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# The Difference of Two Natural Numbers is 3 and the Difference of Their Reciprocals is 3 28 .Find the Numbers. - CBSE Class 10 - Mathematics

ConceptSolutions of Quadratic Equations by Completing the Square

#### Question

The difference of two natural numbers is 3 and the difference of their reciprocals is  $\frac{3}{28}$.Find the numbers.

#### Solution

Let the smaller number be x then the other number be 3 + x.

Then according to question,

$\frac{1}{x} - \frac{1}{3 + x} = \frac{3}{28}$

$\Rightarrow \frac{3 + x - x}{x(3 + x)} = \frac{3}{28}$

$\Rightarrow \frac{3}{3x + x^2} = \frac{3}{28}$

$\Rightarrow 28 = 3x + x^2$

$\Rightarrow x^2 + 3x - 28 = 0$

$\Rightarrow x^2 + 7x - 4x - 28 = 0$

$\Rightarrow x(x + 7) - 4(x + 7) = 0$

$\Rightarrow (x - 4)(x + 7) = 0$

$\Rightarrow x - 4 = 0 \text { or } x + 7 = 0$

$\Rightarrow x = 4 \text { or }x = - 7$

Since, x being a natural number,

Therefore, x = 4.

Then another number will be $3 + x = 3 + 4 = 7$

Thus, the two natural numbers are 7 and 4.

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Solution The Difference of Two Natural Numbers is 3 and the Difference of Their Reciprocals is 3 28 .Find the Numbers. Concept: Solutions of Quadratic Equations by Completing the Square.
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