If x varies directly as y and x = 8 when y = 15, then what is the value of x when y = 10?

Introduction / Context:

This is a straightforward direct variation or direct proportionality question. Such questions are common in school mathematics and competitive exams to test how well you understand proportional relationships between two variables.

Given Data / Assumptions:

• x varies directly as y, so x is proportional to y.
• x = 8 when y = 15.
• We need to find x when y = 10.
• Both x and y are real numbers.

Concept / Approach:

When x varies directly as y, we have x = k * y for some constant of proportionality k. First find k using the given pair (x, y) = (8, 15). Then substitute the new value y = 10 into the same formula to get the required x.

Step-by-Step Solution:

Verification / Alternative check:

You can check the ratio x / y in both cases. Initially, 8 / 15 is the ratio. In the second case, 16/3 divided by 10 is (16/3) * (1/10) = 16 / 30 = 8 / 15, which matches the original ratio, confirming the answer.

Why Other Options Are Wrong:

The values 5, 15/8, 8/15, and 6 do not maintain the same ratio x / y as 8 / 15. For example, 5 / 10 = 1 / 2, which is not equal to 8 / 15. Similarly, 6 / 10 = 3 / 5, which again does not match the required ratio.

Common Pitfalls:

Common mistakes include inverting the proportionality and using y = k * x instead of x = k * y, or trying to set up a complicated proportion instead of directly working with the constant k.

Final Answer:

When y = 10, the corresponding value of x is 16/3.