Of the three numbers, the average of the first number and the second number is 37 more than the average of second number and third number. What is the difference between the first and the third number ?

Options

A. 74
B. 73
C. 37
D. Cannot be determined

Answer

To solve the problem step by step, we need to set up the equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- \( A \) = first number
- \( B \) = second number
- \( C \) = third number

### Step 2: Write the Average of the First and Second Numbers
The average of the first number and the second number is given by:
\[ \text{Average of } A \text{ and } B = \frac{A + B}{2} \]

### Step 3: Write the Average of the Second and Third Numbers
The average of the second number and the third number is given by:
\[ \text{Average of } B \text{ and } C = \frac{B + C}{2} \]

### Step 4: Set Up the Equation
According to the problem, the average of the first and second numbers is 37 more than the average of the second and third numbers. This can be expressed as:
\[ \frac{A + B}{2} = \frac{B + C}{2} + 37 \]

### Step 5: Eliminate the Fractions
To eliminate the fractions, multiply the entire equation by 2:
\[ A + B = B + C + 74 \]

### Step 6: Simplify the Equation
Now, simplify the equation:
\[ A + B - B - C = 74 \]
This simplifies to:
\[ A - C = 74 \]

### Step 7: Find the Difference Between the First and Third Numbers
From the equation \( A - C = 74 \), we can conclude that the difference between the first number and the third number is:
\[ \text{Difference} = A - C = 74 \]

### Final Answer
The difference between the first and the third number is \( 74 \).

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