`Rs1220` is divided, among A, B, C and D, such that B’s share is `(5)/(9) th` of A' s, C' s share is `(7)/(10)th` of B' s and D has `(1)/(3)` as much as B and C together. Find A's share.

Options

A. `Rs540`
B. `Rs802`
C. `Rs100`
D. `Rs650`

Answer

To solve the problem of dividing Rs. 1220 among A, B, C, and D based on the given ratios, we can follow these steps:

### Step 1: Define Variables
Let A's share be \( 18k \), B's share be \( 10k \), C's share be \( 7k \), and D's share be \( \frac{17k}{3} \).

### Step 2: Set Up the Equations
From the problem, we know:
- B's share is \( \frac{5}{9} \) of A's share:
\[ B = \frac{5}{9} A \implies 10k = \frac{5}{9} (18k) \]
- C's share is \( \frac{7}{10} \) of B's share:
\[ C = \frac{7}{10} B \implies C = \frac{7}{10} (10k) = 7k \]
- D has \( \frac{1}{3} \) as much as B and C together:
\[ D = \frac{1}{3} (B + C) = \frac{1}{3} (10k + 7k) = \frac{17k}{3} \]

### Step 3: Total Shares
Now we can write the total shares of A, B, C, and D:
\[ A + B + C + D = 1220 \]
Substituting the values we defined:
\[ 18k + 10k + 7k + \frac{17k}{3} = 1220 \]

### Step 4: Combine the Terms
To combine these terms, we need a common denominator. The common denominator for the fractions is 3:
\[ \frac{54k}{3} + \frac{30k}{3} + \frac{21k}{3} + \frac{17k}{3} = 1220 \]
This simplifies to:
\[ \frac{54k + 30k + 21k + 17k}{3} = 1220 \]
\[ \frac{122k}{3} = 1220 \]

### Step 5: Solve for k
Multiply both sides by 3:
\[ 122k = 3660 \]
Now, divide by 122:
\[ k = \frac{3660}{122} = 30 \]

### Step 6: Find A's Share
Now that we have \( k \), we can find A's share:
\[ A = 18k = 18 \times 30 = 540 \]

### Final Answer
A's share is Rs. 540.

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