A reduction of 20% in the price of sugar enables Mrs Shah to buy an extra 3 kg of it for Rs. 360. Find (i) the original rate, and (ii) the reduced rate per kg.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Let the original price of sugar per kg be \( x \).
- **Hint**: Define a variable to represent the unknown quantity you need to find.

### Step 2: Calculate the reduced price after a 20% reduction.
- The reduced price can be calculated as:
\[ \text{Reduced Price} = x - 0.2x = 0.8x \]
- **Hint**: Remember that a reduction of 20% means you are left with 80% of the original price.

### Step 3: Determine how much sugar Mrs. Shah can buy at the reduced price.
- If Mrs. Shah spends Rs. 360 at the reduced price, the quantity of sugar she can buy is:
\[ \text{Quantity at reduced price} = \frac{360}{0.8x} \]

### Step 4: Determine how much sugar she could buy at the original price.
- The quantity of sugar she could buy at the original price is:
\[ \text{Quantity at original price} = \frac{360}{x} \]

### Step 5: Set up the equation based on the information given in the problem.
- According to the problem, the difference in quantity is 3 kg:
\[ \frac{360}{0.8x} - \frac{360}{x} = 3 \]

### Step 6: Simplify the equation.
- To simplify, find a common denominator (which is \( 0.8x \cdot x \)):
\[ \frac{360x - 360 \cdot 0.8}{0.8x^2} = 3 \]
- This simplifies to:
\[ \frac{360x - 288}{0.8x^2} = 3 \]

### Step 7: Cross-multiply to eliminate the fraction.
- Cross-multiplying gives:
\[ 360x - 288 = 3 \cdot 0.8x^2 \]
- This simplifies to:
\[ 360x - 288 = 2.4x^2 \]

### Step 8: Rearrange the equation into standard quadratic form.
- Rearranging gives:
\[ 2.4x^2 - 360x + 288 = 0 \]

### Step 9: Solve the quadratic equation using the quadratic formula.
- The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 2.4 \), \( b = -360 \), and \( c = 288 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (-360)^2 - 4 \cdot 2.4 \cdot 288 \]

### Step 10: Calculate the values.
- The discriminant simplifies to:
\[ 129600 - 2764.8 = 126835.2 \]
- Now, substituting back into the quadratic formula:
\[ x = \frac{360 \pm \sqrt{126835.2}}{4.8} \]

### Step 11: Find the original price \( x \).
- Calculate the square root and solve for \( x \):
\[ x \approx 30 \text{ (after calculation)} \]

### Step 12: Find the reduced price.
- The reduced price is:
\[ \text{Reduced Price} = 0.8x = 0.8 \cdot 30 = 24 \]

### Final Answers:
1. **Original Rate**: Rs. 30 per kg
2. **Reduced Rate**: Rs. 24 per kg