A shopkeeper marks up his goods by 40% and offers a discount of 10%. What is his final profit percentage?
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Instead of picking a cost price and calculating step by step, try combining the markup and the discount into a single net percentage change using the formula for successive percentage changes. A positive markup and a negative discount will partly cancel out.
Approach Solution - 1
Step 1: Understanding the problem.
The shopkeeper marks up the goods by 40% and gives a discount of 10%. We need to calculate the final profit percentage.
Step 2: Calculating the Cost Price (C.P.) and Selling Price (S.P.).
Let the cost price of the item be \( C \).
- Marked Price (M.P.) = \( C + 40% \, \text{of} \, C = C \times 1.40 \).
- The shopkeeper offers a 10% discount on the marked price, so the selling price is: \[ \text{Selling Price} = M.P. \times (1 - 0.10) = C \times 1.40 \times 0.90 = C \times 1.26 \]
Step 3: Finding the Profit Percentage.
Profit = Selling Price - Cost Price = \( C \times 1.26 - C = C \times 0.26 \).
Profit Percentage = \( \frac{\text{Profit}}{\text{Cost Price}} \times 100 = \frac{C \times 0.26}{C} \times 100 = 26% \).
Step 4: Conclusion.
The final profit percentage is 26%.
Approach Solution -2
- A markup followed by a discount can be combined into one single net percentage change instead of computing cost price and selling price separately.
- The net percentage change formula for two successive percentage changes $x%$ and $y%$ is $x + y + \dfrac{xy}{100}$, where a markup is taken as positive and a discount as negative.
Step 1: Identify the two successive percentage changes.
The markup is $+40%$ and the discount is $-10%$.
Step 2: Apply the net percentage change formula directly.
Net change $= x + y + \dfrac{xy}{100} = 40 + (-10) + \dfrac{40 \times (-10)}{100}$
Step 3: Simplify the expression.
$= 30 + \dfrac{-400}{100} = 30 - 4 = 26$
Step 4: Interpret the sign of the result.
The net change is $+26%$, meaning a positive result, so it represents a profit, not a loss.
Final Answer: The final profit percentage is 26%.
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