Question

The sum of two numbers is 2 and the sum of their reciprocals is 2.25. Find the numbers.

Answer 1

Let the two numbers be $$x$$ and $$(2 - x)$$.

Sum of their reciprocals is $$2.25 = \frac{225}{100} = \frac{9}{4}$$.

According to the condition: $$\frac{1}{x} + \frac{1}{2 - x} = \frac{9}{4}$$

$$\frac{(2 - x) + x}{x(2 - x)} = \frac{9}{4}$$

$$\frac{2}{2x - x^2} = \frac{9}{4}$$

$$8 = 9(2x - x^2)$$

$$8 = 18x - 9x^2$$

$$9x^2 - 18x + 8 = 0$$

Factoring: $$9x^2 - 12x - 6x + 8 = 0$$

$$3x(3x - 4) - 2(3x - 4) = 0$$ 

$$(3x - 4)(3x - 2) = 0$$

$$x = \frac{4}{3} \quad \text{or} \quad x = \frac{2}{3}$$

When $$x = \frac{4}{3}$$, the other number is $$2 - \frac{4}{3} = \frac{2}{3}$$.

Hence, the required numbers are $$\frac{4}{3}$$ and $$\frac{2}{3}$$.