If 20% of a number a is equal to b, then b% of 20 is equal to what percentage of a?

Introduction / Context:
This is an algebraic percentage problem that explores the relationship between two variables a and b defined through a percentage condition. The problem states that 20% of a is equal to b and then asks what b% of 20 is in terms of a. Questions like this are common in quantitative aptitude and help test both percentage manipulation skills and algebraic reasoning in a simple setting.

Given Data / Assumptions:
- 20% of a is equal to b.- We need to find the value of b% of 20 in terms of a.- The answer must be expressed as some percentage of a (for example, 4% of a).- No specific numerical value is given for a or b; the relationship is symbolic.

Concept / Approach:
The approach is to translate the verbal percentage information into algebraic equations. First, we express b in terms of a using the equation for 20% of a. Then, we compute b% of 20 by treating b as a percentage value. Finally, we re-express the result back in terms of a and identify which percentage of a it corresponds to. This involves straightforward use of the percentage formula: x% of y = (x / 100) * y.

Step-by-Step Solution:
Step 1: The statement "20% of a is b" can be written algebraically as:b = 20% of a = (20 / 100) * a.Step 2: Simplify: b = (1 / 5) * a or b = a / 5.Step 3: We now need to evaluate "b% of 20".Step 4: b% of 20 means (b / 100) * 20.Step 5: So b% of 20 = (b / 100) * 20 = (20 * b) / 100 = b / 5.Step 6: We already know that b = a / 5, so substitute b in the expression b / 5.Step 7: b / 5 = (a / 5) / 5 = a / 25.Step 8: Now express a / 25 as a percentage of a.Step 9: a / 25 = (1 / 25) * a = (4 / 100) * a (because 1 / 25 = 4 / 100).Step 10: Therefore, a / 25 is equal to 4% of a.

Verification / Alternative check:
Take a specific numerical example to verify.Let a = 100.Then 20% of 100 = 20, so b = 20.Now compute b% of 20: 20% of 20 = (20 / 100) * 20 = 4.Now find 4% of a: 4% of 100 = (4 / 100) * 100 = 4.Both results are equal to 4, which confirms that b% of 20 is indeed 4% of a.

Why Other Options Are Wrong:
- 6% of a: This would correspond to 0.06a, which is 6a / 100. Our calculation gave a / 25 or 4a / 100, not 6a / 100.- 8% of a: Equal to 2a / 25, again not matching the derived value a / 25.- 10% of a: Equal to a / 10, which is much larger than a / 25 and inconsistent with the algebra.

Common Pitfalls:
- Treating b directly as 20 instead of as 20% of a in symbolic form.- Confusing b% of 20 with 20% of b, which would give a different expression.- Dropping or misplacing the factor 1/100 that always appears in percentage calculations.- Not simplifying fractions properly and hence failing to recognize that 1 / 25 equals 4%.

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