A person distributed his total property among his three sons. The ratio of total property, share of first son, second son and share of third son is 24 : 7: 8. Find the central angle of sector, which represents the share of the third son.

Options

A. `72 (1^(@))/(2)`
B. `135 ^(@)`
C. `105^(@)`
D. `67 (1^(@))/(2)`

Answer

To find the central angle of the sector representing the share of the third son, we can follow these steps:

### Step 1: Understand the Ratio
The ratio of the total property and the shares of the three sons is given as 24 : 7 : 8. This means:
- First son’s share = 24 parts
- Second son’s share = 7 parts
- Third son’s share = 8 parts

### Step 2: Calculate the Total Parts
To find the total property in terms of parts, we add the parts of each son:
\[ \text{Total parts} = 24 + 7 + 8 = 39 \]

### Step 3: Determine the Share of the Third Son
The share of the third son is represented by 8 parts out of the total 39 parts.

### Step 4: Calculate the Central Angle for the Third Son
The total angle in a circle is 360 degrees. To find the central angle corresponding to the third son’s share, we can use the formula:
\[ \text{Central Angle} = \left(\frac{\text{Share of Third Son}}{\text{Total Parts}}\right) \times 360 \]
Substituting the values:
\[ \text{Central Angle} = \left(\frac{8}{39}\right) \times 360 \]

### Step 5: Simplify the Calculation
Calculating the above expression:
\[ \text{Central Angle} = \frac{8 \times 360}{39} \]
\[ = \frac{2880}{39} \]

### Step 6: Perform the Division
Now, we divide 2880 by 39:
\[ 2880 \div 39 \approx 73.84615384615384 \]
This can be approximated to:
\[ \text{Central Angle} \approx 73.85 \text{ degrees} \]

### Step 7: Convert to Fraction (if needed)
To express it as a mixed number:
\[ 73.84615384615384 \approx 73 \frac{11}{13} \text{ degrees} \]

### Final Answer
Thus, the central angle of the sector representing the share of the third son is approximately:
\[ \text{Central Angle} \approx 73.85 \text{ degrees or } 73 \frac{11}{13} \text{ degrees} \]

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