If 686 is divided, into four parts, in proportions `1/2 :2/3:3:4` then the first part is ?
A)52
B)42
C)56
D)48

Options

A. 52
B. 42
C. 56
D. 48

Answer

To solve the problem of dividing 686 into four parts in the ratio of \( \frac{1}{2} : \frac{2}{3} : 3 : 4 \), we can follow these steps:

### Step-by-Step Solution:

1. **Convert the Ratios to a Common Denominator:**
The given ratios are \( \frac{1}{2}, \frac{2}{3}, 3, \) and \( 4 \). To compare these ratios easily, we need to convert them to a common denominator. The least common multiple (LCM) of the denominators (2 and 3) is 6.

- Convert \( \frac{1}{2} \) to sixths:
\[ \frac{1}{2} = \frac{3}{6} \]
- Convert \( \frac{2}{3} \) to sixths:
\[ \frac{2}{3} = \frac{4}{6} \]
- Convert \( 3 \) to sixths:
\[ 3 = \frac{18}{6} \]
- Convert \( 4 \) to sixths:
\[ 4 = \frac{24}{6} \]

Now the ratios are:
\[ 3 : 4 : 18 : 24 \]

2. **Express the Ratios in Whole Numbers:**
The ratios can be expressed as:
\[ 3x : 4x : 18x : 24x \]
where \( x \) is a common multiplier.

3. **Calculate the Total Parts:**
Now, we sum the parts:
\[ 3x + 4x + 18x + 24x = 49x \]

4. **Set Up the Equation:**
We know the total amount is 686, so we set up the equation:
\[ 49x = 686 \]

5. **Solve for \( x \):**
To find \( x \), divide both sides by 49:
\[ x = \frac{686}{49} = 14 \]

6. **Calculate the First Part:**
The first part corresponds to \( 3x \):
\[ \text{First part} = 3x = 3 \times 14 = 42 \]

### Conclusion:
Thus, the first part is \( \boxed{42} \).