The marks of 10 students of a class in an examination arranged in ascending order are as follows :
13, 35, 43, 46, x, x + 4, 55, 61, 71, 80
If the median marks is 48, find the value of x. Hence find the mode of the given data.

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Understand the Data
The marks of 10 students arranged in ascending order are:
13, 35, 43, 46, x, x + 4, 55, 61, 71, 80

### Step 2: Identify the Median
The median is given as 48. Since there are 10 students (an even number), the median can be calculated using the formula:
\[ \text{Median} = \frac{\text{(n/2)th term} + \text{(n/2 + 1)th term}}{2} \]
Here, n = 10, so:
\[ \text{Median} = \frac{\text{5th term} + \text{6th term}}{2} \]
We know that the 5th term is \(x\) and the 6th term is \(x + 4\).

### Step 3: Set Up the Equation
From the median formula, we have:
\[ 48 = \frac{x + (x + 4)}{2} \]
This simplifies to:
\[ 48 = \frac{2x + 4}{2} \]
Multiplying both sides by 2 gives:
\[ 96 = 2x + 4 \]

### Step 4: Solve for x
Now, we will isolate \(x\):
\[ 96 - 4 = 2x \]
\[ 92 = 2x \]
\[ x = \frac{92}{2} = 46 \]

### Step 5: Find x + 4
Now that we have \(x\):
\[ x + 4 = 46 + 4 = 50 \]

### Step 6: List the Complete Data
Now we can list the complete data of marks:
13, 35, 43, 46, 46, 50, 55, 61, 71, 80

### Step 7: Find the Mode
The mode is the number that appears most frequently in the data. Looking at the complete data:
- 46 appears twice,
- All other numbers appear once.

Thus, the mode is:
\[ \text{Mode} = 46 \]

### Final Answers
- The value of \(x\) is **46**.
- The mode of the given data is **46**.

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