of a fraction, it becomes ¹/₅.

$:\implies\:\:\bf{\dfrac{x\:+\:1}{y}\:=\:\dfrac{1}{5}\:}$

$:\implies\:\:\bf\purple{y\:=\:5x\:+\:5\:}--(1)$

ƇƛƧЄ - 2 ;-

• If 1 is subtracted to the denominator of the fraction, it becomes ¹/₇.

$:\implies\:\:\bf{\dfrac{x}{y\:-\:1}\:=\:\dfrac{1}{7}\:}$

$:\implies\:\:\bf{y\:-\:1\:=\:7x\:}$

✒ Put the value of y from equation 1, we get

$:\implies\:\:\bf{5x\:+\:5\:-\:1\:=\:7x\:}$

$:\implies\:\:\bf{5x\:+\:4\:=\:7x\:}$

$:\implies\:\:\bf{7x\:-\:5x\:=\:4\:}$

$:\implies\:\:\bf{2x\:=\:4\:}$

$:\implies\:\:\bf{x\:=\:\dfrac{4}{2}\:}$

$:\implies\:\:\bf\orange{x\:=\:2\:}$

✒ Substituting the value of x in equation 1,

$:\implies\:\:\bf{y\:=\:5\times{2}\:+\:5\:}$

$:\implies\:\:\bf{y\:=\:10\:+\:5\:}$

$:\implies\:\:\bf\green{y\:=\:15\:}$

$\Large\bf\pink{Therefore,}$

$\red\checkmark\:\:\bf\blue{The\:fraction\:is\:\dfrac{2}{15}\:}.$