Simplify the expression a% of b plus b% of a. Which of the following is equal to a% of b + b% of a?

Introduction / Context:
This is an algebraic percentage identity problem. It asks you to rewrite a% of b plus b% of a in a simpler, more compact form. Such identities are very useful in mental arithmetic and shortcut methods in quantitative aptitude, particularly in problems involving percentages and ratios between two quantities.

Given Data / Assumptions:

• We have two variables a and b.
• The expression to simplify is: a% of b + b% of a.
• We need to choose the correct equivalent expression from the options.
• All variables are real numbers and percentages are interpreted in the usual way.

Concept / Approach:
The phrase “a% of b” means (a / 100) * b. Similarly, “b% of a” means (b / 100) * a. By converting both terms into algebraic form, we can add them and factor common terms. The core idea is to carefully handle the structures and see what factor emerges from the sum. Once simplified, we compare the resulting expression with the given options to identify an equivalent percentage form.

Step-by-Step Solution:
Step 1: Write a% of b in algebraic form. a% of b = (a / 100) * b = (ab) / 100. Step 2: Write b% of a in algebraic form. b% of a = (b / 100) * a = (ba) / 100. Step 3: Add the two expressions. a% of b + b% of a = (ab / 100) + (ba / 100). Step 4: Since ab = ba, we can combine them: (ab / 100) + (ab / 100) = 2ab / 100. Step 5: Now express 2ab / 100 back in percentage form. 2ab / 100 = (2a / 100) * b = 2a% of b. Step 6: Therefore, a% of b + b% of a = 2a% of b.

Verification / Alternative check:
Take a simple numerical example to check. Let a = 10 and b = 50. Then a% of b = 10% of 50 = 5. b% of a = 50% of 10 = 5. Sum = 10. Now compute 2a% of b = 2 * 10% of 50 = 20% of 50 = 10. The numerical check confirms that 2a% of b is equal to a% of b + b% of a for these sample values. This supports the algebraic derivation.

Why Other Options Are Wrong:
2a% of 2b would be (2a / 100) * 2b = 4ab / 100, which is twice as large as 2ab / 100.
2a% of 2a and 2b% of 2b involve multiplying by a or b again and do not match the required dimensions of ab / 100. They produce completely different algebraic structures and cannot be equal to the sum of a% of b and b% of a.

Common Pitfalls:
A typical mistake is mishandling the percentage notation, for example treating a% as a / 10 instead of a / 100. Another error is to confuse 2a% of b with (a + b)% of something without checking the underlying algebra. When in doubt, always translate percentage statements into basic fraction or decimal context and manipulate them using standard algebraic rules.

Final Answer:
a% of b + b% of a is equal to 2a% of b.