A retailer offers the following discount schemes for buyers on an article.
I. Two successive discounds of 10%. II. A discount of 12% followed by a discount of 8%. III. Successive discounts of 15% and 5%. IV. A discount of 20%.
The selling price will be minimum under the scheme

Options

A. I
B. II
C. III
D. IV

Answer

To solve the problem, we need to calculate the equivalent discount for each of the four discount schemes provided. The equivalent discount can be calculated using the formula for successive discounts.

Let's denote the original price of the article as \( P \).

### Step 1: Calculate the equivalent discount for Scheme I (Two successive discounts of 10%)

1. The first discount is 10%, so the price after the first discount is:
\[ P_1 = P \times (1 - 0.10) = P \times 0.90 \]

2. The second discount is also 10%, so the price after the second discount is:
\[ P_2 = P_1 \times (1 - 0.10) = (P \times 0.90) \times 0.90 = P \times 0.90^2 = P \times 0.81 \]

3. The equivalent discount can be calculated as:
\[ \text{Equivalent Discount} = 1 - \frac{P_2}{P} = 1 - 0.81 = 0.19 \text{ or } 19\% \]

### Step 2: Calculate the equivalent discount for Scheme II (A discount of 12% followed by a discount of 8%)

1. The first discount is 12%, so the price after the first discount is:
\[ P_1 = P \times (1 - 0.12) = P \times 0.88 \]

2. The second discount is 8%, so the price after the second discount is:
\[ P_2 = P_1 \times (1 - 0.08) = (P \times 0.88) \times 0.92 = P \times 0.88 \times 0.92 = P \times 0.8096 \]

3. The equivalent discount can be calculated as:
\[ \text{Equivalent Discount} = 1 - \frac{P_2}{P} = 1 - 0.8096 = 0.1904 \text{ or } 19.04\% \]

### Step 3: Calculate the equivalent discount for Scheme III (Successive discounts of 15% and 5%)

1. The first discount is 15%, so the price after the first discount is:
\[ P_1 = P \times (1 - 0.15) = P \times 0.85 \]

2. The second discount is 5%, so the price after the second discount is:
\[ P_2 = P_1 \times (1 - 0.05) = (P \times 0.85) \times 0.95 = P \times 0.85 \times 0.95 = P \times 0.8075 \]

3. The equivalent discount can be calculated as:
\[ \text{Equivalent Discount} = 1 - \frac{P_2}{P} = 1 - 0.8075 = 0.1925 \text{ or } 19.25\% \]

### Step 4: Calculate the equivalent discount for Scheme IV (A discount of 20%)

1. The discount is 20%, so the price after the discount is:
\[ P_2 = P \times (1 - 0.20) = P \times 0.80 \]

2. The equivalent discount is:
\[ \text{Equivalent Discount} = 20\% \]

### Step 5: Compare the equivalent discounts

Now we compare the equivalent discounts from all schemes:
- Scheme I: 19%
- Scheme II: 19.04%
- Scheme III: 19.25%
- Scheme IV: 20%

### Conclusion

The maximum discount is 20% from Scheme IV. Therefore, the selling price will be minimum under Scheme IV.

**Final Answer: Scheme IV (20% discount)**

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