Understanding the Sugar Price Reduction Problem

This problem involves calculating the original price per kilogram of sugar when a percentage reduction in price allows a consumer to buy more quantity for the same amount of money. We need to find the initial cost per kg before the discount.

Defining Variables and Initial Setup

Let's define the variables needed to solve this problem:

• Let the original price per kg of sugar be denoted by \(P\) (in ₹/kg).
• The total amount of money the purchaser has is ₹160.
• The percentage reduction in the price is 20%.
• The additional quantity of sugar obtained due to the price reduction is 2.5 kg.

The original quantity of sugar that could be purchased for ₹160 is calculated as:

Original Quantity = \(\frac{\text{Total Amount Spent}}{\text{Original Price per kg}} = \frac{160}{P}\)

Calculating the New Price and Quantity

A reduction of 20% in the price means the new price is 80% of the original price.

The new price per kg of sugar is:

New Price = Original Price - (20% of Original Price)

New Price = \(P - 0.20 \times P = (1 - 0.20) \times P = 0.80P\)

With this new, lower price, the quantity of sugar the purchaser can obtain for ₹160 is:

New Quantity = \(\frac{\text{Total Amount Spent}}{\text{New Price per kg}} = \frac{160}{0.80P}\)

Formulating the Equation Based on the Problem Statement

The problem states that the purchaser can obtain 2.5 kg more sugar with the reduced price. This means the difference between the new quantity and the original quantity is 2.5 kg.

New Quantity - Original Quantity = 2.5 kg

Substituting the expressions we derived:

\(\frac{160}{0.80P} - \frac{160}{P} = 2.5\)

Solving the Equation for the Original Price (P)

Let's solve the equation step-by-step:

First, simplify the term \(\frac{160}{0.80P}\):

\(\frac{160}{0.80P} = \frac{160}{\frac{80}{100}P} = \frac{160 \times 100}{80P} = \frac{16000}{80P}\)

Dividing 16000 by 80 gives:

\(\frac{16000}{80P} = \frac{200}{P}\)

Now, substitute this simplified term back into our equation:

\(\frac{200}{P} - \frac{160}{P} = 2.5\)

Since both terms on the left side have the same denominator (\(P\)), we can combine the numerators:

\(\frac{200 - 160}{P} = 2.5\)

\(\frac{40}{P} = 2.5\)

To find \(P\), we can rearrange the equation. Multiply both sides by \(P\) and then divide by 2.5:

\(P = \frac{40}{2.5}\)

To make the division easier, we can remove the decimal by multiplying the numerator and the denominator by 10:

\(P = \frac{40 \times 10}{2.5 \times 10} = \frac{400}{25}\)

Performing the division:

\(P = 16\)

Therefore, the original price per kg of sugar was ₹16.

Verification of the Result

Let's verify our answer by plugging the original price back into the problem's conditions:

• Original Price = ₹16/kg
• Quantity purchased for ₹160 at original price = \(\frac{160}{16} = 10\) kg.
• New Price after 20% reduction = \(16 \times (1 - 0.20) = 16 \times 0.80 = ₹12.8\)/kg.
• Quantity purchased for ₹160 at new price = \(\frac{160}{12.8} = 12.5\) kg.
• Difference in quantity = New Quantity - Original Quantity = 12.5 kg - 10 kg = 2.5 kg.

The calculated difference of 2.5 kg matches the information given in the question, confirming that our calculated original price of ₹16 per kg is correct.