Twenty-four is divided into two parts such that `7` times the first part added to `5` times the second part makes `146`. Find each part.

Options

A. `10,14`
B. `13,11`
C. `12,12`
D. None

Answer

To solve the problem step by step, we will define the two parts and set up equations based on the information given in the question.

### Step 1: Define the Variables
Let the first part be \( x \) and the second part be \( y \).

### Step 2: Set Up the First Equation
According to the problem, the two parts add up to 24. Therefore, we can write the first equation as:
\[ x + y = 24 \quad \text{(1)} \]

### Step 3: Set Up the Second Equation
The problem states that 7 times the first part added to 5 times the second part equals 146. This gives us the second equation:
\[ 7x + 5y = 146 \quad \text{(2)} \]

### Step 4: Solve for One Variable
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 24 - x \quad \text{(3)} \]

### Step 5: Substitute into the Second Equation
Now, substitute equation (3) into equation (2):
\[ 7x + 5(24 - x) = 146 \]

### Step 6: Simplify the Equation
Distributing the 5 in the equation:
\[ 7x + 120 - 5x = 146 \]
Combine like terms:
\[ (7x - 5x) + 120 = 146 \]
This simplifies to:
\[ 2x + 120 = 146 \]

### Step 7: Isolate the Variable \( x \)
Subtract 120 from both sides:
\[ 2x = 146 - 120 \]
\[ 2x = 26 \]
Now, divide by 2:
\[ x = 13 \]

### Step 8: Find the Value of \( y \)
Now that we have \( x \), we can find \( y \) using equation (3):
\[ y = 24 - x = 24 - 13 = 11 \]

### Step 9: Conclusion
The two parts are:
- First part \( x = 13 \)
- Second part \( y = 11 \)

### Final Answer
The first part is 13 and the second part is 11.

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