Question

The ratio of the A.M. and G.M. of two positive numbers a and b is 5: 3. Find the ratio of a to b.

(a) 9:1

(b) 3:5

(c) 1:9

(d) 3:1

I have been asked this question during an online exam.

I want to ask this question from Relationship Between A.M. and G.M. topic in division Sequences and Series of Mathematics – Class 11

Answer 1

Right answer is (a) 9:1

The explanation is: (A.M.)/(G.M.) = 5/3

=> \(\frac{a+b}{2\sqrt{ab}} = \frac{5}{3}\)

Applying componendo and dividendo rule, we get

=> \(\frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}} = \frac{8}{2}\)

=> \((\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}})^2=4 \)

=> \((\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}})^1=2\)

=> \((\frac{\sqrt{a}}{\sqrt{b}})^1=3\)

Again applying componendo and dividendo rule, we get

a/b = (3/1)^2 = 9. So, a:b =9:1.