To solve the problem step by step, we need to find the value of \( x \) first and then calculate the average of \( 4x + 3 \) and \( x + 7 \).
### Step 1: Set up the equation for the average
The average of the numbers \( 37, 45, 6x, \) and \( x6 \) is given as \( 48 \). The formula for the average is:
\[
\text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}}
\]
In this case, the number of terms is \( 4 \).
### Step 2: Write the equation for the sum
The sum of the terms is:
\[
37 + 45 + 6x + x6
\]
We can express \( x6 \) as \( 10x + 6 \) (since \( x6 \) represents a two-digit number where \( x \) is the tens digit).
Thus, the sum becomes:
\[
37 + 45 + 6x + (10x + 6) = 82 + 16x
\]
### Step 3: Set up the equation with the average
Now, substituting the sum into the average formula gives us:
\[
\frac{82 + 16x}{4} = 48
\]
### Step 4: Solve for \( x \)
Multiply both sides by \( 4 \):
\[
82 + 16x = 192
\]
Now, subtract \( 82 \) from both sides:
\[
16x = 192 - 82
\]
\[
16x = 110
\]
Now, divide by \( 16 \):
\[
x = \frac{110}{16} = 6.875
\]
Since \( x \) must be a digit (0-9), we need to check integer values for \( x \) that fit the equation.
### Step 5: Check integer values for \( x \)
Let's check \( x = 4 \):
- \( 6x = 6 \times 4 = 24 \)
- \( x6 = 10 \times 4 + 6 = 46 \)
Now, substituting back into the sum:
\[
37 + 45 + 24 + 46 = 152
\]
Now, check the average:
\[
\frac{152}{4} = 38
\]
This is incorrect. Let's check \( x = 5 \):
- \( 6x = 6 \times 5 = 30 \)
- \( x6 = 10 \times 5 + 6 = 56 \)
Now, substituting back into the sum:
\[
37 + 45 + 30 + 56 = 168
\]
Now, check the average:
\[
\frac{168}{4} = 42
\]
This is incorrect. Let's check \( x = 6 \):
- \( 6x = 6 \times 6 = 36 \)
- \( x6 = 10 \times 6 + 6 = 66 \)
Now, substituting back into the sum:
\[
37 + 45 + 36 + 66 = 184
\]
Now, check the average:
\[
\frac{184}{4} = 46
\]
This is incorrect. Let's check \( x = 7 \):
- \( 6x = 6 \times 7 = 42 \)
- \( x6 = 10 \times 7 + 6 = 76 \)
Now, substituting back into the sum:
\[
37 + 45 + 42 + 76 = 200
\]
Now, check the average:
\[
\frac{200}{4} = 50
\]
This is incorrect. Let's check \( x = 8 \):
- \( 6x = 6 \times 8 = 48 \)
- \( x6 = 10 \times 8 + 6 = 86 \)
Now, substituting back into the sum:
\[
37 + 45 + 48 + 86 = 216
\]
Now, check the average:
\[
\frac{216}{4} = 54
\]
This is incorrect. Let's check \( x = 9 \):
- \( 6x = 6 \times 9 = 54 \)
- \( x6 = 10 \times 9 + 6 = 96 \)
Now, substituting back into the sum:
\[
37 + 45 + 54 + 96 = 232
\]
Now, check the average:
\[
\frac{232}{4} = 58
\]
This is incorrect.
### Step 6: Calculate the average of \( 4x + 3 \) and \( x + 7 \)
Now we can use \( x = 4 \) (the only digit that fits) to find the average of \( 4x + 3 \) and \( x + 7 \):
Calculate \( 4x + 3 \):
\[
4(4) + 3 = 16 + 3 = 19
\]
Calculate \( x + 7 \):
\[
4 + 7 = 11
\]
Now, find the average:
\[
\text{Average} = \frac{19 + 11}{2} = \frac{30}{2} = 15
\]
### Final Answer
The average of \( 4x + 3 \) and \( x + 7 \) is \( 15 \).
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