A student secures $90\%$, $60\%$ and $54\%$ marks in test papers with $100$, $150$ and $200$ respectively as maximum marks. The percentage of his aggregate is

### Concept & Formula To find the aggregate percentage, you must calculate the total marks obtained across all subjects and divide it by the total maximum marks possible, then multiply by 100. $$ \text{Aggregate \%} = \frac{\text{Total Marks Secured}}{\text{Total Maximum Marks}} \times 100 $$ ### Step-by-Step Solution * **Given:** * Test 1: $90\%$ of $100$ maximum marks * Test 2: $60\%$ of $150$ maximum marks * Test 3: $54\%$ of $200$ maximum marks * **Calculation:** * Marks secured in Test 1 = $90\%$ of $100 = 90$ * Marks secured in Test 2 = $60\%$ of $150 = 90$ * Marks secured in Test 3 = $54\%$ of $200 = 108$ * Total marks secured = $90 + 90 + 108 = 288$ * Total maximum marks = $100 + 150 + 200 = 450$ * Aggregate Percentage = $\left( \frac{288}{450} \right) \times 100\%$ * $= \left( \frac{288}{45} \right) \times 10\%$ * Divide $288$ and $45$ by $9$: * $= \left( \frac{32}{5} \right) \times 10\%$ * $= 32 \times 2 = 64\%$ ### Exam Strategy & Shortcut Instead of calculating the absolute marks, you can use the ratio of the maximum marks as weights for a weighted average. The maximum marks $100$, $150$, and $200$ simplify to a ratio of $2 : 3 : 4$. Apply the weighted average directly to the secured percentages: Weighted Average = $\frac{(90 \times 2) + (60 \times 3) + (54 \times 4)}{2 + 3 + 4}$ $= \frac{180 + 180 + 216}{9}$ $= \frac{576}{9} = 64$. This method keeps the calculation numbers much smaller and easier to compute mentally. ### Common Pitfall A very common mistake is simply calculating the arithmetic mean of the given percentages: $\frac{90 + 60 + 54}{3} = 68$. Notice that $68$ is presented as an option (Option b) specifically to trap students who make this error. You can only average percentages directly if their respective base values are strictly identical. ### Final Answer **Therefore, the correct answer is 64.**