If the mean of six consecutive multiples of `4` is `18`, find the sum of `3rd` and `4th` numbers.

Options

A. `36`
B. `12`
C. `48`
D. `14`

Answer

To solve the problem step by step, let's follow the reasoning provided in the video transcript:

### Step 1: Define the First Multiple
Let the first multiple of 4 be represented as \( 4x \).

### Step 2: Write the Consecutive Multiples
The six consecutive multiples of 4 can be expressed as:
- First multiple: \( 4x \)
- Second multiple: \( 4x + 4 \)
- Third multiple: \( 4x + 8 \)
- Fourth multiple: \( 4x + 12 \)
- Fifth multiple: \( 4x + 16 \)
- Sixth multiple: \( 4x + 20 \)

### Step 3: Calculate the Mean
The mean of these six multiples is given as 18. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Substituting the values, we have:
\[ 18 = \frac{(4x + (4x + 4) + (4x + 8) + (4x + 12) + (4x + 16) + (4x + 20))}{6} \]

### Step 4: Simplify the Sum
Now, let's simplify the sum in the numerator:
\[ \text{Sum} = 4x + 4x + 4 + 4x + 8 + 4x + 12 + 4x + 16 + 4x + 20 \]
This simplifies to:
\[ \text{Sum} = 24x + 60 \]

### Step 5: Set Up the Equation
Now we can set up the equation:
\[ 18 = \frac{24x + 60}{6} \]

### Step 6: Multiply Both Sides by 6
To eliminate the fraction, multiply both sides by 6:
\[ 18 \times 6 = 24x + 60 \]
This gives:
\[ 108 = 24x + 60 \]

### Step 7: Solve for \( x \)
Now, isolate \( 24x \):
\[ 24x = 108 - 60 \]
\[ 24x = 48 \]
Now divide by 24:
\[ x = 2 \]

### Step 8: Find the 3rd and 4th Multiples
Now that we have \( x = 2 \), we can find the 3rd and 4th multiples:
- Third multiple: \( 4x + 8 = 4(2) + 8 = 8 + 8 = 16 \)
- Fourth multiple: \( 4x + 12 = 4(2) + 12 = 8 + 12 = 20 \)

### Step 9: Calculate the Sum of the 3rd and 4th Multiples
Now, we can find the sum of the 3rd and 4th multiples:
\[ \text{Sum} = 16 + 20 = 36 \]

### Final Answer
The sum of the 3rd and 4th numbers is \( \boxed{36} \).

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