The media of a set of eight numbers is 4.5 Given that seven of the numbers are 7,2,13,4,8,2 and 1, find the eight number and write down the mode of eight numbers.

Options

A. 4
B. 7
C. 2
D. 1

Answer

To solve the problem step by step, we need to find the eighth number in the set and then determine the mode of the eight numbers.

### Step 1: List the given numbers
The seven numbers provided are:
7, 2, 13, 4, 8, 2, and 1.

### Step 2: Arrange the numbers in increasing order
We will arrange the numbers in ascending order:
1, 2, 2, 4, 7, 8, 13

### Step 3: Let the eighth number be 'x'
We will denote the eighth number as 'x'. Now our list of numbers becomes:
1, 2, 2, 4, 7, 8, 13, x

### Step 4: Determine the position of the median
Since we have a total of 8 numbers (an even number), the median will be the average of the 4th and 5th numbers in the sorted list.

### Step 5: Write the median formula
The median is given as 4.5. Therefore, we can set up the equation:
\[ \text{Median} = \frac{\text{4th number} + \text{5th number}}{2} = 4.5 \]

### Step 6: Identify the 4th and 5th numbers
From our ordered list, the 4th number is 4 and the 5th number is either 7 or x (depending on the value of x). Thus, we can write:
\[ \frac{4 + x}{2} = 4.5 \]

### Step 7: Solve for x
Now, we will solve the equation:
\[ 4 + x = 4.5 \times 2 \]
\[ 4 + x = 9 \]
\[ x = 9 - 4 \]
\[ x = 5 \]

### Step 8: Add the eighth number to the list
Now that we have found x, we can add it to our list:
1, 2, 2, 4, 5, 7, 8, 13

### Step 9: Find the mode
The mode is the number that appears most frequently in the data set. In our list:
- 1 appears 1 time
- 2 appears 2 times
- 4 appears 1 time
- 5 appears 1 time
- 7 appears 1 time
- 8 appears 1 time
- 13 appears 1 time

The number that appears most frequently is 2.

### Final Answers
- The eighth number is **5**.
- The mode of the eight numbers is **2**.