If 50% of a quantity a is equal to b, then b% of 40 is numerically equal to which of the following expressions in terms of a?

Introduction / Context:
This algebraic percentage question asks you to relate a symbolic percentage expression back to the original variable a. The relation between a and b is given by a simple percentage, and you have to use that relation to evaluate b% of a numerical quantity. Questions like this strengthen understanding of how percentages interact with variables and help improve algebraic manipulation skills that are important across many quantitative topics.

Given Data / Assumptions:

• Fifty percent of a is equal to b, written as 50% of a = b.
• We are asked to find b% of 40.
• The final expression must be written in terms of a.
• All percentages are used in a standard arithmetic sense.

Concept / Approach:
First we express 50% of a in algebraic form to obtain b in terms of a. Once we know b as a multiple of a, we can compute b% of 40 by interpreting b% as b divided by 100 times 40. Substituting the expression for b and simplifying leads to a constant coefficient times a. That constant can then be compared with the options, which are given as numerical values indicating the required fraction of a.

Step-by-Step Solution:
Step 1: Express 50% of a. We have 50% of a = (50 / 100) * a = 0.5 * a. Step 2: According to the question, this equals b, so b = 0.5 * a. Step 3: Now we need b% of 40. By definition, b% of 40 = (b / 100) * 40. Step 4: Substitute b = 0.5 * a into the expression: b% of 40 = (0.5 * a / 100) * 40. Step 5: Simplify the constants. First compute 0.5 * 40 = 20. Step 6: So we get b% of 40 = (20 * a) / 100. Step 7: Reduce (20 * a) / 100 to a simpler fraction: (20 / 100) * a = 0.2 * a. Step 8: Therefore, b% of 40 is equal to 0.2 of a.

Verification / Alternative check:
To verify, choose a convenient value for a. Let a = 100. Then 50% of 100 is 50, so b = 50. Now b% of 40 is 50% of 40, which equals 20. On the other hand, 0.2 of a is 0.2 * 100 = 20. The two values match exactly, confirming that b% of 40 is correctly expressed as 0.2 * a, and 0.2 is the coefficient we need from the options.

Why Other Options Are Wrong:
If we choose 0.25 of a, that would correspond to 25% of a, which is larger than the correct 0.2 * a. The value 0.16 of a would correspond to 16% of a, which is smaller than the correct amount. The value 2 * a is far too large and does not come from any realistic interpretation of the given relationships. Since only 0.2 of a matches the algebraic simplification, the other options cannot be correct.

Common Pitfalls:
A common confusion is mixing up b and b% and treating them as the same thing. Another error is failing to divide by 100 when converting percentage expressions into multipliers. Some learners also mistakenly apply 50% to the number 40 instead of correctly following the sequence given in the question. Writing out each step carefully, with explicit fractions for percentages, helps avoid these mistakes and ensures the correct result.

Final Answer:
The expression b% of 40 is equal to 0.2 of a.