Kamal and her sister together have Rs.2,100. On taking Rs.300 from her sister, Kamal will have the same amount as what her sister had earlier. Find the ratio of the initial amounts that Kamal and her sister had.

Options

A. `3:4`
B. `1:1`
C. `4:3`
D. `5:9`

Answer

To solve the problem step by step, let's define the amounts that Kamal and her sister initially had.

1. **Define Variables**:
Let the amount Kamal had initially be \( x \) and the amount her sister had be \( y \).

2. **Set Up the Equation**:
According to the problem, together they have Rs. 2100. Therefore, we can write the equation:
\[ x + y = 2100 \]

3. **Understand the Transaction**:
Kamal takes Rs. 300 from her sister. After this transaction:
- Kamal's new amount becomes \( x + 300 \)
- Her sister's new amount becomes \( y - 300 \)

According to the problem, after taking Rs. 300, Kamal will have the same amount as her sister had earlier. This gives us the second equation:
\[ x + 300 = y \]

4. **Substitute and Solve**:
Now we can substitute \( y \) from the second equation into the first equation. From the second equation, we have:
\[ y = x + 300 \]
Substituting this into the first equation:
\[ x + (x + 300) = 2100 \]
Simplifying this:
\[ 2x + 300 = 2100 \]
\[ 2x = 2100 - 300 \]
\[ 2x = 1800 \]
\[ x = 900 \]

5. **Find the Value of y**:
Now, substitute \( x \) back into the equation for \( y \):
\[ y = x + 300 = 900 + 300 = 1200 \]

6. **Calculate the Ratio**:
Now we have \( x = 900 \) and \( y = 1200 \). The ratio of the initial amounts that Kamal and her sister had is:
\[ \text{Ratio} = \frac{x}{y} = \frac{900}{1200} = \frac{3}{4} \]

Thus, the ratio of the initial amounts that Kamal and her sister had is \( 3:4 \).