Of the three numbers, the sum of the first two is 45, the sum of the second and the third is 55 and the sum of the third and thrice the first is 90. The third number is

Options

A. 20
B. 25
C. 30
D. 3

Answer

To solve the problem, we need to set up equations based on the information given about the three numbers. Let's denote the three numbers as follows:

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

From the problem statement, we can derive the following equations:

1. The sum of the first two numbers is 45:
\[ x + y = 45 \quad \text{(Equation 1)} \]

2. The sum of the second and third numbers is 55:
\[ y + z = 55 \quad \text{(Equation 2)} \]

3. The sum of the third number and thrice the first number is 90:
\[ z + 3x = 90 \quad \text{(Equation 3)} \]

Now we have a system of three equations. We will solve these equations step by step.

### Step 1: Solve for \( y \) in terms of \( x \)
From Equation 1:
\[ y = 45 - x \quad \text{(Equation 4)} \]

### Step 2: Substitute \( y \) in Equation 2
Now, substitute Equation 4 into Equation 2:
\[ (45 - x) + z = 55 \]
Simplifying this gives:
\[ z = 55 - 45 + x \]
\[ z = 10 + x \quad \text{(Equation 5)} \]

### Step 3: Substitute \( z \) in Equation 3
Now, substitute Equation 5 into Equation 3:
\[ (10 + x) + 3x = 90 \]
Combining like terms gives:
\[ 10 + 4x = 90 \]
Subtracting 10 from both sides:
\[ 4x = 80 \]
Dividing by 4:
\[ x = 20 \quad \text{(Value of the first number)} \]

### Step 4: Find \( y \) using \( x \)
Now that we have \( x \), we can find \( y \) using Equation 4:
\[ y = 45 - x = 45 - 20 = 25 \quad \text{(Value of the second number)} \]

### Step 5: Find \( z \) using \( x \)
Now we can find \( z \) using Equation 5:
\[ z = 10 + x = 10 + 20 = 30 \quad \text{(Value of the third number)} \]

### Conclusion
The third number \( z \) is 30.

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