Rani had certain sum of money (Rs x), out of which she pays 800 to the grocer. She paid `1/3` rd of Rs x to her maid and gardener in the respective ratio of 2 : 1. If after all the mentioned expenses she is left with 2200, how much did she pay to her maid?

Options

A. Rs. 900
B. Rs. 500
C. Rs.1000
D. Rs. 1200

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the total amount of money Rani had
Let the total amount of money Rani had be Rs. \( x \).

### Step 2: Calculate the amount paid to the grocer
Rani paid Rs. 800 to the grocer. So, after this payment, the remaining amount is:
\[ \text{Remaining amount} = x - 800 \]

### Step 3: Determine the amount paid to the maid and gardener
Rani paid \( \frac{1}{3} \) of Rs. \( x \) to her maid and gardener in the ratio of 2:1. The total amount paid to both is:
\[ \text{Amount paid to maid and gardener} = \frac{x}{3} \]
Let the amount paid to the maid be \( 2k \) and the amount paid to the gardener be \( k \). Therefore, we have:
\[ 2k + k = \frac{x}{3} \implies 3k = \frac{x}{3} \implies k = \frac{x}{9} \]
Thus, the amount paid to the maid is:
\[ \text{Amount paid to maid} = 2k = 2 \times \frac{x}{9} = \frac{2x}{9} \]

### Step 4: Calculate the amount paid to the gardener
The amount paid to the gardener is:
\[ \text{Amount paid to gardener} = k = \frac{x}{9} \]

### Step 5: Write the equation for the remaining amount
After paying the grocer, the maid, and the gardener, Rani is left with Rs. 2200. Therefore, we can write the equation:
\[ \text{Remaining amount} = x - 800 - \frac{2x}{9} - \frac{x}{9} \]
This simplifies to:
\[ 2200 = x - 800 - \frac{3x}{9} \]
\[ 2200 = x - 800 - \frac{x}{3} \]

### Step 6: Simplify the equation
To eliminate the fraction, we can multiply the entire equation by 3:
\[ 6600 = 3x - 2400 - x \]
\[ 6600 = 2x - 2400 \]
Now, add 2400 to both sides:
\[ 6600 + 2400 = 2x \]
\[ 9000 = 2x \]
Now, divide by 2:
\[ x = 4500 \]

### Step 7: Calculate the amount paid to the maid
Now that we have \( x = 4500 \), we can find the amount paid to the maid:
\[ \text{Amount paid to maid} = \frac{2x}{9} = \frac{2 \times 4500}{9} = \frac{9000}{9} = 1000 \]

### Final Answer
Rani paid Rs. 1000 to her maid.

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