Question

If the AM and GM between two numbers are in the ratio m : n, then what is the ratio between these two numbers?​

Answer 1

Answer:

Let the two numbers be a and b.

To find:

» a/b

Therefore, AM = (a + b)/2 ... (1)

& GM = √(ab) ... (2)

From given, AM/GM = m/n

» [(a + b)/2] / √(ab) = m / n

On squaring

» (a + b)²/4ab = m² / n²

» 4abm² = a²n² + 2abn² + b²n²

» a²n² + b²n² + 2abn² - 4abm² = 0

» n² (a² + b²) + 2ab (n² - 2m²) = 0

» n² (a² + b²) - 2ab (2m² - n²) = 0

» n² (a² + b²) = 2ab (2m² - n²)

» (a² + b²)/2ab = (2m² - n²)/n²

» By componendo & dividendo

» (a + b)²/(a - b)² = m²/(m² - n²)

» (a + b)/(a - b) = m/√(m² - n²)

» By Componendo and Dividendo

» 2a/2b = [m + √(m² - n²)] / [m - √(m² - n²)]

» a/b = [m + √(m² - n²)] / [m - √(m² - n²)]