If the average marks of `(1)/(4)th` class is 85% and that of `(1)/(3)` rd class is 70% and the average marks of the rest class is 56%,then the average of the whole class is :

Options

A. `67.916%`
B. `72.33%`
C. `69.165%`
D. can't be determined

Answer

To find the average marks of the whole class based on the given averages of different segments of the class, we can follow these steps:

### Step 1: Define the total number of students in the class
Let the total number of students in the class be \( N \).

### Step 2: Calculate the number of students in each segment
- The number of students in the \( \frac{1}{4} \)th class is \( \frac{N}{4} \).
- The number of students in the \( \frac{1}{3} \)rd class is \( \frac{N}{3} \).
- The rest of the class will then be \( N - \left(\frac{N}{4} + \frac{N}{3}\right) \).

### Step 3: Find a common denominator for the fractions
To combine the fractions \( \frac{N}{4} \) and \( \frac{N}{3} \), we need a common denominator. The least common multiple of 4 and 3 is 12.

- Convert \( \frac{N}{4} \) to \( \frac{3N}{12} \).
- Convert \( \frac{N}{3} \) to \( \frac{4N}{12} \).

### Step 4: Calculate the number of students in the rest of the class
Now, we can calculate the number of students in the rest of the class:
\[ N - \left(\frac{N}{4} + \frac{N}{3}\right) = N - \left(\frac{3N}{12} + \frac{4N}{12}\right) = N - \frac{7N}{12} = \frac{5N}{12} \]

### Step 5: Calculate the total marks for each segment
- The total marks for the \( \frac{1}{4} \)th class:
\[ \text{Total marks} = \frac{N}{4} \times 85 = \frac{85N}{4} \]

- The total marks for the \( \frac{1}{3} \)rd class:
\[ \text{Total marks} = \frac{N}{3} \times 70 = \frac{70N}{3} \]

- The total marks for the rest of the class:
\[ \text{Total marks} = \frac{5N}{12} \times 56 = \frac{280N}{12} = \frac{70N}{3} \]

### Step 6: Calculate the total marks for the whole class
Now, we can sum up the total marks from all segments:
\[ \text{Total marks} = \frac{85N}{4} + \frac{70N}{3} + \frac{70N}{3} \]

### Step 7: Find a common denominator to combine the total marks
The common denominator for 4 and 3 is 12. We convert each term:
\[ \frac{85N}{4} = \frac{255N}{12}, \quad \frac{70N}{3} = \frac{280N}{12} \]

Now, adding these:
\[ \text{Total marks} = \frac{255N}{12} + \frac{280N}{12} + \frac{280N}{12} = \frac{255N + 560N}{12} = \frac{815N}{12} \]

### Step 8: Calculate the average of the whole class
Finally, the average marks of the whole class is given by:
\[ \text{Average} = \frac{\text{Total marks}}{N} = \frac{\frac{815N}{12}}{N} = \frac{815}{12} \approx 67.92\% \]

### Final Answer:
The average of the whole class is approximately **67.92%**.

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