Question

The sum of two numbers is 9. The sum of their reciprocals is $\dfrac{1}{2}. Find the sum of squares of the number.

Answer 1

Hint: In this question let the numbers be x and y. Using the constraints given in the questions formulate two equations involving two variables. Use these equations to get the squares of numbers that are ${x^2} + {y^2}$.

Complete step-by-step answer:
Let the first number be x.

And the second number be y.

Now it is given that the sum of two numbers is 9.

$ \Rightarrow x + y = 9$ ......................... (1)

Now it is also given that the sum of their reciprocals is (1/2).

$ \Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{2}$

Now simplify this equation we have,

$ \Rightarrow \dfrac{{y + x}}{{xy}} = \dfrac{1}{2}$

Now from equation (1) we have,

$ \Rightarrow \dfrac{9}{{xy}} = \dfrac{1}{2}$

$ \Rightarrow xy = 18$ .............................. (2)

Now as we know that ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ so use this property we have,

$ \Rightarrow {\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy$

Now from equation (1) and (2) we have,

$ \Rightarrow {\left( 9 \right)^2} = {x^2} + {y^2} + 2\left( {18} \right)$

$ \Rightarrow {x^2} + {y^2} = 81 - 36 = 45$

So the sum of the squares of the numbers is 45.

So this is the required answer.

Note: Such type of problems does not require exact solving for separate values of x and y, and that was the tricky part here, the two equations formed could be easily used to find the relationship between the sum of squares of numbers. These things help utilizing the time while solving problems of these kinds.