The ratio of two positive numbers is 3 : 5 and their arithmetic mean (average) is 48. What is the product of these two numbers?

Introduction / Context:
This is a standard ratio and average problem from quantitative aptitude. The question combines the concept of expressing numbers in terms of a common ratio multiplier with the concept of arithmetic mean. By using the given ratio and the average, we can find the actual values of the two numbers and then compute their product. Such questions test algebraic representation skills and comfort with ratios.

Given Data / Assumptions:

• The ratio of the two positive numbers is 3 : 5.
• The average (arithmetic mean) of the two numbers is 48.
• We are asked to find the product of these two numbers.
• Both numbers are positive real numbers, and in this context they will turn out to be integers.

Concept / Approach:
If the ratio of two numbers is 3 : 5, we can represent them as 3k and 5k, where k is a common positive multiplier. The arithmetic mean of these two numbers is given as 48. The mean of 3k and 5k is (3k + 5k) / 2 = 4k. Equating this to 48 allows us to find k. Once k is known, we compute the actual numbers and then multiply them to obtain the product. This approach uses simple algebra and avoids guesswork.

Step-by-Step Solution:
Step 1: Let the two numbers be 3k and 5k, since their ratio is 3 : 5. Step 2: The arithmetic mean of 3k and 5k is (3k + 5k) / 2. Step 3: Simplify the numerator: 3k + 5k = 8k, so the mean is 8k / 2 = 4k. Step 4: We are told the average is 48, so set 4k = 48. Step 5: Solve for k: k = 48 / 4 = 12. Step 6: The two numbers are 3k = 3 * 12 = 36 and 5k = 5 * 12 = 60. Step 7: The product of these numbers is 36 * 60. Step 8: Compute the product: 36 * 60 = 2160. Step 9: Thus, the required product is 2160.

Verification / Alternative check:
We can verify by checking both the ratio and the average. The ratio 36 : 60 simplifies by dividing numerator and denominator by 12, giving 3 : 5, which matches the given ratio. The arithmetic mean of 36 and 60 is (36 + 60) / 2 = 96 / 2 = 48, which matches the given average. Since both conditions hold, the numbers 36 and 60 are correct, and their product of 2160 is confirmed.

Why Other Options Are Wrong:
Option (a) 3365, option (c) 1854, and option (d) 1632 do not equal 36 * 60. If we tried to back-calculate from these products, we would not get numbers in the ratio 3 : 5 with an average of 48. Therefore, these options do not satisfy the given conditions and are incorrect.

Common Pitfalls:
Some learners forget to divide by 2 when using the average formula or incorrectly set 3k + 5k equal to 48 instead of dividing by 2. Another typical error is to miscalculate the product after correctly finding the numbers. Being careful with the mean formula and performing arithmetic step by step helps avoid such mistakes.

Final Answer:
The product of the two numbers is 2160.