Suman buys `2 (3)/(4)` kg of rice for Rs. `126 (1)/(2)` and Madhu buys `4 (1)/(2)` kg of rice for Rs. `166 (1)/(2)`. Who has bought rice at a lower price and by how much?

Answer

To solve the problem, we need to find out who bought rice at a lower price per kilogram between Suman and Madhu. We will calculate the cost per kilogram of rice for both and then compare the two.

### Step 1: Convert the quantities of rice into improper fractions.

**For Suman:**
- Suman buys \(2 \frac{3}{4}\) kg of rice.
- Convert this to an improper fraction:
\[ 2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} \text{ kg} \]

**For Madhu:**
- Madhu buys \(4 \frac{1}{2}\) kg of rice.
- Convert this to an improper fraction:
\[ 4 \frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \text{ kg} \]

### Step 2: Convert the total prices into improper fractions.

**For Suman:**
- Suman pays \(126 \frac{1}{2}\) rupees.
- Convert this to an improper fraction:
\[ 126 \frac{1}{2} = \frac{(126 \times 2) + 1}{2} = \frac{252 + 1}{2} = \frac{253}{2} \text{ rupees} \]

**For Madhu:**
- Madhu pays \(166 \frac{1}{2}\) rupees.
- Convert this to an improper fraction:
\[ 166 \frac{1}{2} = \frac{(166 \times 2) + 1}{2} = \frac{332 + 1}{2} = \frac{333}{2} \text{ rupees} \]

### Step 3: Calculate the cost per kilogram for Suman.

- The cost per kg for Suman is calculated by dividing the total cost by the total weight:
\[ \text{Cost per kg for Suman} = \frac{\text{Total Cost}}{\text{Total Weight}} = \frac{\frac{253}{2}}{\frac{11}{4}} \]
- To divide fractions, multiply by the reciprocal:
\[ = \frac{253}{2} \times \frac{4}{11} = \frac{253 \times 4}{2 \times 11} = \frac{1012}{22} = \frac{506}{11} \approx 46 \text{ rupees} \]

### Step 4: Calculate the cost per kilogram for Madhu.

- The cost per kg for Madhu is calculated similarly:
\[ \text{Cost per kg for Madhu} = \frac{\text{Total Cost}}{\text{Total Weight}} = \frac{\frac{333}{2}}{\frac{9}{2}} \]
- To divide fractions, multiply by the reciprocal:
\[ = \frac{333}{2} \times \frac{2}{9} = \frac{333 \times 2}{2 \times 9} = \frac{333}{9} \approx 37 \text{ rupees} \]

### Step 5: Compare the costs.

- Suman's cost per kg: \(46\) rupees
- Madhu's cost per kg: \(37\) rupees

### Step 6: Determine who bought rice at a lower price and by how much.

- Madhu bought rice at a lower price than Suman.
- The difference in price is:
\[ 46 - 37 = 9 \text{ rupees} \]

### Final Answer:
Madhu bought rice at a lower price by 9 rupees.

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