In aptitude (Algebra and Custom Operations), a custom binary operation is defined as a * b = 2a - 3b + ab. Using this definition, what is the value of (3 * 5) + (5 * 3)?

Introduction / Context:
This problem introduces a custom binary operation defined on real numbers. The question is not about standard arithmetic but about correctly applying the given rule. In many aptitude and reasoning tests, such custom operations are used to check whether you read definitions carefully and substitute values accurately instead of assuming usual multiplication or addition rules.

Given Data / Assumptions:

• The operation is defined as a * b = 2a - 3b + ab.
• We need to compute (3 * 5) + (5 * 3).
• All calculations are done with real numbers using this definition.
• There is no assumption that * is commutative; we must evaluate both orders separately.

Concept / Approach:
The key idea is to treat the definition a * b = 2a - 3b + ab as a formula. To find 3 * 5, substitute a = 3 and b = 5. To find 5 * 3, substitute a = 5 and b = 3. Then add the two results using ordinary addition. It is important not to swap a and b casually because the expression uses a and b in different ways, which generally makes the operation non commutative.

Step-by-Step Solution:
Step 1: Compute 3 * 5 using the definition a * b = 2a - 3b + ab. Step 2: For 3 * 5, take a = 3 and b = 5. Step 3: Substitute: 3 * 5 = 2 * 3 - 3 * 5 + 3 * 5. Step 4: Calculate: 2 * 3 = 6, 3 * 5 = 15. Step 5: So 3 * 5 = 6 - 15 + 15 = 6. Step 6: Now compute 5 * 3 with a = 5 and b = 3. Step 7: Substitute: 5 * 3 = 2 * 5 - 3 * 3 + 5 * 3. Step 8: Calculate: 2 * 5 = 10, 3 * 3 = 9, and 5 * 3 = 15. Step 9: So 5 * 3 = 10 - 9 + 15 = 16. Step 10: Finally, compute (3 * 5) + (5 * 3) = 6 + 16 = 22.

Verification / Alternative check:
We can double check each step carefully. For 3 * 5, plug directly into the formula: 2a - 3b + ab = 2 * 3 - 3 * 5 + 3 * 5. The last two terms cancel, leaving 6. For 5 * 3, apply 2a - 3b + ab = 2 * 5 - 3 * 3 + 5 * 3 = 10 - 9 + 15 = 16. There is no simplification mistake, so the sum 6 + 16 = 22 is confirmed.

Why Other Options Are Wrong:
23, 24, 25, and 28 all arise from possible arithmetic or substitution errors such as mixing up a and b, miscomputing products, or using 3 * 5 as standard multiplication instead of the defined operation. Once the definition is applied correctly to both pairs, only the value 22 fits the calculation.

Common Pitfalls:
A frequent mistake is to treat * as ordinary multiplication, quickly computing 3 * 5 = 15 and 5 * 3 = 15 and then adding to get 30. Others reverse the roles of a and b or forget to distribute the minus sign correctly in the expression 2a - 3b + ab. Always rewrite the formula and substitute step by step for each pair to avoid confusion in custom operation problems.

Final Answer:
The value of (3 * 5) + (5 * 3) under the given custom operation is 22.