To solve the problem step by step, we will first define the six numbers and then use the given averages to find the difference between the first and last numbers.
### Step 1: Define the six numbers
Let the six numbers be \( A, B, C, D, E, F \), arranged in decreasing order such that \( A > B > C > D > E > F \).
### Step 2: Calculate the sum of the first five numbers
The average of the first five numbers \( A, B, C, D, E \) is given as 30. We can use the formula for average:
\[
\text{Average} = \frac{\text{Sum of numbers}}{\text{Total numbers}}
\]
Thus, we have:
\[
30 = \frac{A + B + C + D + E}{5}
\]
Multiplying both sides by 5 gives:
\[
A + B + C + D + E = 150 \quad \text{(Equation 1)}
\]
### Step 3: Calculate the sum of the last five numbers
The average of the last five numbers \( B, C, D, E, F \) is given as 25. Using the average formula again:
\[
25 = \frac{B + C + D + E + F}{5}
\]
Multiplying both sides by 5 gives:
\[
B + C + D + E + F = 125 \quad \text{(Equation 2)}
\]
### Step 4: Find the value of \( A \) and \( F \)
Now, we can use Equations 1 and 2 to find \( A \) and \( F \). From Equation 1, we know:
\[
A + B + C + D + E = 150
\]
From Equation 2, we know:
\[
B + C + D + E + F = 125
\]
Subtract Equation 2 from Equation 1:
\[
(A + B + C + D + E) - (B + C + D + E + F) = 150 - 125
\]
This simplifies to:
\[
A - F = 25
\]
### Step 5: Conclusion
The difference between the first number \( A \) and the last number \( F \) is:
\[
A - F = 25
\]
Thus, the difference of the first and last numbers is **25**.
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