To find the value of \( x \) such that the mode of the data set \( 13, 27, 24, 13, 17, 21, 22, x, 13, 17 \) is \( 17 \), we can follow these steps:
### Step 1: Understand the concept of mode
The mode of a data set is the number that appears most frequently. In this case, we want the mode to be \( 17 \).
### Step 2: Count the occurrences of the numbers in the data set
Let's list out the numbers and their frequencies:
- \( 13 \): appears 3 times
- \( 27 \): appears 1 time
- \( 24 \): appears 1 time
- \( 17 \): appears 2 times (currently)
- \( 21 \): appears 1 time
- \( 22 \): appears 1 time
- \( x \): frequency depends on its value
### Step 3: Determine the required frequency for the mode
For \( 17 \) to be the mode, it must appear more frequently than any other number. Currently, \( 13 \) appears 3 times, while \( 17 \) appears only 2 times. Therefore, to make \( 17 \) the mode, it needs to appear at least 3 times.
### Step 4: Set the value of \( x \)
To increase the frequency of \( 17 \) to 3, we can set \( x = 17 \). This will add one more occurrence of \( 17 \) to the data set.
### Step 5: Verify the frequencies after setting \( x \)
After setting \( x = 17 \), the frequencies will be:
- \( 13 \): 3 times
- \( 27 \): 1 time
- \( 24 \): 1 time
- \( 17 \): 3 times (after including \( x \))
- \( 21 \): 1 time
- \( 22 \): 1 time
Now, both \( 13 \) and \( 17 \) appear 3 times. However, since we want \( 17 \) to be the mode, we need to ensure it appears more than 3 times.
### Step 6: Conclusion
Thus, the value of \( x \) must be \( 17 \) to ensure that \( 17 \) has the highest frequency.
**Final Answer: \( x = 17 \)**
