S1, is a series of 5 consecutive positive multiples of 4 whose sum is 100. S2, is another series of 4 consecutive even numbers, whose 2nd lowest number is 6 less than the highest number of S1, What is the average of S2,?

Options

A. 21
B. 19
C. 23
D. 25

Answer

To solve the problem step by step, we will break down the information given about the two series, S1 and S2.

### Step 1: Determine the series S1
S1 consists of 5 consecutive positive multiples of 4. Let's denote the first multiple as \( x \). Therefore, the series can be represented as:
- First number: \( x \)
- Second number: \( x + 4 \)
- Third number: \( x + 8 \)
- Fourth number: \( x + 12 \)
- Fifth number: \( x + 16 \)

### Step 2: Set up the equation for the sum of S1
The sum of these numbers is given to be 100:
\[ x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 100 \]
This simplifies to:
\[ 5x + (4 + 8 + 12 + 16) = 100 \]
Calculating the constant terms:
\[ 4 + 8 + 12 + 16 = 40 \]
Thus, we have:
\[ 5x + 40 = 100 \]

### Step 3: Solve for x
Now, we can solve for \( x \):
\[ 5x = 100 - 40 \]
\[ 5x = 60 \]
\[ x = 12 \]

### Step 4: Find the numbers in S1
Now that we have \( x = 12 \), we can find the actual numbers in S1:
- First number: \( 12 \)
- Second number: \( 16 \)
- Third number: \( 20 \)
- Fourth number: \( 24 \)
- Fifth number: \( 28 \)

### Step 5: Identify the highest number in S1
The highest number in S1 is:
\[ x + 16 = 12 + 16 = 28 \]

### Step 6: Determine the series S2
S2 consists of 4 consecutive even numbers. Let’s denote the first number in S2 as \( y \). The numbers in S2 will be:
- First number: \( y \)
- Second number: \( y + 2 \)
- Third number: \( y + 4 \)
- Fourth number: \( y + 6 \)

### Step 7: Set up the relationship for S2
According to the problem, the second lowest number in S2 (which is \( y + 2 \)) is 6 less than the highest number in S1 (which is 28):
\[ y + 2 = 28 - 6 \]
This simplifies to:
\[ y + 2 = 22 \]

### Step 8: Solve for y
Now we can solve for \( y \):
\[ y = 22 - 2 \]
\[ y = 20 \]

### Step 9: Find the numbers in S2
Now that we have \( y = 20 \), we can find the actual numbers in S2:
- First number: \( 20 \)
- Second number: \( 22 \)
- Third number: \( 24 \)
- Fourth number: \( 26 \)

### Step 10: Calculate the average of S2
To find the average of S2, we sum the numbers and divide by the number of observations:
\[ \text{Sum of S2} = 20 + 22 + 24 + 26 = 92 \]
The number of observations is 4, so the average is:
\[ \text{Average} = \frac{92}{4} = 23 \]

### Final Answer
The average of S2 is \( 23 \).

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