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If the third proportional between 2√3 and 6 is A and the third proportional to 3√2 and 6 is B. What is the mean proportional between A√3 and B√2?

If the third proportional between 2√3 and 6 is A and the third proportional to 3√2 and 6 is B. What is the mean proportional between A√3 and B√2? Correct Answer 6√6

Let the third proportional between two numbers 2√3 and 6 be A. Therefore, the definition of the third proportional, we have

2√3 ∶ 6 ∶∶ 6 ∶ A

We can write in terms of the fraction

2√3/6 = 6/A

A = (6 × 6)/2√3 = 6√3

Similarly,

If the third proportional to 3√2 and 6 is B, then

3√2 ∶ 6 ∶∶ 6 ∶ B

3√2/6 = 6/B

B = (6 × 6)/3√2 = 6√2

Let the mean proportional between A√3 and B√2 be x. Therefore, the definition of the mean proportional

A√3 ∶ x ∶∶ x ∶ B√2

A√3/x = x/B√2

x2 = A√3 × B√2 = 6√3 × √3 × 6√2 × √2 = 6 × 3 × 6 × 2 = 216

x = √216 = 6√6 

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