A man sells `3 (1)/(2)` bags of potatoes for Rs. `65 (2)/(5)`. How much will he earn if he sells `8 (3)/(4)` bags of potatoes?

Answer

To solve the problem step by step, we need to find out how much the man will earn if he sells \(8 \frac{3}{4}\) bags of potatoes, given that he sells \(3 \frac{1}{2}\) bags for \(Rs. 65 \frac{2}{5}\).

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions for easier calculations.

- For \(3 \frac{1}{2}\):
\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]

- For \(65 \frac{2}{5}\):
\[ 65 \frac{2}{5} = \frac{65 \times 5 + 2}{5} = \frac{325 + 2}{5} = \frac{327}{5} \]

- For \(8 \frac{3}{4}\):
\[ 8 \frac{3}{4} = \frac{8 \times 4 + 3}{4} = \frac{32 + 3}{4} = \frac{35}{4} \]

### Step 2: Find the Cost of One Bag of Potatoes
Now, we will find the cost of one bag of potatoes by dividing the total cost by the number of bags.

\[ \text{Cost of 1 bag} = \frac{\text{Total Cost}}{\text{Number of Bags}} = \frac{\frac{327}{5}}{\frac{7}{2}} \]

To divide fractions, we multiply by the reciprocal:

\[ \text{Cost of 1 bag} = \frac{327}{5} \times \frac{2}{7} = \frac{327 \times 2}{5 \times 7} = \frac{654}{35} \]

### Step 3: Calculate the Earnings from Selling \(8 \frac{3}{4}\) Bags
Now we will calculate how much he will earn by selling \(8 \frac{3}{4}\) bags.

\[ \text{Earnings} = \text{Number of Bags} \times \text{Cost of 1 Bag} = \frac{35}{4} \times \frac{654}{35} \]

The \(35\) in the numerator and denominator cancels out:

\[ \text{Earnings} = \frac{654}{4} \]

### Step 4: Simplify the Earnings
Now we simplify \(\frac{654}{4}\):

\[ \frac{654}{4} = 163.5 \]

### Final Answer
Thus, the man will earn \(Rs. 163.5\) if he sells \(8 \frac{3}{4}\) bags of potatoes.

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