P,Q and R hire a meadow for X2920. P puts 10 cows for 20 days, Q puts 30 cows for 8 days and R puts 16 cows for 9 days. Find the rent paid by R.

Options

A. X 735
B. X 820
C. X 575
D. X 720

Answer

To find the rent paid by R, we will follow these steps:

### Step 1: Calculate the total work done by each partner
We need to calculate the work done by each partner in terms of cow-days, which is the product of the number of cows and the number of days they were kept.

- **For P**:
\[ \text{Work by P} = 10 \text{ cows} \times 20 \text{ days} = 200 \text{ cow-days} \]

- **For Q**:
\[ \text{Work by Q} = 30 \text{ cows} \times 8 \text{ days} = 240 \text{ cow-days} \]

- **For R**:
\[ \text{Work by R} = 16 \text{ cows} \times 9 \text{ days} = 144 \text{ cow-days} \]

### Step 2: Find the ratio of work done by P, Q, and R
Now, we will express the work done by each partner in a ratio:
\[ \text{Ratio} = P : Q : R = 200 : 240 : 144 \]

### Step 3: Simplify the ratio
To simplify the ratio, we can divide each term by the greatest common divisor (GCD). The GCD of 200, 240, and 144 is 16.

\[ \text{Simplified Ratio} = \frac{200}{16} : \frac{240}{16} : \frac{144}{16} = 12.5 : 15 : 9 \]
To avoid fractions, we can multiply through by 2:
\[ \text{Ratio} = 25 : 30 : 18 \]

### Step 4: Calculate the total parts
Now, we will add the parts of the ratio:
\[ \text{Total parts} = 25 + 30 + 18 = 73 \]

### Step 5: Calculate R's share of the rent
The total rent is X2920. We need to find R's share based on the ratio:
\[ \text{R's share} = \frac{\text{R's part}}{\text{Total parts}} \times \text{Total Rent} \]
\[ \text{R's share} = \frac{18}{73} \times 2920 \]

### Step 6: Perform the calculation
Calculating R's share:
\[ \text{R's share} = \frac{18 \times 2920}{73} \]
Calculating \( \frac{2920}{73} \):
\[ \frac{2920}{73} = 40 \]
Now, multiplying by 18:
\[ \text{R's share} = 40 \times 18 = 720 \]

### Final Answer
Thus, the rent paid by R is **X720**.

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