The sum of two numbers is 30. If the sum of their reciprocals is $$\tfrac{6}{25}$$​, find the two numbers.

Correct option is B

Given:

Sum of two numbers = 30

Sum of their reciprocals = $$\frac{ 6}{25}$$​​

Solution:

Let the two numbers be 'x' and 'y'.

According to the given information:

x + y = 30

$$\frac{1}{x} + \frac{1}{y} =\frac{ 6}{25}$$​​

From the first equation, express 'y' in terms of 'x':

y = 30 - x

Substitute the value of 'y' in the second equation:

​$$\frac{1}{x} + \frac{1}{(30 - x)} =\frac{ 6}{25}$$​​

Solve for 'x':

​$$\frac{30 }{(30x - x^2)} = \frac{6}{25}$$​​

​$$6(30x - x^2) = 30 \times 25$$​​

​$$180x - 6x^2$$​= 750

​$$6x^2 - 180x + 750$$​= 0

​$$x^2 - 30x + 125 = 0$$​ (Divide the equation by 6)

Factor the quadratic equation:

(x - 25)(x - 5) = 0

Find the possible values for 'x':

x = 25 or x = 5

Find the corresponding values for 'y':

If x = 25, then y = 30 - 25 = 5

If x = 5, then y = 30 - 5 = 25

Therefore, the two numbers are 25 and 5.

Option (b) is right.