Due to a reduction of `20%` in the price of wheat. Arya can purchase 5kg more wheat for `Rs.320`. Then find original price

Options

A. `16` kg,
B. `18` kg,
C. `20` kg
D. `22` kg

Answer

To solve the problem step by step, we need to find the original price of wheat based on the given information.

### Step 1: Understand the problem
We know that due to a reduction of 20% in the price of wheat, Arya can purchase 5 kg more wheat for Rs. 320. We need to find the original price of wheat per kg.

### Step 2: Define the variables
Let the original price of wheat be \( P \) (in Rs. per kg). After a reduction of 20%, the new price becomes:
\[ \text{New Price} = P - 0.2P = 0.8P \]

### Step 3: Set up the equation for the amount of wheat purchased
The amount of wheat Arya can buy at the original price for Rs. 320 is:
\[ \text{Quantity at original price} = \frac{320}{P} \]

The amount of wheat Arya can buy at the new price for Rs. 320 is:
\[ \text{Quantity at new price} = \frac{320}{0.8P} \]

### Step 4: Relate the quantities
According to the problem, the difference in the quantity purchased due to the price reduction is 5 kg:
\[ \frac{320}{0.8P} - \frac{320}{P} = 5 \]

### Step 5: Simplify the equation
To simplify the left side, we can find a common denominator:
\[ \frac{320P - 320 \cdot 0.8P}{0.8P^2} = 5 \]
This simplifies to:
\[ \frac{320P - 256P}{0.8P^2} = 5 \]
\[ \frac{64P}{0.8P^2} = 5 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 64P = 5 \cdot 0.8P^2 \]
\[ 64P = 4P^2 \]

### Step 7: Rearrange the equation
Rearranging the equation gives:
\[ 4P^2 - 64P = 0 \]
Factoring out \( 4P \):
\[ 4P(P - 16) = 0 \]

### Step 8: Solve for \( P \)
Setting each factor to zero gives:
\[ 4P = 0 \quad \text{or} \quad P - 16 = 0 \]
Since \( P \) cannot be zero, we have:
\[ P = 16 \]

### Conclusion
The original price of wheat is Rs. 16 per kg.

---