The average of four terms is 30 and the first term is 1/3 of the sum of the remaining terms. What is the first term?

Options

A. 40
B. 30
C. 20
D. 60

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the given information
We know that the average of four terms is 30. Let’s denote the four terms as \( x_1, x_2, x_3, \) and \( x_4 \).

### Step 2: Set up the equation for the average
The average of these four terms can be expressed as:
\[ \text{Average} = \frac{x_1 + x_2 + x_3 + x_4}{4} = 30 \]
Multiplying both sides by 4 gives:
\[ x_1 + x_2 + x_3 + x_4 = 120 \]

### Step 3: Use the information about the first term
We are also given that the first term \( x_1 \) is one-third of the sum of the remaining terms \( x_2, x_3, \) and \( x_4 \). This can be expressed as:
\[ x_1 = \frac{1}{3}(x_2 + x_3 + x_4) \]

### Step 4: Express the sum of the remaining terms
From the equation \( x_1 + x_2 + x_3 + x_4 = 120 \), we can express the sum of the remaining terms:
\[ x_2 + x_3 + x_4 = 120 - x_1 \]

### Step 5: Substitute this into the equation for \( x_1 \)
Now we can substitute \( x_2 + x_3 + x_4 \) into the equation for \( x_1 \):
\[ x_1 = \frac{1}{3}(120 - x_1) \]

### Step 6: Solve for \( x_1 \)
Multiplying both sides by 3 to eliminate the fraction gives:
\[ 3x_1 = 120 - x_1 \]
Now, add \( x_1 \) to both sides:
\[ 3x_1 + x_1 = 120 \]
\[ 4x_1 = 120 \]
Now, divide both sides by 4:
\[ x_1 = 30 \]

### Conclusion
The first term \( x_1 \) is 30.