The product of two numbers is 9375 and when the larger number is divided by the smaller number the quotient is 15. What is the sum of the two numbers?

Introduction / Context:
This is a number theory question about two numbers whose product and quotient are known. We are told the product of the numbers and the quotient when the larger is divided by the smaller. From this information, we are required to find the sum of the two numbers. Questions like this are good practice for applying basic algebra and the relationships among product, quotient, and factors.

Given Data / Assumptions:
- There are two numbers, a larger one and a smaller one.
- The product of the two numbers is 9375.
- When the larger number is divided by the smaller number, the quotient is 15.
- The remainder is zero because the problem does not mention any remainder.
- We must find the sum of the two numbers.

Concept / Approach:
Let the smaller number be S and the larger number be L. The quotient condition tells us that L = 15S. The product condition tells us that L * S = 9375. Substituting L = 15S into the product equation allows us to solve for S. Once S is known, L can be found easily and their sum can be computed.

Step-by-Step Solution:
Step 1: Let the smaller number be S and the larger number be L.Step 2: From the quotient condition, L divided by S gives a quotient of 15, so L = 15S.Step 3: From the product condition, L * S = 9375.Step 4: Substitute L = 15S into L * S = 9375 to get 15S * S = 9375.Step 5: This simplifies to 15S^2 = 9375.Step 6: Divide both sides by 15 to obtain S^2 = 9375 / 15.Step 7: Compute 9375 / 15 = 625, so S^2 = 625.Step 8: Take the positive square root (since we consider positive numbers here) to get S = 25.Step 9: Then the larger number is L = 15 * 25 = 375.Step 10: The sum of the two numbers is S + L = 25 + 375 = 400.

Verification / Alternative check:
Check the product: 25 * 375 = 9375, which matches the given product. Check the quotient: when 375 is divided by 25, the quotient is 15 and the remainder is 0, which matches the given information. The sum 25 + 375 is 400, so all conditions are satisfied and the result is consistent.

Why Other Options Are Wrong:
If the sum of the numbers were 100, 200, 300, or 250, then the numbers would have to be very different, and it would be impossible for them to have both a product of 9375 and a quotient of 15. For example, a sum of 200 would imply average around 100, but 100 * 100 is only 10000, and making the product exactly 9375 with a quotient of 15 is not possible under that constraint. Only the sum 400 corresponds to the exact pair 25 and 375 that satisfies both product and quotient conditions.

Common Pitfalls:
Some learners may misinterpret the quotient and think that S = 15L instead of L = 15S, which reverses the relationship. Others may try to guess numbers randomly instead of using the algebraic relationship. Another mistake is to forget to take the positive square root when solving for S. Following the systematic approach of expressing one variable in terms of the other ensures a correct and efficient solution.

Final Answer:
The sum of the two numbers is 400.