A shopkeeper offers the following three schemes. Which scheme has the maximum discount percent ?
I. Two successive discount of 20% and 25%.
II. Buy 5 get 3 free
III. Buy 5 get 6

Options

A. Only II
B. Only I
C. Only I and II
D. Only I and III

Answer

To determine which scheme has the maximum discount percent, we will analyze each scheme step by step.

### Step 1: Calculate the effective discount for Scheme I (Two successive discounts of 20% and 25%).

1. **First Discount**: 20% off means you pay 80% of the original price.
2. **Second Discount**: 25% off means you pay 75% of the remaining price after the first discount.

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

- After the first discount:
\[ \text{Price after 1st discount} = P \times (1 - 0.20) = P \times 0.80 \]

- After the second discount:
\[ \text{Price after 2nd discount} = (P \times 0.80) \times (1 - 0.25) = P \times 0.80 \times 0.75 = P \times 0.60 \]

- Therefore, the effective discount is:
\[ \text{Effective Discount} = P - P \times 0.60 = P \times 0.40 \]

- The discount percentage is:
\[ \text{Discount Percentage} = \left(\frac{0.40P}{P}\right) \times 100 = 40\% \]

### Step 2: Calculate the effective discount for Scheme II (Buy 5 get 3 free).

1. **Total items received**: When you buy 5 items, you get 3 additional items for free, making a total of \( 5 + 3 = 8 \) items.
2. **Cost of 5 items**: Let's denote the price of one item as \( P \). Therefore, the cost for 5 items is \( 5P \).
3. **Effective discount calculation**:
\[ \text{Effective Discount} = \frac{\text{Total items received} - \text{Items paid for}}{\text{Total items received}} \times 100 \]
\[ = \frac{8 - 5}{8} \times 100 = \frac{3}{8} \times 100 = 37.5\% \]

### Step 3: Calculate the effective discount for Scheme III (Buy 5 get 6).

1. **Total items received**: When you buy 5 items, you get 6 additional items for free, making a total of \( 5 + 6 = 11 \) items.
2. **Cost of 5 items**: Again, the cost for 5 items is \( 5P \).
3. **Effective discount calculation**:
\[ \text{Effective Discount} = \frac{\text{Total items received} - \text{Items paid for}}{\text{Total items received}} \times 100 \]
\[ = \frac{11 - 5}{11} \times 100 = \frac{6}{11} \times 100 \approx 54.55\% \]

### Conclusion

Now, we can summarize the effective discounts for each scheme:
- Scheme I: 40%
- Scheme II: 37.5%
- Scheme III: 54.55%

The scheme with the maximum discount percent is **Scheme III (Buy 5 get 6)**.

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