In an algebraic percentage relation, 20% of a quantity a is equal to b.\nBased on this relation, b% of 20 is numerically equal to what fraction or percentage of a?

Introduction / Context:
This question tests the connection between percentage expressions and algebra. Instead of inserting numbers directly, the problem uses variables and symbolic expressions like 20% of a equals b. Many competitive exams include such questions to see whether candidates can translate verbal percentage statements into formulas and then simplify them systematically. Mastery of this type of question helps in simplifying more complicated percentage and ratio problems later on.

Given Data / Assumptions:

• 20% of a is equal to b, written as 20% of a = b.
• We are asked to compute b% of 20.
• We must express b% of 20 as some fraction or percentage of a.
• All percentages are interpreted in their usual arithmetic sense.

Concept / Approach:
We need to convert percentage statements into algebraic equations using decimal or fraction forms. First, we express 20% of a in algebraic form. Next, we use that relation to express b in terms of a. Once we know b in terms of a, we can compute b% of 20 by substituting this expression and simplifying. The goal is to express the final result as k% of a or as k times a, where k is a numerical constant.

Step-by-Step Solution:
Step 1: Write 20% of a as an algebraic expression: 20% of a = (20 / 100) * a = 0.2 * a. Step 2: This is given to be equal to b, so we have b = 0.2 * a. Step 3: Now consider b% of 20. By definition, b% of 20 = (b / 100) * 20. Step 4: Substitute b = 0.2 * a into that expression: b% of 20 = (0.2 * a / 100) * 20. Step 5: Simplify the constants: 0.2 * 20 = 4, so the numerator becomes 4 * a. Step 6: We then have b% of 20 = (4 * a) / 100 = 0.04 * a. Step 7: The quantity 0.04 * a is equal to 4% of a. Step 8: Therefore, b% of 20 is 4% of a.

Verification / Alternative check:
As a quick numerical check, suppose a = 100. Then 20% of 100 is 20, so b = 20. Now b% of 20 means 20% of 20, which equals 4. On the other hand, 4% of a is 4% of 100 which is also 4. The two values match perfectly, confirming that the expression b% of 20 is indeed equal to 4% of a and that no algebraic step has been mishandled in the reasoning.

Why Other Options Are Wrong:
If we choose 8% of a, that would correspond to 0.08 * a, which is double the correct value. Choosing 40% of a would give 0.4 * a, which is far too large compared to 0.04 * a. Selecting 80% of a would give an even larger 0.8 * a. None of these match the simplified algebraic expression we derived, so they cannot represent b% of 20 under the given relation.

Common Pitfalls:
A common mistake is to mix up 20% of a and b% of 20 or to treat b% as simply b divided by 100 without applying it to 20. Another frequent error is misplacing the percentage divisor, for example writing 20b instead of 20 * b or 20% * b. Careful stepwise substitution and simplification prevent such mistakes and lead reliably to the correct 4% result.

Final Answer:
The expression b% of 20 is equal to 4% of a.