Divide Rs 1,260 among A,B and C so that the ratio between the shares of A and B is `2:3` and the ratio between the shares of B and C is `4:5.`

Answer

To solve the problem of dividing Rs 1,260 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:
- The ratio of A to B is 2:3.
- The ratio of B to C is 4:5.

### Step 2: Express the Ratios in Terms of a Common Variable
We can express the shares of A, B, and C in terms of a common variable:
- Let the share of A be \( 2x \).
- Let the share of B be \( 3x \) (from the ratio A:B = 2:3).

Now, we need to express C in terms of B. Since B is also part of the second ratio:
- From the ratio B:C = 4:5, we can express B as \( 4y \) and C as \( 5y \).

### Step 3: Equate the Shares of B
Since B is represented in both ratios, we can equate the two expressions for B:
\[ 3x = 4y \]

### Step 4: Solve for One Variable
From the equation \( 3x = 4y \), we can express y in terms of x:
\[ y = \frac{3x}{4} \]

### Step 5: Express C in Terms of x
Now substitute \( y \) into the expression for C:
\[ C = 5y = 5 \left(\frac{3x}{4}\right) = \frac{15x}{4} \]

### Step 6: Write the Total Share Equation
Now we can express the total amount distributed among A, B, and C:
\[ A + B + C = 1260 \]
Substituting the values:
\[ 2x + 3x + \frac{15x}{4} = 1260 \]

### Step 7: Find a Common Denominator and Simplify
To simplify the equation, we can convert all terms to have a common denominator (which is 4):
\[ \frac{8x}{4} + \frac{12x}{4} + \frac{15x}{4} = 1260 \]
Combining the fractions gives:
\[ \frac{35x}{4} = 1260 \]

### Step 8: Solve for x
To solve for x, multiply both sides by 4:
\[ 35x = 5040 \]
Now divide by 35:
\[ x = \frac{5040}{35} = 144 \]

### Step 9: Calculate the Shares
Now we can calculate the individual shares:
- Share of A: \( 2x = 2 \times 144 = 288 \)
- Share of B: \( 3x = 3 \times 144 = 432 \)
- Share of C: \( \frac{15x}{4} = \frac{15 \times 144}{4} = 540 \)

### Final Shares
Thus, the shares are:
- A = Rs 288
- B = Rs 432
- C = Rs 540