Question

In a competitive exam there were 5 sections. 10% of the total number of students cleared the cut off in all the sections and 5% cleared none of the sections. From the remaining candidates 30% cleared only section 1, 20% cleared only section 2, 10% cleared only section 3 and remaining 1020 candidates cleared only section 4. How many students appeared in the competitive exam?

• A. 2550
• B. 2800
• C. 3000
• D. 3200

Hint:

Subtract the fixed percentages first, then split the remaining group by the given shares.

Answer 1

The Correct Option is C

To solve this question, let's analyze the information given and calculate the total number of students who appeared for the exam step-by-step.

1. Let the total number of students who appeared in the exam be \(T\). 
2. It is given that 10% of the students cleared the cut-off in all sections. Therefore: \(0.10T\) students cleared all sections.
3. 5% of the students did not clear any section, which gives us: \(0.05T\) students cleared none of the sections.
4. The remaining students are those who cleared at least one but not all sections. Thus, the remaining students are: \(T - 0.10T - 0.05T = 0.85T\).
5. From these remaining students:
   • 30% cleared only section 1, which is \(0.30 \times 0.85T\).
   • 20% cleared only section 2, which is \(0.20 \times 0.85T\).
   • 10% cleared only section 3, which is \(0.10 \times 0.85T\).
   • The number of students who cleared only section 4 is given as 1020.
6. Let's set up an equation based on the above information: \(0.30 \times 0.85T + 0.20 \times 0.85T + 0.10 \times 0.85T + 1020 = 0.85T\)
7. Solving the equation:
   • \((0.30 + 0.20 + 0.10) \times 0.85T + 1020 = 0.85T\)
   • \(0.60 \times 0.85T + 1020 = 0.85T\)
   • \(0.51T + 1020 = 0.85T\)
   • Simplifying, we have \(1020 = 0.85T - 0.51T\)
   • \(1020 = 0.34T\)
   • \(T = \frac{1020}{0.34} = 3000\)

Thus, the total number of students who appeared in the exam is 3000.

Therefore, the correct answer is: 3000.

Answer 2

Step 1: Represent the total number of students algebraically.
Let the total number of students be \(100x\).

Step 2: Remove the students already accounted for.
10% cleared all sections, that is \(10x\) students. 5% cleared none, that is \(5x\) students. The remaining students are \(100x - 10x - 5x = 85x\).

Step 3: Split the remaining students by section.
Of these \(85x\), 30% cleared only section 1, 20% cleared only section 2 and 10% cleared only section 3. Together this is \(30 + 20 + 10 = 60\%\) of \(85x\), leaving \(40\%\) of \(85x\) for section 4 only.

Step 4: Use the given count for section 4 to solve for x.
We know \(0.40 \times 85x = 1020\), so \(34x = 1020\), giving \(x = 30\).

Final Answer:
Total students \(= 100x = 100 \times 30 = 3000\). \[ \boxed{3000} \]