Question

the sum of the numerator and denominator of a certain fraction is 8.if 2 is added to the numerator and to the denominator, the value of the fraction increased by 4/35.find the fraction.​

Answer 1

Answer :

›»› The fraction fraction will be 3/5.

Given :

• The sum of the numerator and denominator of a certain fraction is 8. If 2 is added to the numerator and to the denominator, the value of the fraction increased by 4/35.

To Find :

• The fraction.

Solution :

Let us assume that the numerator is "x" and the denominator is "y" respectively.

As it is given that the sum of the numerator and denominator of a certain fraction is 8.

⟶ x + y = 8 .......(1)

⟶ y = 8 - x ........(2)

As it is also given that If 2 is added to the numerator and to the denominator, the value of the fraction increased by 4/35.

$\footnotesize\displaystyle \tt{: \implies \frac{x + 2}{y + 2} = \frac{x}{y} + \frac{4}{35} }$

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$\footnotesize\displaystyle{\tt{: \implies \frac{x + 2}{8 - x + 2} = \frac{x}{8 - x} + \frac{4}{35} }}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{x + 2}{8 + 2 - x} = \frac{x}{8 - x} + \frac{4}{35} }}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{x + 2}{10 - x} = \frac{x}{8 - x} + \frac{4}{35} }}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{x + 2}{8 + 2 - x} - \frac{x}{8 - x} + \frac{4}{35} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{35(8 - x) \times (x + 2) - 35x \times (10 - x) \times (8 - x)}{35(10 - x) \times (8 - x)} }}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{(280 - 35x) \times (x + 2) - 350x + {35x + ( - 40 + 4x) \times (8 - x)}^{2} }{35(10 - x) \times (8 - x)} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{280x + 560 - {35x}^{2} - 70x - 350x + {35x}^{2} - 320 + 40x + 32x - {4x}^{2} }{35(10 - x) \times (8 - x)} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{280x + 560 - 70x - 350x - 320 + 40x + 32x - {4x}^{2} }{35(10 - x) \times (8 - x)} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ \frac{ - 68x + 560 - 320 - {4x}^{2}}{35(10 - x) \times (8 - x)} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ - 68x + 240 - {4x}^{2} = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ { - 4x}^{2} - 68x + 240 = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ {x}^{2} + 17x - 60 = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{ {x}^{2} + 20x - 3x - 60 = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{x \times (x + 20) - 3(x + 20) = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{(x + 20) \times (x - 3) = 0}}}}$

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$\footnotesize{\displaystyle{\tt{: \implies{x + 20 =0 \quad| \quad x - 3 = 0}}}}$

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$\footnotesize{\displaystyle{\bf{: \implies{x = -20 \quad| \quad x = 3}}}}$

Since x cannot be negative so, x = 3

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Now, put the value of x in equation (2)

$\displaystyle{\tt{: \implies{y = 8 - x}}}$

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$\displaystyle{\tt{: \implies{y = 8 - 3}}}$

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$\displaystyle{\bf{: \implies{y = 5}}}$

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⟶ The fraction = x/y

⟶ The fraction = 3/5

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Hence, the fraction fraction will be 3/5.

Answer 2

Refer to the attached images ❤️