A school has 4 sections of Chemistry in Class X having 40, 35, 45 and 42 students. The mean marks obtained in Chemistry test are 50, 60, 55 and 45 respectively for the 4 sections. Determine the overall average of marks per student.

### Concept & Formula This problem is a classic **Weighted Average** calculation involving four distinct groups. You need to find the grand total of all marks and divide it by the total number of students. $$Weighted \ Average = \frac{\sum (Number \ of \ Students \times Average \ Marks)}{\sum (Number \ of \ Students)}$$ ### Step-by-Step Solution * **Given Data:** * Section 1: 40 students, Mean = 50 * Section 2: 35 students, Mean = 60 * Section 3: 45 students, Mean = 55 * Section 4: 42 students, Mean = 45 * **Calculate Total Marks per Section:** * Section 1 Total = $40 \times 50 = 2000$ * Section 2 Total = $35 \times 60 = 2100$ * Section 3 Total = $45 \times 55 = 2475$ * Section 4 Total = $42 \times 45 = 1890$ * **Calculate Overall Totals:** * Grand Total Marks = $2000 + 2100 + 2475 + 1890 = 8465$ * Total Students = $40 + 35 + 45 + 42 = 162$ * **Calculate Overall Average:** * Overall Average = $\frac{8465}{162} \approx 52.253...$ * Rounded to two decimal places, this is 52.25. ### Exam Strategy & Shortcut Use the **Deviation (Assumed Mean) Method** to keep the numbers manageable. 1. Assume a base mean of 50. 2. Calculate deviations from 50 for each section's mean: * Sec 1: $50 \rightarrow$ deviation $0 \implies 40 \times 0 = 0$ * Sec 2: $60 \rightarrow$ deviation $+10 \implies 35 \times 10 = +350$ * Sec 3: $55 \rightarrow$ deviation $+5 \implies 45 \times 5 = +225$ * Sec 4: $45 \rightarrow$ deviation $-5 \implies 42 \times (-5) = -210$ 3. Sum of total deviations = $0 + 350 + 225 - 210 = +365$. 4. Average deviation = $\frac{+365}{162} \approx +2.253$. 5. Final Average = Assumed Mean ($50$) + Average Deviation ($2.25$) = 52.25. ### Common Pitfall Attempting the massive additions and divisions ($8465 \div 162$) manually is prone to severe calculation errors. Furthermore, a lazy student might just average the four means: $\frac{50 + 60 + 55 + 45}{4} = 52.5$, which is conceptually wrong and leads to choosing an incorrect option if it were provided. ### Final Answer **Therefore, the correct answer is 52.25.**