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Home » The sum of two natural numbers is 15 and the sum of their reciprocals is . Find the numbers.
The sum of two natural numbers is 15 and the sum of their reciprocals is \frac{3}{10}. Find the numbers.
CBSE | Class 10 | Exercise 10e | Maths | Quadratic Equations | RS Aggarwal
The sum of two natural numbers is 15 and the sum of their reciprocals is \frac{3}{10}. Find the numbers.

Let the required natural numbers be x and (15-x).

According to the given condition,

\begin{array}{l} \frac{1}{x}+\frac{1}{15-x}=\frac{3}{10} \\ \Rightarrow \frac{15-x+x}{x(15-x)}=\frac{3}{10} \\ \Rightarrow \frac{15}{15 x-x^{2}}=\frac{3}{10} \\ \Rightarrow 15 x-x^{2}=50 \\ \Rightarrow x^{2}-15 x+50=0 \\ \Rightarrow x^{2}-10 x-5 x+50=0 \\ \Rightarrow x(x-10)-5(x-10)=0 \\ \Rightarrow(x-5)(x-10)=0 \\ \Rightarrow x-5=0 \text { or } x-10=0 \\ \Rightarrow x=5 \text { or } x=10 \end{array}

When x=5

15-x=15-5=10

When x=10

15-x=15-10=5

Hence, the required natural numbers are 5 and 10 .

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