Difficulty: Easy
Correct Answer: 25
Explanation:
Introduction / Context:
This question focuses on the classical definition of the third proportional of two numbers. It is a direct application of ratio and proportion, used frequently in elementary algebra and aptitude tests. Understanding these terms is important because they appear in many old style aptitude questions and also help in dealing with geometric mean and related topics.
Given Data / Assumptions:
Concept / Approach:
If a, b and c are three numbers such that a : b = b : c, then c is called the third proportional to a and b. This definition gives the relation b^2 = a * c, or c = b^2 / a. In this problem, a = 9 and b = 15, so c will be equal to (15^2) / 9. We just need to compute this value and then match it with one of the options provided.
Step-by-Step Solution:
Let the third proportional to 9 and 15 be c.By definition, 9 : 15 = 15 : c.This implies 9 * c = 15 * 15.So 9 * c = 225.Therefore, c = 225 / 9.Compute 225 / 9 = 25.Hence, the third proportional is 25.
Verification / Alternative check:
Check the proportion: 9 : 15 should equal 15 : 25.Compute 9 / 15 = 3 / 5 = 0.6.Compute 15 / 25 = 3 / 5 = 0.6.Since both ratios are equal, the proportion holds and 25 is correct as the third proportional.
Why Other Options Are Wrong:
Values such as 21, 30, 45 or 36 do not satisfy the proportion 9 : 15 = 15 : c. If we take c = 21, we would get 15 / 21 = 5 / 7, which is not equal to 3 / 5. Similar checks show that 30, 45 and 36 also fail. Only c = 25 gives the correct equality of ratios.
Common Pitfalls:
A frequent mistake is to confuse third proportional with arithmetic progression or with geometric mean. Some students compute c as 15 * 9 or as (9 + 15) / 2 without using the proportion definition. The safest method is to always start from the equality a : b = b : c and cross multiply to obtain a * c = b^2.
Final Answer:
The third proportional to 9 and 15 is 25.
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