Question

The average of marks obtained by 120 candidates in a certain examination is 35. If the average marks of passed candidates is 39 and that of the failed candidates is 15, what is the number of candidates who passed the examination?

• A. 90
• B. 85
• C. 100
• D. 120

Hint:

When two groups have different averages, use total \(=\) (avg\(_1\)\(\times\)size\(_1\)) \(+\) (avg\(_2\)\(\times\)size\(_2\)), or set up a weighted-average equation and solve for the group size.

Answer 1

The Correct Option is C

Step 1: Convert averages to total marks. 
Overall average \(=35\) for \(120\) candidates \(\Rightarrow\) total marks \(=120\times 35=4200\). 

Step 2: Let the number of passed candidates be \(p\). 
Then failed candidates \(=120-p\). 
Total marks \(=\) (passed total) \(+\) (failed total) 
\[ 39p + 15(120-p) = 4200. \] 

Step 3: Solve for \(p\). 
\[ 39p + 1800 - 15p = 4200 \Rightarrow 24p = 2400 \Rightarrow p = 100. \] \[ \boxed{100} \]

Answer 2

The alligation (weighted average) method gives a fast alternative for splitting the 120 candidates into passed and failed groups.

1. 90: Checking the total marks: \(39\times90+15\times(120-90)=3510+450=3960\), which does not equal the required total of \(4200\).
2. 85: Here total marks would be \(39\times85+15\times35=3315+525=3840\), also short of 4200.
3. 100: Here total marks are \(39\times100+15\times20=3900+300=4200\), matching exactly.
4. 120: This would mean nobody failed, which is inconsistent with the problem stating a separate average for failed candidates, and the total would be \(39\times120=4680\), not 4200.

By alligation, the ratio of passed to failed candidates is \((35-15):(39-35)=20:4=5:1\), so out of 120 candidates split into 6 equal parts of 20 each, the number who passed is \(5\times20=100\).

So the correct answer is 100.