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 number of candidates who failed the examination?

Answer

To solve the problem step by step, we will use the concept of averages and the method of allegations.

### Step 1: Understand the given data
We know:
- Total number of candidates = 120
- Average marks of all candidates = 35
- Average marks of passed candidates = 39
- Average marks of failed candidates = 15

### Step 2: Calculate the total marks obtained by all candidates
To find the total marks obtained by all candidates, we can use the formula:
\[ \text{Total Marks} = \text{Average Marks} \times \text{Number of Candidates} \]
Substituting the values:
\[ \text{Total Marks} = 35 \times 120 = 4200 \]

### Step 3: Set up the equations for passed and failed candidates
Let:
- \( p \) = number of passed candidates
- \( f \) = number of failed candidates

From the problem, we know:
\[ p + f = 120 \]

### Step 4: Express total marks in terms of passed and failed candidates
The total marks can also be expressed as:
\[ \text{Total Marks} = (\text{Average Marks of Passed} \times p) + (\text{Average Marks of Failed} \times f) \]
Substituting the values:
\[ 4200 = 39p + 15f \]

### Step 5: Solve the equations
Now we have a system of equations:
1. \( p + f = 120 \)
2. \( 39p + 15f = 4200 \)

From the first equation, we can express \( p \) in terms of \( f \):
\[ p = 120 - f \]

Substituting \( p \) in the second equation:
\[ 39(120 - f) + 15f = 4200 \]
Expanding this:
\[ 4680 - 39f + 15f = 4200 \]
Combining like terms:
\[ 4680 - 24f = 4200 \]
Now, isolate \( f \):
\[ 4680 - 4200 = 24f \]
\[ 480 = 24f \]
\[ f = \frac{480}{24} = 20 \]

### Step 6: Conclusion
The number of candidates who failed the examination is \( f = 20 \).

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