Let's break down this problem step by step to find the original price of the item per kg. The problem involves a percentage reduction in price which leads to being able to buy a larger quantity for the same total amount.

Understanding the Price Reduction

We are told that the price of the item is reduced by 20%. If the original price per kg is $P$ rupees, then the reduced price per kg will be 20% less than $P$.

Percentage reduction = $20\%$

Reduced price = Original price $-$ $20\%$ of Original price

Reduced price = $P - \left(\frac{20}{100}\right)P$

Reduced price = $P - 0.20P$

Reduced price = $0.80P$ rupees per kg.

Relating Price and Quantity

The total amount spent is fixed at $₹360$. We know that Total Cost = Price per kg $\times$ Quantity in kg. Therefore, Quantity in kg = $\frac{\text{Total Cost}}{\text{Price per kg}}$.

Let the original quantity of the item bought for $₹360$ be $Q_{original}$ kg.

Using the original price $P$, we have:

$Q_{original} = \frac{360}{P}$ kg.

When the price is reduced to $0.80P$, the new quantity of the item bought for $₹360$ is $Q_{new}$ kg.

Using the reduced price $0.80P$, we have:

$Q_{new} = \frac{360}{0.80P}$ kg.

Setting up the Equation

The problem states that as a result of the price reduction, customers can get 2 kg more of the item for $₹360$. This means the new quantity is 2 kg more than the original quantity.

$Q_{new} = Q_{original} + 2$

Substitute the expressions for $Q_{new}$ and $Q_{original}$ into this equation:

$\frac{360}{0.80P} = \frac{360}{P} + 2$

Solving for the Original Price

Now, we need to solve this equation for $P$, the original price per kg.

Rearrange the equation to group terms with $P$:

$\frac{360}{0.80P} - \frac{360}{P} = 2$

To subtract the fractions, find a common denominator, which is $0.80P$.

$\frac{360 \times 1}{0.80P} - \frac{360 \times 0.80}{0.80P} = 2$

$\frac{360 - 360 \times 0.80}{0.80P} = 2$

Calculate $360 \times 0.80 = 288$.

$\frac{360 - 288}{0.80P} = 2$

$\frac{72}{0.80P} = 2$

Now, multiply both sides by $0.80P$:

$72 = 2 \times 0.80P$

$72 = 1.60P$

To find $P$, divide both sides by $1.60$:

$P = \frac{72}{1.60}$

To make the division easier, multiply the numerator and the denominator by 10:

$P = \frac{720}{16}$

Now, perform the division:

$720 \div 16$

$16 \times 4 = 64$

$72 - 64 = 8$

Bring down the 0, making it 80.

$16 \times 5 = 80$

$80 - 80 = 0$

So, $P = 45$.

The original price per kg of the item was $₹45$.

Verification

Let's verify the answer:

• Original price = $₹45$ per kg. Original quantity for $₹360 = \frac{360}{45} = 8$ kg.
• Reduced price = $₹45 - 20\%$ of $₹45 = 45 - 0.20 \times 45 = 45 - 9 = ₹36$ per kg.
• New quantity for $₹360$ at reduced price = $\frac{360}{36} = 10$ kg.

The difference in quantity is $10$ kg $-$ $8$ kg $= 2$ kg, which matches the information given in the problem. So, the calculated original price of $₹45$ per kg is correct.

Revision Table: Key Concepts

This problem involves understanding percentage change and how price and quantity are related when the total cost is fixed.

• Percentage Reduction: To find a value after a percentage reduction, subtract the percentage multiplied by the original value from the original value. $New Value = Original Value \times (1 - \frac{Percent Reduction}{100})$.
• Price, Quantity, and Total Cost: Total Cost = Price $\times$ Quantity. If Total Cost is constant, Price is inversely proportional to Quantity. When price decreases, quantity increases for the same total cost.

Additional Information: Inverse Proportion

When the total cost of buying an item is fixed, the price per unit and the quantity bought are inversely proportional. This means if the price goes down, the quantity you can buy with the same amount of money goes up, and vice versa. In this problem, the price decreased by 20%, which means the new price is 80% of the original price. Since Price $\times$ Quantity = Constant (₹360), the new quantity will be $\frac{1}{0.80}$ times the original quantity. $\frac{1}{0.80} = \frac{1}{4/5} = \frac{5}{4} = 1.25$. This means the new quantity is 1.25 times (or 125%) the original quantity. The increase in quantity is $1.25 - 1 = 0.25$ times the original quantity, which is 25% more. This 25% increase in quantity corresponds to 2 kg, allowing us to find the original quantity and subsequently the original price.

• New Quantity = $1.25 \times$ Original Quantity
• New Quantity $-$ Original Quantity = $0.25 \times$ Original Quantity
• We are given that $Q_{new} - Q_{original} = 2$ kg.
• So, $0.25 \times Q_{original} = 2$ kg.
• $Q_{original} = \frac{2}{0.25} = \frac{2}{1/4} = 2 \times 4 = 8$ kg.

The original quantity was 8 kg. Since the total cost was $₹360$ for 8 kg, the original price per kg was $\frac{360}{8} = ₹45$. This confirms the result obtained from solving the equation.