Successive discounts of p% and q% on the catalogue price of an article is equivalent to a single discount of :

Options

A. `(x-y - (xy)/(100))%`
B. `(p-q -(pq)/(100))%`
C. `(p+q - (pq)/(100))%`
D. `(p+q+(pq)/(100))%`

Answer

To find the equivalent single discount of successive discounts of p% and q% on the catalogue price of an article, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Discounts**:
- Let the catalogue price (marked price) of the article be assumed as \( MP = 100 \) (this simplifies calculations).
- The first discount is \( p\% \) and the second discount is \( q\% \).

2. **Calculate the Selling Price after the First Discount**:
- After applying the first discount of \( p\% \), the selling price becomes:
\[ SP_1 = MP - (p\% \text{ of } MP) = 100 - \frac{p}{100} \times 100 = 100 - p \]

3. **Calculate the Selling Price after the Second Discount**:
- Now, apply the second discount of \( q\% \) on the new selling price \( SP_1 \):
\[ SP_2 = SP_1 - (q\% \text{ of } SP_1) = (100 - p) - \frac{q}{100} \times (100 - p) \]
- Simplifying this:
\[ SP_2 = (100 - p) \left(1 - \frac{q}{100}\right) = (100 - p) \left(\frac{100 - q}{100}\right) \]

4. **Calculate the Final Selling Price**:
- Therefore, the final selling price \( SP_2 \) can be expressed as:
\[ SP_2 = \frac{(100 - p)(100 - q)}{100} \]

5. **Calculate the Equivalent Single Discount**:
- The equivalent single discount can be found using the formula:
\[ \text{Discount Percentage} = \frac{MP - SP_2}{MP} \times 100 \]
- Substituting \( MP = 100 \) and \( SP_2 \):
\[ \text{Discount Percentage} = \frac{100 - \frac{(100 - p)(100 - q)}{100}}{100} \times 100 \]
- Simplifying this expression:
\[ = 100 - \frac{(100 - p)(100 - q)}{100} \]
- Expanding \( (100 - p)(100 - q) \):
\[ = 10000 - 100p - 100q + pq \]
- Therefore:
\[ \text{Discount Percentage} = 100 - \left(100 - \frac{100p + 100q - pq}{100}\right) \]
- This simplifies to:
\[ = p + q - \frac{pq}{100} \]

6. **Final Result**:
- The equivalent single discount of successive discounts of \( p\% \) and \( q\% \) is:
\[ \text{Equivalent Single Discount} = p + q - \frac{pq}{100} \]