Question

If the AM of two positive numbers be three times their geometric mean then the ratio of the numbers is

Answer 1

Given:

The AM of two positive numbers is three times their geometric mean.

To find:

The ratio of the numbers is?

Solution:

Let the two positive numbers be a and b.

From given, we have,

The AM of two positive numbers is three times their geometric mean.

AM = 3 × GM

⇒ (a + b)/2 = 3√ab

⇒ a + b = 6√ab

squaring on both the sides, we get,

(a + b)² = (6√ab)²

a² + b² + 2ab = 36ab

a² + b² - 34 ab = 0

⇒ a²/b² + 1 - 34a/b = 0

⇒ (a/b)² - 34 (a/b) + 1 = 0

the above equation represents a quadratic equation,

so we have,

a/b = { 34 ± √ [34² - 4 × 1 × 1] }/ 2

a/b = { 34 ± √ [1156 - 4] }/ 2

a/b = { 34 ± √ [1152] }/ 2

a/b = 17 ± 24√2/2

a/b = 17 ± 12√2

Hence the ratio is a/b = 17 ± 12√2