The average of the 2-digit numbers `49, 6x, x4` and `70` is `65`. What is the average of `(x+8)` and `(x+12)` ?

Options

A. 16
B. 17
C. 19
D. 18

Answer

To solve the problem step by step, we will first find the value of \( x \) using the information given about the average of the numbers \( 49, 6x, x4, \) and \( 70 \). Then, we will calculate the average of \( (x + 8) \) and \( (x + 12) \).

### Step 1: Set up the equation for the average
The average of the numbers \( 49, 6x, x4, \) and \( 70 \) is given as \( 65 \). The formula for the average is:

\[ \text{Average} = \frac{\text{Sum of all terms}}{\text{Number of terms}} \]

In this case, the number of terms is \( 4 \), so we have:

\[ 65 = \frac{49 + 6x + x4 + 70}{4} \]

### Step 2: Express \( 6x \) and \( x4 \) in terms of \( x \)
The number \( 6x \) can be expressed as:

\[ 6x = 60 + x \]

The number \( x4 \) can be expressed as:

\[ x4 = 10x + 4 \]

### Step 3: Substitute the expressions into the average equation
Now, substituting \( 6x \) and \( x4 \) into the average equation gives:

\[ 65 = \frac{49 + (60 + x) + (10x + 4) + 70}{4} \]

### Step 4: Simplify the equation
Now, simplify the equation:

\[ 65 = \frac{49 + 60 + x + 10x + 4 + 70}{4} \]

Combine the constant terms:

\[ 65 = \frac{183 + 11x}{4} \]

### Step 5: Multiply both sides by 4
To eliminate the fraction, multiply both sides by \( 4 \):

\[ 260 = 183 + 11x \]

### Step 6: Solve for \( x \)
Now, isolate \( x \):

\[ 11x = 260 - 183 \]
\[ 11x = 77 \]
\[ x = \frac{77}{11} = 7 \]

### Step 7: Find the average of \( (x + 8) \) and \( (x + 12) \)
Now that we have \( x = 7 \), we can find the average of \( (x + 8) \) and \( (x + 12) \):

\[ \text{Average} = \frac{(x + 8) + (x + 12)}{2} \]

Substituting \( x = 7 \):

\[ \text{Average} = \frac{(7 + 8) + (7 + 12)}{2} \]
\[ = \frac{15 + 19}{2} \]
\[ = \frac{34}{2} = 17 \]

### Final Answer
The average of \( (x + 8) \) and \( (x + 12) \) is \( \boxed{17} \).