Of the three numbers, the sum of the first two is 55, sum of the second and third is 65 and sum of third with thrice of the first is 110. The third number is

Options

A. 25
B. 30
C. 35
D. 28

Answer

To solve the problem step by step, we will define the three numbers as follows:

Let:
- The first number be \( x \)
- The second number be \( y \)
- The third number be \( z \)

We are given the following equations based on the problem statement:

1. The sum of the first two numbers is 55:
\[ x + y = 55 \quad \text{(Equation 1)} \]

2. The sum of the second and third numbers is 65:
\[ y + z = 65 \quad \text{(Equation 2)} \]

3. The sum of the third number and thrice the first number is 110:
\[ z + 3x = 110 \quad \text{(Equation 3)} \]

### Step 1: Solve for \( y \) in terms of \( x \)

From Equation 1:
\[ y = 55 - x \]

**Hint:** Substitute \( y \) in the other equations to reduce the number of variables.

### Step 2: Substitute \( y \) into Equation 2

Substituting \( y \) in Equation 2:
\[ (55 - x) + z = 65 \]
Simplifying this gives:
\[ z = 65 - 55 + x \]
\[ z = 10 + x \quad \text{(Equation 4)} \]

**Hint:** Now we have \( z \) in terms of \( x \).

### Step 3: Substitute \( z \) into Equation 3

Now substitute \( z \) from Equation 4 into Equation 3:
\[ (10 + x) + 3x = 110 \]
This simplifies to:
\[ 10 + 4x = 110 \]
Subtracting 10 from both sides:
\[ 4x = 100 \]
Dividing by 4:
\[ x = 25 \quad \text{(Value of the first number)} \]

**Hint:** Now that you have \( x \), use it to find \( y \) and \( z \).

### Step 4: Find \( y \) using \( x \)

Using the value of \( x \) in Equation 1:
\[ y = 55 - 25 = 30 \quad \text{(Value of the second number)} \]

**Hint:** Now substitute \( x \) into Equation 4 to find \( z \).

### Step 5: Find \( z \) using \( x \)

Using the value of \( x \) in Equation 4:
\[ z = 10 + 25 = 35 \quad \text{(Value of the third number)} \]

### Final Answer

The third number \( z \) is:
\[ \boxed{35} \]