A reduction of 20% in the price of sugar enables a purchaser to obtain `2 (1)/(2)` kg more for Rs. 160. Find the original price per kg of sugar.

Options

A. Rs. 12
B. Rs. 20
C. Rs. 16
D. Rs. 18

Answer

To find the original price per kg of sugar, we can follow these steps:

### Step 1: Understand the Reduction in Price
The price of sugar is reduced by 20%. This means that the new price is 80% of the original price.

### Step 2: Let the Original Price be 'x'
Let the original price per kg of sugar be \( x \) rupees.

### Step 3: Calculate the New Price
Since the price is reduced by 20%, the new price per kg of sugar will be:
\[ \text{New Price} = x - 0.2x = 0.8x \]

### Step 4: Determine the Quantity of Sugar Purchased
The problem states that the purchaser can buy an additional \( 2 \frac{1}{2} \) kg (which is \( \frac{5}{2} \) kg) of sugar for Rs. 160 after the price reduction.

### Step 5: Calculate the Quantity Before the Price Reduction
Let’s denote the quantity of sugar that could be purchased before the price reduction as \( Q_1 \) kg:
\[ Q_1 = \frac{160}{x} \]
And after the price reduction, the quantity purchased is \( Q_2 \) kg:
\[ Q_2 = \frac{160}{0.8x} = \frac{160}{\frac{4}{5}x} = \frac{160 \times 5}{4x} = \frac{200}{x} \]

### Step 6: Set Up the Equation
According to the problem, the difference in quantity purchased before and after the price reduction is \( \frac{5}{2} \) kg:
\[ Q_2 - Q_1 = \frac{5}{2} \]
Substituting the expressions for \( Q_1 \) and \( Q_2 \):
\[ \frac{200}{x} - \frac{160}{x} = \frac{5}{2} \]

### Step 7: Simplify the Equation
Combine the fractions:
\[ \frac{200 - 160}{x} = \frac{5}{2} \]
\[ \frac{40}{x} = \frac{5}{2} \]

### Step 8: Cross Multiply to Solve for x
Cross multiplying gives:
\[ 40 \cdot 2 = 5 \cdot x \]
\[ 80 = 5x \]

### Step 9: Solve for x
Now, divide both sides by 5:
\[ x = \frac{80}{5} = 16 \]

### Conclusion
The original price per kg of sugar is \( \text{Rs. } 16 \).

---