41 In a test the mean mark of four students was 61 . The median was 60 and the mode was 57 . What was the highest mark?

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UpStudy AI · Accepted Answer

To solve this problem, we need to extract the known conditions and then work through the information step by step.

### Known Conditions:
1. Mean mark of four students = 61
2. Median mark = 60
3. Mode mark = 57

### Step 1: Calculate the Total Marks
The mean is calculated as the total marks divided by the number of students. Therefore, we can find the total marks of the four students:

$$
\text{Total Marks} = \text{Mean} \times \text{Number of Students} = 61 \times 4 = 244
$$

### Step 2: Understanding the Median
The median is the middle value when the marks are arranged in order. Since there are four students, the median will be the average of the two middle values. Given that the median is 60, we can denote the marks in ascending order as $$a, b, c, d$$. Thus:

$$
\frac{b + c}{2} = 60 \implies b + c = 120
$$

### Step 3: Understanding the Mode
The mode is the value that appears most frequently. Since the mode is 57, at least two of the students must have scored 57. We can assume:

$$
a = 57 \quad \text{and} \quad b = 57
$$

### Step 4: Setting Up the Equations
Now we have:

- $$a = 57$$
- $$b = 57$$
- $$b + c = 120 \implies 57 + c = 120 \implies c = 63$$

Now we have three marks: $$57, 57, 63$$. We need to find the fourth mark $$d$$.

### Step 5: Calculate the Fourth Mark
Using the total marks calculated earlier:

$$
57 + 57 + 63 + d = 244
$$

Calculating the sum of the known marks:

$$
57 + 57 + 63 = 177
$$

Now substituting this into the total marks equation:

$$
177 + d = 244 \implies d = 244 - 177 = 67
$$

### Conclusion
The marks of the four students are $$57, 57, 63, 67$$. The highest mark among these is:

$$
\text{Highest Mark} = 67
$$

Thus, the highest mark is **67**.
The highest mark is 67.

41 In a test the mean mark of four students was 61 . The median was 60 and the mode was 57 . What was the highest mark?

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