The mean of `8` numbers is `11`. If `3` of these numbers are in the ratio `1:2:3` and the remaining `5` numbers are `9, 10, 13, 14,` and `18`, find the `3` numbers.

Options

A. `4,8,12`
B. `4,9,13`
C. `6,10,13`
D. `7,10,13`

Answer

To solve the problem, we need to find three numbers that are in the ratio of 1:2:3, given that the mean of eight numbers is 11. Here’s a step-by-step solution:

### Step 1: Calculate the Total Sum of the Numbers
The mean of the 8 numbers is given as 11. The formula for the mean is:

\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Observations}} \]

Thus, we can calculate the total sum of the 8 numbers:

\[ \text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 11 \times 8 = 88 \]

### Step 2: Set Up the Equation for the Three Numbers
Let the three numbers in the ratio 1:2:3 be represented as \(x\), \(2x\), and \(3x\). The remaining five numbers are given as 9, 10, 13, 14, and 18.

### Step 3: Write the Equation
The total sum of all eight numbers can be expressed as:

\[ x + 2x + 3x + 9 + 10 + 13 + 14 + 18 = 88 \]

### Step 4: Simplify the Equation
Combine like terms:

\[ 6x + (9 + 10 + 13 + 14 + 18) = 88 \]

Calculate the sum of the remaining numbers:

\[ 9 + 10 + 13 + 14 + 18 = 64 \]

So the equation becomes:

\[ 6x + 64 = 88 \]

### Step 5: Solve for \(x\)
Now, isolate \(x\):

\[ 6x = 88 - 64 \]
\[ 6x = 24 \]
\[ x = \frac{24}{6} = 4 \]

### Step 6: Find the Three Numbers
Now that we have \(x\), we can find the three numbers:

1. First number: \(x = 4\)
2. Second number: \(2x = 2 \times 4 = 8\)
3. Third number: \(3x = 3 \times 4 = 12\)

Thus, the three numbers are **4, 8, and 12**.

### Summary of the Solution
The three numbers in the ratio 1:2:3 are **4, 8, and 12**.

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