### Solution
Let's break down the problem step-by-step to find the arithmetic mean of the five numbers given the following:
- The arithmetic mean of two numbers is 28.
- The arithmetic mean of three other numbers is 58.
### Step-by-Step Solution
1. Understanding the given arithmetic means:
For two numbers:
[tex]\[
\text{Mean of two numbers} = 28
\][/tex]
For three numbers:
[tex]\[
\text{Mean of three numbers} = 58
\][/tex]
2. Calculating the total sum of those numbers:
For the two numbers, let’s denote them by [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
[tex]\[
\frac{a + b}{2} = 28
\][/tex]
Multiplying both sides by 2 to find the sum:
[tex]\[
a + b = 28 \times 2 = 56
\][/tex]
For the three numbers, let’s denote them by [tex]\( c \)[/tex], [tex]\( d \)[/tex], and [tex]\( e \)[/tex].
[tex]\[
\frac{c + d + e}{3} = 58
\][/tex]
Multiplying both sides by 3 to find the sum:
[tex]\[
c + d + e = 58 \times 3 = 174
\][/tex]
3. Finding the total sum of the five numbers:
Summing up the totals from the two sets of numbers:
[tex]\[
a + b + c + d + e = 56 + 174 = 230
\][/tex]
4. Calculating the arithmetic mean of the five numbers:
To find the arithmetic mean of the five numbers, we divide the total sum by 5:
[tex]\[
\text{Mean of five numbers} = \frac{a + b + c + d + e}{5} = \frac{230}{5} = 46.0
\][/tex]
### Final Answer:
The arithmetic mean of the five numbers is [tex]\( 46.0 \)[/tex].
