Two numbers x and y are such that their mean proportion (geometric mean) is 9 and their third proportion is 243. What are the values of x and y?

Introduction / Context:
This question uses the concepts of mean proportion (geometric mean) and third proportion of two numbers. It tests understanding of proportional relationships and the ability to set up and solve simultaneous equations using these geometric definitions. Such questions are often seen in quantitative aptitude sections under the topic of ratio and proportion.

Given Data / Assumptions:

• Mean proportion (geometric mean) of x and y is 9.
• Third proportion to x and y is 243.
• We must find x and y.
• x and y are positive numbers.

Concept / Approach:
The geometric mean of x and y is sqrt(xy), so we have sqrt(xy) = 9, which implies xy = 81. The third proportion to x and y is defined as a number t such that x : y = y : t, or equivalently y^2 = x * t. Here, t is given as 243, so y^2 = 243x. We now have two equations: xy = 81 and y^2 = 243x. Solving these simultaneously yields x and y.

Step-by-Step Solution:

Verification / Alternative check:
Check mean proportion: sqrt(xy) = sqrt(3 * 27) = sqrt(81) = 9, which matches the given value. Check third proportion: with x = 3 and y = 27, the ratio x : y = 3 : 27 = 1 : 9, and y : t should be 1 : 9 as well, so t must be 243 (since 27 : 243 = 1 : 9). This confirms the correctness of x and y.

Why Other Options Are Wrong:

• 3 and 9 give xy = 27, not 81, and do not satisfy both conditions.
• 6 and 27 give xy = 162, not 81.
• 6 and 81 give xy = 486, again not 81, and they also fail the third proportion condition.

Common Pitfalls:
Some students mix up the definitions of mean and third proportion or think both are arithmetic instead of geometric. Others may incorrectly square or cube during manipulation and introduce algebraic errors. Writing down the precise definitions and carefully substituting from one equation into another is the safest route.

Final Answer:
The values of the numbers are 3 and 27.