Question

If b is the mean proportion to a and c with a common ration \(\sqrt[3]{\dfrac{2}{3}}\)then (a - b)³ ∶ (b - c)³ is∶ 
1. 1 ∶ 2
2. 2 ∶ 3
3. 1 ∶ 1
4. 3 ∶ 2

Answer 1

Correct Answer - Option 4 : 3 ∶ 2

Given:

b is the mean proportion to a and c

common ratio = \(\sqrt[3]{\dfrac{2}{3}}\)

Calculation:

b = √ac

Then, (a - b)3 ∶ (b - c)3 = (a - √ac)³ ∶ (√ac - c)³

⇒ [√a(√a - √c)]³ ∶ [√c(√a - √c]³

⇒ (√a/√c)³

We know that, a × (common ratio)² = c

Or, (√2/√3)^2/3 = c/a

Then, (√a/√c)3 = [(√3/√2)2/3]³

⇒ 3 ∶ 2

∴ The required solution is 3 ∶ 2.