A Sum of Rs. 21000 is divided among A,B,C such that shares of A and B are in the ratio of 2:3 and those of B and C are in the ratio 4:5. The amount received by A is:

Options

A. Rs. 6000
B. Rs. 4500
C. Rs. 4800
D. Rs. 8400

Answer

To solve the problem of dividing Rs. 21,000 among A, B, and C based on the given ratios, we can follow these steps:

### Step 1: Understand the Ratios
We are given two ratios:
1. The ratio of A to B is 2:3.
2. The ratio of B to C is 4:5.

### Step 2: Express the Shares in Terms of a Common Variable
Let:
- A's share = 2k
- B's share = 3k (from the first ratio)

From the second ratio, we can express B and C as:
- B's share = 4α
- C's share = 5α

### Step 3: Equate the Shares of B
Since both expressions represent B's share, we can set them equal to each other:
\[ 3k = 4α \]

### Step 4: Solve for k in Terms of α
From the equation \( 3k = 4α \), we can express k in terms of α:
\[ k = \frac{4α}{3} \]

### Step 5: Express A, B, and C in Terms of α
Now we can substitute k back into the expressions for A and B:
- A's share = \( 2k = 2 \left(\frac{4α}{3}\right) = \frac{8α}{3} \)
- B's share = \( 4α \)
- C's share = \( 5α \)

### Step 6: Write the Total Share Equation
The total amount shared among A, B, and C is Rs. 21,000:
\[ A + B + C = 21000 \]
Substituting the expressions we found:
\[ \frac{8α}{3} + 4α + 5α = 21000 \]

### Step 7: Combine the Terms
To combine the terms, convert \( 4α \) and \( 5α \) into fractions with a common denominator:
\[ 4α = \frac{12α}{3} \]
\[ 5α = \frac{15α}{3} \]
Now, we can rewrite the equation:
\[ \frac{8α}{3} + \frac{12α}{3} + \frac{15α}{3} = 21000 \]
Combine the fractions:
\[ \frac{35α}{3} = 21000 \]

### Step 8: Solve for α
To eliminate the fraction, multiply both sides by 3:
\[ 35α = 63000 \]
Now, divide by 35:
\[ α = \frac{63000}{35} = 1800 \]

### Step 9: Find A's Share
Now that we have α, we can find A's share:
\[ A = \frac{8α}{3} = \frac{8 \times 1800}{3} = \frac{14400}{3} = 4800 \]

### Conclusion
The amount received by A is Rs. 4800.

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