The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b ?

Options

A. a=0, b=7
B. a=5, b=2
C. a=1, b =6
D. a=3, b=4

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) given the mean and variance of the numbers \( a, b, 8, 5, 10 \).

### Step 1: Calculate the Mean
The mean of the numbers is given as 6. The formula for the mean is:

\[ \text{Mean} = \frac{a + b + 8 + 5 + 10}{5} \]

Substituting the known values:

\[ 6 = \frac{a + b + 23}{5} \]

Multiplying both sides by 5:

\[ 30 = a + b + 23 \]

Rearranging gives us:

\[ a + b = 30 - 23 = 7 \quad \text{(Equation 1)} \]

### Step 2: Calculate the Variance
The variance is given as 6.80. The formula for variance is:

\[ \text{Variance} = \frac{(a - \text{mean})^2 + (b - \text{mean})^2 + (8 - \text{mean})^2 + (5 - \text{mean})^2 + (10 - \text{mean})^2}{5} \]

Substituting the mean:

\[ 6.80 = \frac{(a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2}{5} \]

Calculating the known squared differences:

\[ 8 - 6 = 2 \quad \Rightarrow \quad (2)^2 = 4 \]
\[ 5 - 6 = -1 \quad \Rightarrow \quad (-1)^2 = 1 \]
\[ 10 - 6 = 4 \quad \Rightarrow \quad (4)^2 = 16 \]

Now substituting these values back into the variance equation:

\[ 6.80 = \frac{(a - 6)^2 + (b - 6)^2 + 4 + 1 + 16}{5} \]

This simplifies to:

\[ 6.80 = \frac{(a - 6)^2 + (b - 6)^2 + 21}{5} \]

Multiplying both sides by 5:

\[ 34 = (a - 6)^2 + (b - 6)^2 + 21 \]

Rearranging gives us:

\[ (a - 6)^2 + (b - 6)^2 = 34 - 21 = 13 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( b \) from Equation 1 into Equation 2
From Equation 1, we have \( b = 7 - a \). Substitute this into Equation 2:

\[ (a - 6)^2 + ((7 - a) - 6)^2 = 13 \]

This simplifies to:

\[ (a - 6)^2 + (1 - a)^2 = 13 \]

Expanding both squares:

\[ (a - 6)^2 = a^2 - 12a + 36 \]
\[ (1 - a)^2 = 1 - 2a + a^2 \]

Combining these gives:

\[ a^2 - 12a + 36 + 1 - 2a + a^2 = 13 \]

This simplifies to:

\[ 2a^2 - 14a + 37 = 13 \]

Rearranging gives:

\[ 2a^2 - 14a + 24 = 0 \]

### Step 4: Simplify the Quadratic Equation
Dividing the entire equation by 2:

\[ a^2 - 7a + 12 = 0 \]

### Step 5: Factor the Quadratic Equation
Factoring gives:

\[ (a - 3)(a - 4) = 0 \]

Thus, the solutions for \( a \) are:

\[ a = 3 \quad \text{or} \quad a = 4 \]

### Step 6: Find Corresponding Values of \( b \)
Using \( b = 7 - a \):

- If \( a = 3 \), then \( b = 7 - 3 = 4 \).
- If \( a = 4 \), then \( b = 7 - 4 = 3 \).

### Conclusion
The possible values of \( (a, b) \) are \( (3, 4) \) and \( (4, 3) \).