A data collection has 10 numbers, 9 of which are 35, 70, 15, 59, 19, 15, 57, 26, and 34. If the median of this data is 32, find the tenth number.

Answer

To find the tenth number in the data collection, we will follow these steps:

### Step 1: List the given numbers
The numbers provided are: 35, 70, 15, 59, 19, 15, 57, 26, and 34.

### Step 2: Arrange the numbers in ascending order
We need to arrange these numbers from the smallest to the largest:
- The sorted order is: 15, 15, 19, 26, 34, 35, 57, 59, 70.

### Step 3: Identify the median position
Since there are 10 numbers (including the unknown tenth number), the median will be the average of the 5th and 6th numbers in the sorted list.

### Step 4: Introduce the unknown variable
Let the tenth number be represented as \( x \). Now we will include \( x \) in our sorted list and find the new median.

### Step 5: Determine the median formula
For an even number of observations, the median is calculated as:
\[ \text{Median} = \frac{\text{5th term} + \text{6th term}}{2} \]
Given that the median is 32, we can set up the equation:
\[ \frac{\text{5th term} + \text{6th term}}{2} = 32 \]
This implies:
\[ \text{5th term} + \text{6th term} = 64 \]

### Step 6: Identify the 5th and 6th terms
In our sorted list, if \( x \) is less than or equal to 34, the 5th term will be 34 and the 6th term will be 35. If \( x \) is greater than 34, the 5th term will be 35 and the 6th term will be 57.

### Step 7: Solve for \( x \)
Assuming \( x \) is less than or equal to 34:
- The equation becomes:
\[ 34 + 35 = 64 \]
This is true. Therefore, we can find \( x \):
\[ x + 34 = 64 \implies x = 64 - 34 = 30 \]

### Step 8: Conclusion
Thus, the tenth number is \( \boxed{30} \).

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