Twenty successive discount of 5%, 10%, 15%, 20%,... is equivalent to what single discount?

Question: Twenty successive discount of 5%, 10%, 15%, 20%,... is equivalent to what single discount?

• 20%
• 50%
• 100% 
• None of the above 

Answer: 

successive discount formula:

$$D_{\text{eff}} = A + B - \frac{A \times B}{100}$$

(where A and B are the successive discounts.)

The successive discounts are:

$$5\%, 10\%, 15\%, ..., 100\%$$

This forms an arithmetic sequence with:

• First term ($a$) = 5%

• Common difference ($d$) = 5%

• Number of terms ($n$) = 20

*We apply the formula step by step, merging two successive discounts at a time.

First Iteration:

Merging 5% and 10%:

$$D_1 = 5 + 10 - \frac{5 \times 10}{100} = 15 - 0.5 = 14.5\%$$

Second Iteration:

Merging 14.5% and 15%:

$$D_2=14.5+15-\frac{14.5\times 15}{100}=29.5-2.175=27.325\mathrm{\%}$$

If we continue this iterative process for all 20 discounts, we get:

$$D_{\text{final}}=100\mathrm{\%}$$

Additional Method :

 Successive discount shortcut formula:

$$D_{\text{final}} = 100 \times \left(1 - \prod_{i=1}^{n} (1 - \frac{D_i}{100}) \right)$$

where $D_i$ are the successive discounts.

**Identify the Given Data

• Discounts: 5%, 10%, 15%, ..., 100% (20 terms)

• Shortcut approach: Instead of calculating step-by-step, we calculate the final price factor:

$$\text{Final Price Factor} = (1 - 0.05) \times (1 - 0.10) \times (1 - 0.15) \times \dots \times (1 - 1.00)$$

**Direct Calculation

Since the last discount is 100%, the price factor becomes zero because:

$$(1 - 1.00) = 0$$

Thus, the final price is zero, meaning the equivalent single discount is 100%.

If you find this content helpful, consider supporting us by clicking on any of the ads on this page.

Mock Test Click Here

Madhyamik Previous Year Solution Click Here

CBSE Class 10 History Notes Click Here

You Can Translate It In Your Language: