If the mean of 8` `consecutive multiples of `6` is `45`, find the `3^(rd)` and `6^(th)` numbers.

Options

A. `30,42`
B. `30,54`
C. `36,54`
D. `24,48`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Problem
We need to find the 3rd and 6th numbers from 8 consecutive multiples of 6, given that their mean is 45.

### Step 2: Define the Multiples
Let the first multiple of 6 be represented as \(6x\). The 8 consecutive multiples of 6 can then be expressed as:
1. \(6x\) (1st term)
2. \(6x + 6\) (2nd term)
3. \(6x + 12\) (3rd term)
4. \(6x + 18\) (4th term)
5. \(6x + 24\) (5th term)
6. \(6x + 30\) (6th term)
7. \(6x + 36\) (7th term)
8. \(6x + 42\) (8th term)

### Step 3: Calculate the Mean
The mean is calculated as the sum of all observations divided by the number of observations.

The sum of the 8 terms is:
\[ (6x) + (6x + 6) + (6x + 12) + (6x + 18) + (6x + 24) + (6x + 30) + (6x + 36) + (6x + 42) \]

This simplifies to:
\[ 8 \cdot 6x + (0 + 6 + 12 + 18 + 24 + 30 + 36 + 42) \]
Calculating the sum of the constants:
\[ 0 + 6 + 12 + 18 + 24 + 30 + 36 + 42 = 168 \]
Thus, the total sum is:
\[ 48x + 168 \]

### Step 4: Set Up the Mean Equation
Given that the mean is 45, we can set up the equation:
\[ \frac{48x + 168}{8} = 45 \]

### Step 5: Solve for \(x\)
Multiply both sides by 8:
\[ 48x + 168 = 360 \]
Now, subtract 168 from both sides:
\[ 48x = 360 - 168 \]
Calculating the right side:
\[ 48x = 192 \]
Now, divide by 48:
\[ x = \frac{192}{48} = 4 \]

### Step 6: Find the 3rd and 6th Terms
Now that we have \(x = 4\), we can find the 3rd and 6th terms:

- **3rd term**:
\[ 6x + 12 = 6 \cdot 4 + 12 = 24 + 12 = 36 \]

- **6th term**:
\[ 6x + 30 = 6 \cdot 4 + 30 = 24 + 30 = 54 \]

### Final Answer
The 3rd term is **36** and the 6th term is **54**.

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