Sure, let's break down the problem step by step to find the fifth number.
1. Calculate the total sum of all 9 numbers:
The average of all 9 numbers is given as 23. To find the total sum, we multiply the average by the count of numbers.
[tex]\[
\text{Total sum of all 9 numbers} = 23 \times 9 = 207
\][/tex]
2. Calculate the total sum of the first five numbers:
The average of the first five numbers is given as 20. To find the total sum of these first five numbers, we multiply the average by the count of first five numbers.
[tex]\[
\text{Total sum of the first 5 numbers} = 20 \times 5 = 100
\][/tex]
3. Calculate the total sum of the last five numbers:
The average of the last five numbers is given as 25. To find the total sum of these last five numbers, we multiply the average by the count of last five numbers.
[tex]\[
\text{Total sum of the last 5 numbers} = 25 \times 5 = 125
\][/tex]
4. Find the overlap (the fifth number):
Notice that the fifth number is included in both the first five numbers and the last five numbers. Consequently, it has been added twice in our total calculations. To correct this, we compute the sum of all 10 counted numbers (5 first + 5 last) and then subtract the total sum of all numbers (which only includes each number once).
[tex]\[
\text{Total sum including double-counted fifth number} = 100 + 125 = 225
\][/tex]
Now subtract the total sum of all 9 numbers:
[tex]\[
\text{Sum of the fifth number} = 225 - 207 = 18
\][/tex]
Thus, the fifth number is [tex]\( \boxed{18} \)[/tex]. By following these steps, we have determined the value of the fifth number to be 18.
