Out of three numbers the sum of the first and the second number is 73 and the sum of the second and the third number is 77. The sum of third and thrice the first number is 104. What is the third number?

Options

A. 25
B. 29
C. 48
D. None of these

Answer

To solve the problem step by step, we will first define the three numbers and then set up the equations based on the information provided in the question.

### Step 1: Define the Variables
Let:
- \( x \) = first number
- \( y \) = second number
- \( z \) = third number

### Step 2: Set Up the Equations
From the problem statement, we have the following equations:
1. The sum of the first and second numbers is 73:
\[ x + y = 73 \quad \text{(Equation 1)} \]
2. The sum of the second and third numbers is 77:
\[ y + z = 77 \quad \text{(Equation 2)} \]
3. The sum of the third and thrice the first number is 104:
\[ z + 3x = 104 \quad \text{(Equation 3)} \]

### Step 3: Solve for One Variable
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 73 - x \quad \text{(Equation 4)} \]

### Step 4: Substitute \( y \) in Equation 2
Now substitute Equation 4 into Equation 2:
\[ (73 - x) + z = 77 \]
Simplifying this gives:
\[ z = 77 - 73 + x \]
\[ z = 4 + x \quad \text{(Equation 5)} \]

### Step 5: Substitute \( z \) in Equation 3
Now substitute Equation 5 into Equation 3:
\[ (4 + x) + 3x = 104 \]
Combining like terms:
\[ 4 + 4x = 104 \]
Subtracting 4 from both sides:
\[ 4x = 100 \]
Dividing by 4:
\[ x = 25 \quad \text{(Value of the first number)} \]

### Step 6: Find \( y \) and \( z \)
Now we can find \( y \) using Equation 4:
\[ y = 73 - x = 73 - 25 = 48 \quad \text{(Value of the second number)} \]

Next, we find \( z \) using Equation 5:
\[ z = 4 + x = 4 + 25 = 29 \quad \text{(Value of the third number)} \]

### Final Answer
The third number \( z \) is:
\[ \boxed{29} \]