Understanding Shopkeeper Discount Schemes

Shopkeepers offer various schemes to attract customers. These can include successive discounts, buy-one-get-one offers, or getting extra items free when buying a certain quantity. For a customer, the best scheme is the one that provides the maximum effective discount.

We need to analyze the three schemes provided and calculate the effective discount percentage for each to determine which one is most beneficial for the customer.

Calculating Effective Discount for Scheme - I

Scheme - I offers two successive discounts: 15% and 25%.

When two successive discounts, $d_1$ and $d_2$, are applied, the effective single discount ($d_{eff}$) is calculated using the formula:

\(d_{eff} = d_1 + d_2 - \frac{d_1 \times d_2}{100}\)

Substituting the given values, \(d_1 = 15\%\) and \(d_2 = 25\%\):

\(d_{eff} = 15\% + 25\% - \frac{15 \times 25}{100}\)

\(d_{eff} = 40\% - \frac{375}{100}\)

\(d_{eff} = 40\% - 3.75\%\)

\(d_{eff} = 36.25\%\)

So, Scheme - I is equivalent to a single discount of 36.25%.

Calculating Effective Discount for Scheme - II

Scheme - II is "Buy 5, get 3 free".

In this type of scheme, the customer pays for a certain number of items but receives more items in total. The discount comes from the items received for free.

• Number of items bought (paid for) = 5
• Number of items received free = 3
• Total number of items received = Number of items bought + Number of free items = 5 + 3 = 8

The discount percentage is calculated based on the free items as a proportion of the total items received.

\(\text{Discount Percentage} = \left( \frac{\text{Number of free items}}{\text{Total number of items received}} \right) \times 100\%\)

\(\text{Discount Percentage} = \left( \frac{3}{8} \right) \times 100\%\)

\(\text{Discount Percentage} = \frac{300}{8}\%\)

\(\text{Discount Percentage} = 37.5\%\)

So, Scheme - II offers an effective discount of 37.5%.

Calculating Effective Discount for Scheme - III

Scheme - III is "Buy 4, get 6".

Based on the context of similar problems and common interpretations, "Buy 4, get 6" typically means the customer pays for 4 items and receives a total of 6 items. This implies getting some items free.

• Number of items paid for = 4
• Total number of items received = 6
• Number of items received free = Total items received - Items paid for = 6 - 4 = 2

Now, we calculate the discount percentage using the same formula as for Scheme II:

\(\text{Discount Percentage} = \left( \frac{\text{Number of free items}}{\text{Total number of items received}} \right) \times 100\%\)

\(\text{Discount Percentage} = \left( \frac{2}{6} \right) \times 100\%\)

\(\text{Discount Percentage} = \left( \frac{1}{3} \right) \times 100\%\)

\(\text{Discount Percentage} \approx 33.33\%\)

So, Scheme - III offers an effective discount of approximately 33.33%.

Note: If Scheme III were interpreted as "Buy 4, get 6 free" (meaning 6 additional items are free), the total items received would be 4 + 6 = 10, with 6 free. This would give a discount of \(\frac{6}{10} \times 100\% = 60\%\). However, based on the given correct answer, the interpretation of getting a total of 6 items by paying for 4 is the correct one for this problem.

Comparing the Schemes for Maximum Discount

Let's compare the effective discount percentages for all three schemes:

Comparing the percentages, we see that:

• Scheme - I offers 36.25% discount.
• Scheme - II offers 37.5% discount.
• Scheme - III offers ≈ 33.33% discount.

The highest effective discount is offered by Scheme - II (37.5%).

Conclusion: Best Scheme for Customers

For a customer, the best scheme is the one that provides the highest discount. Comparing the calculated effective discounts:

\(37.5\% (\text{Scheme - II}) > 36.25\% (\text{Scheme - I}) > 33.33\% (\text{Scheme - III})\)

Therefore, Scheme - II is the best scheme for customers.

Discount Scheme Analysis Revision

Let's quickly recap the calculation methods for different types of discount schemes often encountered:

• Successive Discounts: Calculate effective discount using the formula \(d_{eff} = d_1 + d_2 - \frac{d_1 \times d_2}{100}\). This compounds the discounts.
• Buy X, Get Y Free: Calculate the percentage of free items out of the total number of items received (\(\frac{\text{Y}}{\text{X+Y}} \times 100\%\)). This represents the proportion of the total value the customer gets for free.
• Buy X, Get Y Total (paying for X, receiving Y): Calculate the percentage of free items (which is \(Y-X\)) out of the total number of items received (\(\frac{\text{Y-X}}{\text{Y}} \times 100\%\)). This also represents the value saved.

Additional Information on Discounts and Savings

Understanding how different discount schemes work is crucial for making smart purchasing decisions. Successive discounts are common, but the second discount is applied to the price after the first discount, not the original price. "Buy N, Get M Free" offers can sometimes seem more attractive than percentage discounts, especially when the number of free items is high relative to the number bought. Always calculate the effective percentage discount to make a true comparison between different types of offers.

For example, a 50% discount sounds great, but "Buy 1, Get 1 Free" is equivalent to a 50% discount (\(\frac{1}{1+1} \times 100\% = 50\%\)). "Buy 2, Get 1 Free" is a 33.33% discount (\(\frac{1}{2+1} \times 100\% \approx 33.33\%\)). Being able to convert these offers to a single percentage makes comparison easy.