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

Answer

To solve the problem step-by-step, we need to find the values of \( a \) and \( b \) given that the mean of the numbers \( a, b, 8, 5, 10 \) is 6 and the variance is 6.80.

### Step 1: Calculate the Mean
The mean of the numbers is given by the formula:

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

We know that the mean is 6, so we can set up the equation:

\[ 6 = \frac{a + b + 8 + 5 + 10}{5} \]

### Step 2: Simplify the Mean Equation
Cross-multiplying gives us:

\[ 30 = a + b + 8 + 5 + 10 \]

Now, simplifying the right side:

\[ 30 = a + b + 23 \]

Subtracting 23 from both sides:

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

### Step 3: Calculate the Variance
The variance is given by the formula:

\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]

Where \( x_i \) are the numbers and \( \bar{x} \) is the mean. We know the variance is 6.80, so we can set up the equation:

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

### Step 4: Simplify the Variance Equation
Cross-multiplying gives us:

\[ 34 = (a - 6)^2 + (b - 6)^2 + (8 - 6)^2 + (5 - 6)^2 + (10 - 6)^2 \]

Calculating the squares:

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

Substituting these values into the equation:

\[ 34 = (a - 6)^2 + (b - 6)^2 + 4 + 1 + 16 \]

Adding the constants:

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

Subtracting 21 from both sides:

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

### Step 5: Substitute Equation 1 into Equation 2
From Equation 1, we have \( b = 7 - a \). Substitute this into Equation 2:

\[ 13 = (a - 6)^2 + ((7 - a) - 6)^2 \]
\[ 13 = (a - 6)^2 + (1 - a)^2 \]

### Step 6: Expand and Simplify
Expanding both squares:

\[ 13 = (a^2 - 12a + 36) + (1 - 2a + a^2) \]
\[ 13 = 2a^2 - 14a + 37 \]

Rearranging gives:

\[ 2a^2 - 14a + 37 - 13 = 0 \]
\[ 2a^2 - 14a + 24 = 0 \]

Dividing the entire equation by 2:

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

### Step 7: Factor the Quadratic
Factoring the quadratic:

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

Thus, we have:

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

### Step 8: Find Corresponding Values of \( b \)
Using \( a + b = 7 \):

1. If \( a = 4 \):
\[ b = 7 - 4 = 3 \]

2. If \( a = 3 \):
\[ b = 7 - 3 = 4 \]

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

### Possible Options
From the options given:
- Option A: \( a = 0, b = 7 \) (not valid)
- Option B: \( a = 5, b = 2 \) (not valid)
- Option C: \( a = 2, b = 5 \) (not valid)
- Option D: \( a = 3, b = 4 \) (valid)

Thus, the correct answer is **Option D**.