Divide `"Rs". 1,290` into `A`, `B` and `C` such that `A` is `(2)/(5)` of `B` and `B : C = 4 : 3`.

Answer

To divide Rs. 1,290 into A, B, and C such that A is \( \frac{2}{5} \) of B and the ratio of B to C is \( 4:3 \), we can follow these steps:

### Step 1: Define Variables
Let:
- A = X
- B = Y
- C = Z

### Step 2: Write the Total Equation
According to the problem, the total amount is:
\[ X + Y + Z = 1290 \]
This is our first equation.

### Step 3: Express A in terms of B
We know that A is \( \frac{2}{5} \) of B:
\[ X = \frac{2}{5} Y \]
This is our second equation.

### Step 4: Express C in terms of B
The ratio of B to C is given as \( 4:3 \), which means:
\[ \frac{Y}{Z} = \frac{4}{3} \]
From this, we can express Z in terms of Y:
\[ Z = \frac{3}{4} Y \]
This is our third equation.

### Step 5: Substitute the Expressions into the Total Equation
Now, substitute the expressions for X and Z from equations 2 and 3 into the first equation:
\[ \frac{2}{5} Y + Y + \frac{3}{4} Y = 1290 \]

### Step 6: Find a Common Denominator
The common denominator for 5 and 4 is 20. We can rewrite each term:
- \( \frac{2}{5} Y = \frac{8}{20} Y \)
- \( Y = \frac{20}{20} Y \)
- \( \frac{3}{4} Y = \frac{15}{20} Y \)

Now, combine these:
\[ \frac{8}{20} Y + \frac{20}{20} Y + \frac{15}{20} Y = 1290 \]
This simplifies to:
\[ \frac{43}{20} Y = 1290 \]

### Step 7: Solve for Y
To find Y, multiply both sides by 20:
\[ 43Y = 1290 \times 20 \]
\[ 43Y = 25800 \]
Now, divide by 43:
\[ Y = \frac{25800}{43} = 600 \]

### Step 8: Calculate A and C
Now that we have Y, we can find A and C:
- For A:
\[ X = \frac{2}{5} Y = \frac{2}{5} \times 600 = 240 \]

- For C:
\[ Z = \frac{3}{4} Y = \frac{3}{4} \times 600 = 450 \]

### Final Result
Thus, the amounts are:
- A = Rs. 240
- B = Rs. 600
- C = Rs. 450