A shopkeeper earns a profit of `40%` on the cost price of an article after giving three consecutive discounts of `5%, 10% and 15%` to a customer. What would have been the profit percentage, had the shopkeeper given discounts of `5% and 10%` only?

Options

A. `59.71%`
B. `66.71%`
C. `64.71%`
D. `74.71%`

Answer

To solve the problem step by step, let's denote the following:

- Let the Cost Price (CP) of the article be \( x \).
- The shopkeeper earns a profit of \( 40\% \) on the Cost Price.

### Step 1: Calculate the Selling Price after all discounts
The Selling Price (SP) after applying the profit is given by:
\[ SP = CP + Profit = x + 0.4x = 1.4x \]

### Step 2: Calculate the Market Price (MP)
Let’s denote the Market Price (MP) as \( MP \). The shopkeeper gives three consecutive discounts of \( 5\% \), \( 10\% \), and \( 15\% \).

To find the effective discount, we can use the formula for consecutive discounts:
\[ \text{Effective Discount} = D_1 + D_2 + D_3 - \left( \frac{D_1 \times D_2}{100} + \frac{D_2 \times D_3}{100} + \frac{D_1 \times D_3}{100} \right) \]
Where \( D_1 = 5\% \), \( D_2 = 10\% \), and \( D_3 = 15\% \).

Calculating the effective discount:
1. First, calculate the sum of the discounts:
\[ D_1 + D_2 + D_3 = 5 + 10 + 15 = 30\% \]
2. Now calculate the pairwise products:
\[ \frac{5 \times 10}{100} = 0.5, \quad \frac{10 \times 15}{100} = 1.5, \quad \frac{5 \times 15}{100} = 0.75 \]
Adding these gives:
\[ 0.5 + 1.5 + 0.75 = 2.75 \]
3. Now, the effective discount is:
\[ \text{Effective Discount} = 30 - 2.75 = 27.25\% \]

### Step 3: Calculate the Selling Price after all discounts
The Selling Price after applying the effective discount on the Market Price is:
\[ SP = MP \times \left(1 - \frac{27.25}{100}\right) = MP \times 0.7275 \]

### Step 4: Relate Market Price to Selling Price
Since we know that:
\[ SP = 1.4x \]
We can set up the equation:
\[ 1.4x = MP \times 0.7275 \]
From this, we can express the Market Price:
\[ MP = \frac{1.4x}{0.7275} \]

### Step 5: Calculate Selling Price with 5% and 10% discounts
Now, we need to find the Selling Price if only \( 5\% \) and \( 10\% \) discounts are given. The effective discount for these two discounts is:
\[ \text{Effective Discount} = 5 + 10 - \left( \frac{5 \times 10}{100} \right) = 15 - 0.5 = 14.5\% \]

Thus, the Selling Price with these discounts is:
\[ SP' = MP \times \left(1 - \frac{14.5}{100}\right) = MP \times 0.855 \]

### Step 6: Substitute Market Price
Substituting for \( MP \):
\[ SP' = \frac{1.4x}{0.7275} \times 0.855 \]

### Step 7: Calculate Profit
Now, we can find the profit:
\[ \text{Profit} = SP' - CP = \left(\frac{1.4x \times 0.855}{0.7275}\right) - x \]

### Step 8: Calculate Profit Percentage
Finally, the profit percentage is given by:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{CP} \times 100 \]

### Conclusion
By calculating the above expressions, we can find the profit percentage when only \( 5\% \) and \( 10\% \) discounts are applied.