To solve this question, we need to determine the cost of [tex]\(8 \frac{1}{4} \, \text{kg}\)[/tex] of sugar given the cost for [tex]\(5 \frac{1}{2}\, \text{kg}\)[/tex] of sugar is [tex]\(₹ 206 \frac{1}{4}\)[/tex].
### Step-by-Step Solution:
1. Convert the given mixed fractions to improper fractions for easier calculations:
- [tex]\(5 \frac{1}{2} \, \text{kg} = 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2} \, \text{kg}\)[/tex]
- [tex]\(206 \frac{1}{4} \, \text{rupees} = 206 + \frac{1}{4} = \frac{824}{4} + \frac{1}{4} = \frac{825}{4} \, \text{rupees}\)[/tex]
- [tex]\(8 \frac{1}{4} \, \text{kg} = 8 + \frac{1}{4} = \frac{32}{4} + \frac{1}{4} = \frac{33}{4} \, \text{kg}\)[/tex]
2. Calculate the cost per kilogram of sugar:
- The cost for [tex]\(5 \frac{1}{2} \, \text{kg}\)[/tex] ([tex]\(\frac{11}{2}\, \text{kg}\)[/tex]) of sugar is [tex]\(₹ 206 \frac{1}{4}\)[/tex] ([tex]\(\frac{825}{4} \, \text{rupees}\)[/tex]).
- Cost per kg = [tex]\(\frac{\text{total cost}}{\text{total weight}} = \frac{\frac{825}{4} \, \text{rupees}}{\frac{11}{2} \, \text{kg}} = \frac{825}{4} \cdot \frac{2}{11} \text{rupees/kg}\)[/tex]
3. Simplify the cost per kg calculation:
- [tex]\(\frac{825}{4} \cdot \frac{2}{11} = \frac{825 \cdot 2}{4 \cdot 11} = \frac{1650}{44} = 37.50 \, \text{rupees/kg}\)[/tex]
4. Calculate the cost for [tex]\(8 \frac{1}{4} \, \text{kg}\)[/tex] using the cost per kg:
- The required amount of sugar is [tex]\(8 \frac{1}{4}\, \text{kg} = \frac{33}{4}\, \text{kg}\)[/tex].
- Total cost for [tex]\(8 \frac{1}{4}\, \text{kg}\)[/tex] = cost per kg [tex]\(\times\)[/tex] required amount
- Total cost = [tex]\(37.50 \, \text{rupees/kg} \times \frac{33}{4}\, \text{kg}\)[/tex]
5. Simplify the total cost calculation:
- [tex]\(37.50 \times \frac{33}{4} = 37.50 \times 8.25 = 309.375 \, \text{rupees}\)[/tex]
Therefore, the cost of [tex]\(8 \frac{1}{4}\, \text{kg}\)[/tex] of sugar is [tex]\(₹ 309.375\)[/tex].
