The sum of two positive numbers is 40. If the GM of these two numbers is lower than their AM by 20%, then what is the difference between the two numbers?

Correct option is C

Given:

Sum of two positive numbers = 40

The GM of these two numbers is 20% lower than their AM

Concept:

AM (Arithmetic Mean) = (a + b) / 2

GM (Geometric Mean) = √(a × b)

GM = 0.8 × AM

Solution:

Let the two numbers be a and b.

a + b = 40

AM = (a + b) / 2 = 40 / 2 = 20

GM = √(a × b)

Given GM = 0.8 × AM

=> √(a × b) = 0.8 × 20

=> √(a × b) = 16

=> a × b = 16²

=> a × b = 256

We have the equations:

1. a + b = 40

2. a × b = 256

Solving these two equations:

Let a and b be the roots of the quadratic equation x²- (a+b)x + ab = 0

=> x²- 40x + 256 = 0

Solving for x using the quadratic formula x = [ -b ± √(b²- 4ac) ] / 2a

=> x = [ 40 ± √(1600 - 1024) ] / 2

=> x = [ 40 ± √576 ] / 2

=> x = [ 40 ± 24 ] / 2

=> x = 32 or x = 8

So, the two numbers are 32 and 8.

Difference between the two numbers = 32 - 8 = 24

Hence, the difference between the two numbers is 24.