#### Question

The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.

#### Solution

Let one numbers be *x *then other (9 - *x*).

Then according to question

`1/x + 1(9-x)=1/2`

`rArr((9-x)+x)/(x(9-x))=1/2`

`rArr9/(x(9-x))=1/2`

By cross multiplication

⇒ 18 = x(9 - x)

⇒ x^{2} - 9x + 18 = 0

⇒ x^{2} - 6x - 3x + 18 = 0

⇒ x(x - 6) - 3(x - 6) = 0

⇒ (x - 6)(x - 3) = 0

⇒ x - 6 = 0

⇒ x = 6

Or

⇒ x - 3 = 0

⇒ x = 3

Since, x being a number,

Therefore,

When x = 6 then

(9 - x) = (9 - 6) = 3

When x = 3 then

(9 - x) = (9 - 3) = 6

Thus, two consecutive number be either 3, 6 or 6, 3.

Is there an error in this question or solution?

Solution The Sum of Two Numbers is 9. the Sum of Their Reciprocals is 1/2. Find the Numbers. Concept: Solutions of Quadratic Equations by Factorization.