Comparing Discount Schemes to Find Maximum Selling Price

The question asks us to determine under which discount scheme the selling price of an article will be the maximum. A higher selling price means the customer receives a lower overall discount.

To find the maximum selling price, we need to calculate the effective final price for each scheme or compare the effective total discounts. The lower the total discount, the higher the selling price.

Let's assume the original price of the article is $100 for easy calculation. We can calculate the final price after applying the discounts for each scheme.

Analyzing Each Discount Scheme

Scheme i: Two successive discounts of 15% each

First discount = 15%

Price after first discount = $100 - 15% of $100 = $100 - $15 = $85

Second discount = 15% on the reduced price ($85)

Discount amount = 15% of $85 = \( \frac{15}{100} \times 85 = 0.15 \times 85 = \$12.75 \)

Final Selling Price under Scheme i = $85 - $12.75 = $72.25

Alternatively, the remaining percentage after a 15% discount is \( (1 - 0.15) = 0.85 \). After two successive 15% discounts, the final price is \( 100 \times (1 - 0.15) \times (1 - 0.15) = 100 \times 0.85 \times 0.85 = 100 \times 0.7225 = \$72.25 \).

Scheme ii: A discount of 25% followed by a discount of 5%

First discount = 25%

Price after first discount = $100 - 25% of $100 = $100 - $25 = $75

Second discount = 5% on the reduced price ($75)

Discount amount = 5% of $75 = \( \frac{5}{100} \times 75 = 0.05 \times 75 = \$3.75 \)

Final Selling Price under Scheme ii = $75 - $3.75 = $71.25

Alternatively, the final price is \( 100 \times (1 - 0.25) \times (1 - 0.05) = 100 \times 0.75 \times 0.95 = 100 \times 0.7125 = \$71.25 \).

Scheme iii: Two successive discounts of 20% and 10%

First discount = 20%

Price after first discount = $100 - 20% of $100 = $100 - $20 = $80

Second discount = 10% on the reduced price ($80)

Discount amount = 10% of $80 = \( \frac{10}{100} \times 80 = 0.10 \times 80 = \$8 \)

Final Selling Price under Scheme iii = $80 - $8 = $72.00

Alternatively, the final price is \( 100 \times (1 - 0.20) \times (1 - 0.10) = 100 \times 0.80 \times 0.90 = 100 \times 0.72 = \$72.00 \).

Scheme iv: A discount of 30%

Single discount = 30%

Final Selling Price under Scheme iv = $100 - 30% of $100 = $100 - $30 = $70.00

Alternatively, the final price is \( 100 \times (1 - 0.30) = 100 \times 0.70 = \$70.00 \).

Comparing Final Selling Prices

Let's compare the final selling prices calculated for each scheme:

Comparing the final selling prices:

• Scheme i: $72.25
• Scheme ii: $71.25
• Scheme iii: $72.00
• Scheme iv: $70.00

The maximum selling price is $72.25, which is under Scheme i.

Therefore, the selling price will be maximum under Scheme i.

Understanding Effective Discount Rate

We can also compare the schemes by calculating the effective single discount rate for each successive discount scheme. The effective single discount (D) for two successive discounts \(d_1\) and \(d_2\) is given by the formula:

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

• Scheme i: \( d_1 = 15\% \), \( d_2 = 15\% \) \( D = 15 + 15 - \frac{15 \times 15}{100} = 30 - \frac{225}{100} = 30 - 2.25 = 27.75\% \) Effective discount = 27.75%. Selling price = \( 100 - 27.75 = 72.25\% \) of original price.
• Scheme ii: \( d_1 = 25\% \), \( d_2 = 5\% \) \( D = 25 + 5 - \frac{25 \times 5}{100} = 30 - \frac{125}{100} = 30 - 1.25 = 28.75\% \) Effective discount = 28.75%. Selling price = \( 100 - 28.75 = 71.25\% \) of original price.
• Scheme iii: \( d_1 = 20\% \), \( d_2 = 10\% \) \( D = 20 + 10 - \frac{20 \times 10}{100} = 30 - \frac{200}{100} = 30 - 2 = 28\% \) Effective discount = 28%. Selling price = \( 100 - 28 = 72\% \) of original price.
• Scheme iv: Single discount of 30%. Effective discount = 30%. Selling price = \( 100 - 30 = 70\% \) of original price.

Comparing the effective discounts:

• Scheme i: 27.75%
• Scheme ii: 28.75%
• Scheme iii: 28.00%
• Scheme iv: 30.00%

The lowest effective discount is 27.75% under Scheme i. A lower discount results in a higher selling price. Therefore, Scheme i gives the maximum selling price.

Revision Table: Comparing Discount Schemes

Scheme i results in the highest final price percentage (72.25%), thus yielding the maximum selling price.

Additional Information on Successive Discounts

When two successive discounts \(d_1\) and \(d_2\) are offered, the order of the discounts does not matter. Applying a 20% discount then a 10% discount gives the same final price as applying a 10% discount then a 20% discount. The final price factor is always \( (1 - d_1/100) \times (1 - d_2/100) \). The effective single discount is \( d_1 + d_2 - \frac{d_1 d_2}{100} \).

To get the lowest selling price (which is best for the buyer and results from the highest effective discount), the shopkeeper would offer the scheme with the largest effective discount. In this case, Scheme iv (30%) has the highest effective discount. To get the maximum selling price (which is best for the shopkeeper and results from the lowest effective discount), the shopkeeper would offer the scheme with the smallest effective discount, which is Scheme i (27.75%).