A single discount equivalent to a discount seres of 5%, 10% , 15% and 20%, is :

Options

A. `43.86%`
B. `42.86%`
C. `40.86%`
D. `41.86%`

Answer

To find the single discount equivalent to a series of discounts of 5%, 10%, 15%, and 20%, we can use the formula for calculating the equivalent discount when multiple discounts are applied sequentially. The formula is:

\[ D = A + B + \frac{AB}{100} \]

Where \( A \) and \( B \) are the two discounts being combined. We will apply this formula iteratively to combine all four discounts.

### Step 1: Combine the first two discounts (5% and 10%)
Let \( A = 5 \) and \( B = 10 \).

Using the formula:
\[ D_1 = 5 + 10 + \frac{5 \times 10}{100} \]
\[ D_1 = 5 + 10 + \frac{50}{100} \]
\[ D_1 = 5 + 10 + 0.5 \]
\[ D_1 = 15.5\% \]

### Step 2: Combine the result with the next discount (15%)
Now, let \( A = 15.5 \) and \( B = 15 \).

Using the formula:
\[ D_2 = 15.5 + 15 + \frac{15.5 \times 15}{100} \]
\[ D_2 = 15.5 + 15 + \frac{232.5}{100} \]
\[ D_2 = 15.5 + 15 + 2.325 \]
\[ D_2 = 32.825\% \]

### Step 3: Combine the result with the last discount (20%)
Now, let \( A = 32.825 \) and \( B = 20 \).

Using the formula:
\[ D_3 = 32.825 + 20 + \frac{32.825 \times 20}{100} \]
\[ D_3 = 32.825 + 20 + \frac{656.5}{100} \]
\[ D_3 = 32.825 + 20 + 6.565 \]
\[ D_3 = 59.39\% \]

### Step 4: Final calculation
Now we need to find the equivalent single discount. We will apply the formula again to find the final equivalent discount.

Using the formula:
\[ D = 59.39 + 20 + \frac{59.39 \times 20}{100} \]
\[ D = 59.39 + 20 + 11.878 \]
\[ D = 59.39 + 20 + 11.878 \]
\[ D = 91.268\% \]

However, this value is not correct as we need to find the equivalent discount from the original price.

To find the single discount equivalent to the series of discounts, we can also use the formula:
\[ D = 1 - (1 - d_1)(1 - d_2)(1 - d_3)(1 - d_4) \]
Where \( d_1, d_2, d_3, d_4 \) are the decimal forms of the percentages.

### Step 5: Convert percentages to decimals and calculate
Convert the discounts to decimals:
- \( d_1 = 0.05 \)
- \( d_2 = 0.10 \)
- \( d_3 = 0.15 \)
- \( d_4 = 0.20 \)

Now calculate:
\[ D = 1 - (1 - 0.05)(1 - 0.10)(1 - 0.15)(1 - 0.20) \]
\[ D = 1 - (0.95)(0.90)(0.85)(0.80) \]
\[ D = 1 - (0.95 \times 0.90 \times 0.85 \times 0.80) \]
\[ D = 1 - (0.5788) \]
\[ D = 0.4212 \]

Converting back to percentage:
\[ D = 42.12\% \]

### Conclusion
The single discount equivalent to a discount series of 5%, 10%, 15%, and 20% is approximately **42.12%**.

### Final Answer
The closest option is **42.86%**.