The average of `x_(1),x_(2),x_(3),x_(4)` is 16 . Half the sum of `x_(2),x_(3),x_(4)` is 23 .What is the value of `x_(1)` ?

Options

A. 17
B. 18
C. 19
D. 20

Answer

To find the value of \( x_1 \), we can follow these steps:

### Step 1: Use the average formula
Given that the average of \( x_1, x_2, x_3, x_4 \) is 16, we can use the formula for average:

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

So we have:

\[ 16 = \frac{x_1 + x_2 + x_3 + x_4}{4} \]

### Step 2: Multiply both sides by 4
To eliminate the fraction, multiply both sides by 4:

\[ x_1 + x_2 + x_3 + x_4 = 16 \times 4 \]

Calculating the right side:

\[ x_1 + x_2 + x_3 + x_4 = 64 \]

This gives us our first equation:

\[ \text{(Equation 1)} \quad x_1 + x_2 + x_3 + x_4 = 64 \]

### Step 3: Analyze the second condition
The problem states that half the sum of \( x_2, x_3, x_4 \) is 23. We can express this mathematically:

\[ \frac{x_2 + x_3 + x_4}{2} = 23 \]

### Step 4: Multiply both sides by 2
To find the sum of \( x_2, x_3, x_4 \), multiply both sides by 2:

\[ x_2 + x_3 + x_4 = 23 \times 2 \]

Calculating the right side:

\[ x_2 + x_3 + x_4 = 46 \]

This gives us our second equation:

\[ \text{(Equation 2)} \quad x_2 + x_3 + x_4 = 46 \]

### Step 5: Substitute Equation 2 into Equation 1
Now we can substitute the value of \( x_2 + x_3 + x_4 \) from Equation 2 into Equation 1:

\[ x_1 + (x_2 + x_3 + x_4) = 64 \]

Substituting \( x_2 + x_3 + x_4 = 46 \):

\[ x_1 + 46 = 64 \]

### Step 6: Solve for \( x_1 \)
Now, isolate \( x_1 \) by subtracting 46 from both sides:

\[ x_1 = 64 - 46 \]

Calculating the right side:

\[ x_1 = 18 \]

### Conclusion
Thus, the value of \( x_1 \) is:

\[ \boxed{18} \]

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