The mean proportion between the numbers $a$ and $b$ is $6$. Which of the following pairs of numbers can be the

Question

The mean proportion between the numbers $a$ and $b$ is $6$. Which of the following pairs of numbers can be the values of $a$ and $b$?

Examoogle · Accepted Answer

The correct answer is Option B: 12 and 3. Check the options by multiplying the two numbers given in each pair. The pair whose product is the square of the mean proportion (i.e., $6^{2} = 36$) is the correct answer.

Step-by-Step Solution & Concepts:
Shortcut Trick: Check the options by multiplying the two numbers given in each pair. The pair whose product is the square of the mean proportion (i.e., $6^{2} = 36$) is the correct answer.

Formula Used: $$b = \sqrt{a 	imes c} \implies b^{2} = a 	imes c$$

Given: Mean proportion between two numbers a and b is 6.

Steps: Define Mean Proportion: If the mean proportion between two numbers $a$ and $b$ is $x$, then by definition, $a:x = x:b$. This implies $x^{2} = a 	imes b$.. Substitute Given Value: Given the mean proportion $x = 6$, we substitute this into the equation $x^{2} = a 	imes b$.. Evaluate Options: We test each option by calculating the product of the two numbers to see if it equals 36: A) $16 	imes 4 = 64$, B) $12 	imes 3 = 36$, C) $10 	imes 6 = 60$, D) $12 	imes 16 = 192$.

Answer: B is correct because the product of 12 and 3 is 36, which is the square of the mean proportion 6.

Exam Tip: This concept is a specific case of geometric progression where three terms $a, x, b$ are in G.P., so the middle term is the geometric mean of the outer terms.

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