By a reduction of re.-1 per kg in the price of sugar Mohan can buy one kg more for rs.56 . find the original price of sugar per kg ?

Answer

To solve the problem step by step, we will follow a structured approach:

### Step 1: Define the Variables
Let the original price of sugar per kg be \( x \) rupees.

### Step 2: Calculate the Quantity of Sugar
The quantity of sugar Mohan can buy for Rs. 56 at the original price is:
\[ \text{Quantity} = \frac{56}{x} \text{ kg} \]

### Step 3: Determine the New Price
If the price of sugar is reduced by Rs. 1, the new price becomes:
\[ \text{New Price} = x - 1 \text{ rupees per kg} \]

### Step 4: Calculate the New Quantity of Sugar
The quantity of sugar Mohan can buy for Rs. 56 at the new price is:
\[ \text{New Quantity} = \frac{56}{x - 1} \text{ kg} \]

### Step 5: Set Up the Equation
According to the problem, with the reduced price, Mohan can buy 1 kg more sugar:
\[ \frac{56}{x - 1} = \frac{56}{x} + 1 \]

### Step 6: Clear the Fractions
To eliminate the fractions, we can multiply through by \( x(x - 1) \):
\[ 56x = 56(x - 1) + x(x - 1) \]

### Step 7: Expand and Simplify
Expanding both sides gives:
\[ 56x = 56x - 56 + x^2 - x \]
Now, simplify the equation:
\[ 0 = x^2 - x - 56 \]

### Step 8: Rearrange into Standard Form
Rearranging gives us the standard form of the quadratic equation:
\[ x^2 - x - 56 = 0 \]

### Step 9: Factor the Quadratic Equation
We need to factor \( x^2 - x - 56 \). We look for two numbers that multiply to -56 and add to -1. These numbers are -8 and 7:
\[ (x - 8)(x + 7) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
1. \( x - 8 = 0 \) → \( x = 8 \)
2. \( x + 7 = 0 \) → \( x = -7 \)

Since the price cannot be negative, we discard \( x = -7 \).

### Step 11: Conclusion
Thus, the original price of sugar per kg is:
\[ \text{Original Price} = 8 \text{ rupees per kg} \]