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Read the sentences and solve the following problems.

a) The arithmetic mean of two numbers is 17, and the arithmetic mean of another three numbers is 52. Find the arithmetic mean of all five numbers.

Solution: Let [tex]\( a, b, c, d, \)[/tex] and [tex]\( e \)[/tex] be the five numbers. Then [tex]\((a + b) / 2 = 17\)[/tex], so [tex]\(a + b = 34\)[/tex] and [tex]\((c + d + e) / 3 = 52\)[/tex], so [tex]\(c + d + e = 156\)[/tex]. Therefore, the arithmetic mean of the five numbers is [tex]\((a + b + c + d + e) / 5 = (34 + 156) / 5 = 190 / 5 = 38\)[/tex].

b) The arithmetic mean of two numbers is 28, and the arithmetic mean of another three numbers is 58. Find the arithmetic mean of all five numbers.



Answer :

GinnyAnswer
### 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].

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