If 40% of x is equal to y, then y% of 40 is equal to what percentage of x?

Introduction / Context:
This question tests your understanding of how percentages of variables relate to one another. It combines two percentage statements: 40 percent of x equals y, and then asks what y percent of 40 is in terms of x. Problems like this improve algebraic thinking and the ability to translate word statements into mathematical expressions involving percentages.

Given Data / Assumptions:

• 40% of x is equal to y.
• We are asked to compute y% of 40.
• The final expression should be written as a percentage of x.
• All variables represent real numbers and no additional conditions are imposed.

Concept / Approach:
We translate percentage statements into algebraic equations. If 40% of x equals y, then y = (40 / 100) * x. To find y% of 40, we write it as (y / 100) * 40. Then we substitute y in terms of x from the first equation into this second expression. Finally, we simplify the result so that it becomes k% of x, and compare k with the options given in the question.

Step-by-Step Solution:
Step 1: Express 40% of x in algebraic form. 40% of x = (40 / 100) * x = 0.4x. Step 2: According to the question, 40% of x equals y, so y = 0.4x. Step 3: Now we need to find y% of 40. By definition, y% of 40 = (y / 100) * 40. Step 4: Substitute y = 0.4x into this expression. So y% of 40 = (0.4x / 100) * 40. Step 5: Simplify the multiplication: (0.4x / 100) * 40 = 0.4x * 40 / 100. Step 6: Evaluate 0.4 * 40 = 16, so we get 16x / 100. Step 7: Therefore, y% of 40 = 16x / 100, which is 16% of x.

Verification / Alternative check:
To verify, assume a simple numerical value for x. Let x = 100. Then 40% of 100 is 40, so y = 40. Now compute y% of 40, that is 40% of 40, which equals (40 / 100) * 40 = 16. As a percentage of x = 100, this value 16 is exactly 16% of x. This confirms that the expression is indeed 16% of x.

Why Other Options Are Wrong:
4% of x would be 4x / 100, which is much smaller than 16x / 100. 8% of x is also only half of the correct value. 160% of x would give 1.6x, which is far larger than the simplified expression 16x / 100. Therefore none of these match the computed result, and only 16% of x is correct.

Common Pitfalls:
Learners often confuse which quantity is being expressed as a percentage of which. Another common error is to treat y% as simply y / 100 without multiplying by the base quantity (here 40). It is also easy to make mistakes when simplifying fractions or decimals. Carefully writing each algebraic step and simplifying gradually helps avoid such mistakes.

Final Answer:
y% of 40 is equal to 16% of x.