Question

The arithmetic mean of two positive numbers a and b is greater than their geometric mean by 5. Their geometric mean is increased by 4 from their harmonic mean. Find a - b.
1. 20
2. 30
3. 40
4. 15

Answer 1

Correct Answer - Option 2 : 30

Given 

Difference in AM and GM = 5 

Formula used 

Arithmatic mean = (a + b)/2 

Geometric mean = √ab 

Harmonic mean = 2ab/(a + b)

Calculation 

According to the question 

⇒ (a + b)/2 = √ab + 5      ---(i)

⇒ √ab = 2ab/(a + b) + 4      ----(ii) 

From (i) and (ii) 

⇒ √ab = 2ab/(2√ab + 10) + 4

⇒ √ab(2√ab + 10) = 2ab + 4(2√ab + 10) 

⇒ 2ab + 10√ab = 2ab + 8√ab + 40

⇒ 2√ab = 40 

⇒ √ab = 20

⇒ ab = 400

From equation (1)

⇒ a + b = 2√ab + 10

⇒ a + b = 2 × 20 + 10 

⇒ a + b = 50

Squaring both side

⇒ (a + b)² = 2500

(a + b)² = a² + b² + 2ab 

⇒ a2 + b2 = 2500 - 2ab

(a - b)² = a² + b² - 2ab

⇒ (a - b)2 = 2500 - 4ab

⇒ (a - b)2 = 2500 - 1600 

⇒ (a - b)² = 900

⇒ (a - b) = 30

∴ a - b is 30.