Question

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HELP ME SOLVE THIS QUESTION :

⭐FIND THE POSITIVE FRACTION LESS THAN 1 WHOSE SUM OF THE NUMERATOR AND THE DENOMINATOR IS 10,AND THE DIFFERENCE OF THE FRACTION AND ITS RECIPROCAL IS 40/21.⭐

CLASS:10 CHAPTER :QUADRATIC EQUATIONS

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Answer 1

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Let the numerator be 10-x, then the denominator will be x

$\frac{10 - x}{x} - \frac{x}{10 - x} = \frac{40}{21} \\ \frac{ {(10 - x)}^{2} - {x}^{2} }{x(x - 10)} = \frac{40}{21} \\ \frac{100 - 20x + {x}^{2} - {x}^{2} }{ {x}^{2} - 10x } = \frac{40}{21} \\ 2100 - 420x = 40 {x}^{2} - 400x \\ 40 {x}^{2} + 20x - 2100 = 0 \\ 2 {x}^{2} + x - 105 = 0 \\ 2 {x}^{2} - 14x + 15x - 105 = 0 \\ 2x(x - 7) + 15(x - 7) = 0 \\ (x - 7)(2x + 15) = 0$
Since the numerator cannot be negative in this case as that will make the denominator positive and the fraction negative, x = 7

The fraction will be 3/7

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Answer 2

Let the numerator be x and the denominator be 10 - x.

The original fraction is x/10 - x.

Given that Difference of the fraction and its reciprocal is 40/21.

⇒ (x/10 - x) + (10 - x/x) = 40/21

⇒ 21x^2 - 21(10 - x)^2 = 40x(-x + 10)

⇒ 21x^2 - 21(100 + x^2 - 20x) = -40x^2 + 400x

⇒ 21x^2 - 2100 - 21x^2 + 420x = -40x^2 + 400x

⇒ -40x^2 - 20x + 2100 = 0

⇒ 40x^2 + 20x - 2100 = 0

⇒ 20(2x^2 + x - 105) = 0

⇒ 2x^2 + x - 105 = 0

⇒ 2x^2 + 15x - 14x - 105 = 0

⇒ x(2x + 15) - 7(2x + 15) = 0

⇒ (x - 7)(2x + 15) = 0

⇒ x = 7, -15/2{Neglect}

∴ Required fraction is 7/3, Which is not valid because it is not < 1.

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Let the numerator be 10 - x and the denominator be x.

The original fraction is 10 - x/x.

The reciprocal = (x/10 - x).

Given that Difference of the fraction and its reciprocal is 40/21.

⇒ (10 - x/x) - (x/10 - x) = 40/21

⇒ 21(10 - x)^2 - 21x^2 = 40x(-x + 10)

⇒ 21(100 + x^2 - 20x) - 21x^2 = -40x + 400x

⇒ 2100 + 21x^2 - 420x - 21x^2 = -40x + 400x

⇒ -40x^2 + 820x = 2100

⇒ -40x^2 + 820x - 2100 = 0

⇒ 40x^2 - 820x + 2100 = 0

⇒ 20(2x^2 - 41x + 105) = 0

⇒ 2x^2 - 41x + 105 = 0

⇒ 2x^2 - 35x - 6x + 105 = 0

⇒ x(2x - 35) - 3(2x - 35) = 0

⇒ (x - 3)(2x - 35) = 0

⇒ x = 3, 35/2.

⇒ x = 3.

∴ Required fraction = 3/7.

Hope this helps!