54 is to be divided into two parts such that the sum of 10 times the first and 22 times the second is 780. The bigger part is:

Options

A. 34
B. 36
C. 30
D. 32

Answer

To solve the problem, we need to divide 54 into two parts such that the sum of 10 times the first part and 22 times the second part equals 780. Let's denote the first part as \( x \) and the second part as \( 54 - x \).

### Step-by-Step Solution:

1. **Define the Parts**:
Let the first part be \( x \) and the second part be \( 54 - x \).

2. **Set Up the Equation**:
According to the problem, we have:
\[ 10 \times \text{(first part)} + 22 \times \text{(second part)} = 780 \]
Substituting the values, we get:
\[ 10x + 22(54 - x) = 780 \]

3. **Expand the Equation**:
Now, expand the equation:
\[ 10x + 22 \times 54 - 22x = 780 \]
Calculate \( 22 \times 54 \):
\[ 22 \times 54 = 1188 \]
So the equation becomes:
\[ 10x + 1188 - 22x = 780 \]

4. **Combine Like Terms**:
Combine the \( x \) terms:
\[ (10x - 22x) + 1188 = 780 \]
This simplifies to:
\[ -12x + 1188 = 780 \]

5. **Isolate the Variable**:
Now, isolate \( x \) by moving 1188 to the other side:
\[ -12x = 780 - 1188 \]
Calculate \( 780 - 1188 \):
\[ 780 - 1188 = -408 \]
So we have:
\[ -12x = -408 \]

6. **Solve for \( x \)**:
Divide both sides by -12:
\[ x = \frac{-408}{-12} = 34 \]

7. **Find the Second Part**:
Now, substitute \( x \) back to find the second part:
\[ \text{Second part} = 54 - x = 54 - 34 = 20 \]

8. **Identify the Bigger Part**:
The two parts are \( x = 34 \) and \( 54 - x = 20 \). Therefore, the bigger part is:
\[ \text{Bigger part} = 34 \]

### Final Answer:
The bigger part is **34**.