Three equal successive discounts% allows on an article affect as single discount of 78.4%. Find each equal discount %?

Options

A. `38.33%`
B. `37.5%`
C. `40%`
D. `44.44%`

Answer

To find the equal successive discount percentage that results in a single discount of 78.4%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have three equal successive discounts that together give a total discount of 78.4%. We need to find the value of each equal discount percentage.

2. **Set Up the Equation**: Let the equal discount percentage be \( D\% \). If the marked price of the article is 100, then after applying the discount, the selling price (SP) can be expressed as:
\[ SP = 100 \times \left(1 - \frac{D}{100}\right)^3 \]
This is because the discount is applied successively three times.

3. **Calculate the Selling Price with the Total Discount**: A single discount of 78.4% means that the selling price is:
\[ SP = 100 \times (1 - 0.784) = 100 \times 0.216 = 21.6 \]

4. **Set the Two Selling Prices Equal**: Now, we can set the two expressions for SP equal to each other:
\[ 100 \times \left(1 - \frac{D}{100}\right)^3 = 21.6 \]

5. **Simplify the Equation**: Dividing both sides by 100 gives:
\[ \left(1 - \frac{D}{100}\right)^3 = 0.216 \]

6. **Take the Cube Root**: To solve for \( 1 - \frac{D}{100} \), we take the cube root of both sides:
\[ 1 - \frac{D}{100} = \sqrt[3]{0.216} \]

7. **Calculate the Cube Root**: The cube root of 0.216 is 0.6:
\[ 1 - \frac{D}{100} = 0.6 \]

8. **Solve for \( D \)**: Rearranging gives:
\[ \frac{D}{100} = 1 - 0.6 = 0.4 \]
Thus,
\[ D = 0.4 \times 100 = 40 \]

9. **Conclusion**: Each equal discount percentage is \( D = 40\% \).

### Final Answer:
The equal successive discount percentage is **40%**.

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