Rs. 68000 is divided among A,B and C in the ratio of ` 1/2 : (1)/(4) : (5)/(16)`. The difference of the greastest and the smallest part is :

Options

A. Rs. 6000
B. Rs. 14440
C. Rs. 9200
D. Rs. 16000

Answer

To solve the problem of dividing Rs. 68000 among A, B, and C in the ratio of \( \frac{1}{2} : \frac{1}{4} : \frac{5}{16} \), we will follow these steps:

### Step 1: Convert the Ratios to a Common Denominator
The given ratios are:
- A: \( \frac{1}{2} \)
- B: \( \frac{1}{4} \)
- C: \( \frac{5}{16} \)

To compare these ratios easily, we find a common denominator. The least common multiple (LCM) of the denominators (2, 4, and 16) is 16.

Now, we convert each ratio:
- A: \( \frac{1}{2} = \frac{8}{16} \)
- B: \( \frac{1}{4} = \frac{4}{16} \)
- C: \( \frac{5}{16} = \frac{5}{16} \)

So, the ratios become:
- A : B : C = 8 : 4 : 5

### Step 2: Express the Ratios in Terms of a Variable
Let the common multiple be \( x \). Then we can express the shares of A, B, and C as:
- A's share = \( 8x \)
- B's share = \( 4x \)
- C's share = \( 5x \)

### Step 3: Set Up the Equation Based on Total Amount
According to the problem, the total amount shared among A, B, and C is Rs. 68000. Therefore, we can set up the equation:
\[ 8x + 4x + 5x = 68000 \]
This simplifies to:
\[ 17x = 68000 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{68000}{17} = 4000 \]

### Step 5: Calculate Each Person's Share
Now we can find the individual shares:
- A's share = \( 8x = 8 \times 4000 = 32000 \)
- B's share = \( 4x = 4 \times 4000 = 16000 \)
- C's share = \( 5x = 5 \times 4000 = 20000 \)

### Step 6: Find the Difference Between the Greatest and Smallest Share
The greatest share is A's share (32000) and the smallest share is B's share (16000). Now, we find the difference:
\[ \text{Difference} = \text{Greatest share} - \text{Smallest share} = 32000 - 16000 = 16000 \]

### Final Answer
The difference between the greatest and the smallest part is Rs. 16000.

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