The average of 4-digit numbers 1x44, x345, 3356 and 41x3 is 2767. What is the average of 6x, 5x + 3 and 7x +3?

Options

A. 15
B. 14
C. 17
D. 12

Answer

To solve the problem step by step, we will first find the value of \( x \) from the given average of the four 4-digit numbers and then use that value to calculate the average of the three expressions involving \( x \).

### Step 1: Calculate the total sum of the four numbers
Given that the average of the four numbers \( 1x44, x345, 3356, \) and \( 41x3 \) is \( 2767 \), we can find the total sum of these numbers.

\[ \text{Average} = \frac{\text{Total Sum}}{\text{Number of Terms}} \]

\[ 2767 = \frac{\text{Total Sum}}{4} \]

Multiplying both sides by \( 4 \):

\[ \text{Total Sum} = 2767 \times 4 = 11068 \]

### Step 2: Set up the equation for the sum of the numbers
Now, we can express the total sum of the numbers in terms of \( x \):

\[ 1x44 + x345 + 3356 + 41x3 = 11068 \]

### Step 3: Substitute known values
We know that \( 3356 \) is a constant, so we can subtract it from \( 11068 \):

\[ 1x44 + x345 + 41x3 = 11068 - 3356 \]

Calculating the right side:

\[ 11068 - 3356 = 7712 \]

Thus, we have:

\[ 1x44 + x345 + 41x3 = 7712 \]

### Step 4: Express the numbers in terms of \( x \)
Now we can express each number:

- \( 1x44 = 1000 + 100x + 40 + 4 = 1000 + 100x + 44 \)
- \( x345 = 1000x + 300 + 40 + 5 = 1000x + 345 \)
- \( 41x3 = 4100 + 10x + 3 = 4100 + 10x + 3 \)

### Step 5: Combine the equations
Now we can combine these expressions:

\[ (1000 + 100x + 44) + (1000x + 345) + (4100 + 10x + 3) = 7712 \]

Combining like terms:

\[ (1000 + 1000x + 4100) + (100x + 10x) + (44 + 345 + 3) = 7712 \]

This simplifies to:

\[ (5100 + 1000x) + (110x) + 392 = 7712 \]

### Step 6: Solve for \( x \)
Now, combine the constants:

\[ 5100 + 392 + 1000x + 110x = 7712 \]

This simplifies to:

\[ 5492 + 1110x = 7712 \]

Subtract \( 5492 \) from both sides:

\[ 1110x = 7712 - 5492 \]

Calculating the right side:

\[ 7712 - 5492 = 2220 \]

Thus, we have:

\[ 1110x = 2220 \]

Dividing both sides by \( 1110 \):

\[ x = \frac{2220}{1110} = 2 \]

### Step 7: Calculate the average of the new expressions
Now that we have \( x = 2 \), we can substitute this value into the expressions \( 6x, 5x + 3, \) and \( 7x + 3 \):

1. \( 6x = 6 \times 2 = 12 \)
2. \( 5x + 3 = 5 \times 2 + 3 = 10 + 3 = 13 \)
3. \( 7x + 3 = 7 \times 2 + 3 = 14 + 3 = 17 \)

### Step 8: Calculate the average of these three values
Now we find the average:

\[ \text{Average} = \frac{12 + 13 + 17}{3} = \frac{42}{3} = 14 \]

### Final Answer
The average of \( 6x, 5x + 3, \) and \( 7x + 3 \) is \( 14 \).