A retailer allows a trade discount of 20% and a cash discount of $6 \frac{1}{4}\%$ on the market price of the products and gets a net profit of 20% on the cost. By how much above the cost, should the products be labelled for sale?

### Concept & Successive Discounts When multiple discounts are given (trade discount followed by cash discount), they act successively on the Marked Price. Calculate the final Selling Price as a fraction of the Marked Price. ### Step-by-Step Solution - Let Cost Price (CP) = Rs. 100. - Desired net profit is $20\%$ on CP. - Selling Price (SP) = $100 + 20 = \text{Rs. } 120$. - Let the Market Price (MP) be $x$. - Trade discount = $20\%$. Price after trade discount = $x \times (1 - 0.20) = 0.8x$. - Cash discount = $6 \frac{1}{4}\% = 6.25\%$. Converting to fraction: $6.25\% = \frac{6.25}{100} = \frac{1}{16}$. - Final Selling Price after cash discount = $0.8x \times (1 - \frac{1}{16}) = 0.8x \times \frac{15}{16}$. - $0.8x$ is $\frac{4}{5}x$. So, SP = $(\frac{4}{5}x) \times (\frac{15}{16}) = \frac{60}{80}x = \frac{3}{4}x = 0.75x$. - Equating the two Selling Prices: $0.75x = 120$. - $x = \frac{120}{0.75} = \frac{120}{\frac{3}{4}} = \frac{120 \times 4}{3} = 160$. - The Marked Price must be Rs. 160, which is Rs. 60 above the CP of Rs. 100. - The markup percentage is $60\%$. ### Exam Strategy & Shortcut Convert percentages to fractions: $20\%$ discount = $\frac{4}{5}$ multiplier. $6.25\%$ discount = $\frac{15}{16}$ multiplier. Total discount multiplier = $\frac{4}{5} \times \frac{15}{16} = \frac{3}{4}$. We know $\text{MP} \times \frac{3}{4} = \text{SP}$. For a $20\%$ profit, $\text{SP} = 1.2 \times \text{CP}$. So, $\text{MP} \times \frac{3}{4} = \frac{6}{5} \times \text{CP} \Rightarrow \frac{\text{MP}}{\text{CP}} = \frac{6}{5} \times \frac{4}{3} = \frac{8}{5} = 1.6$. Thus, MP is $60\%$ above CP. ### Common Pitfall Avoid adding the successive discounts linearly (i.e., $20\% + 6.25\% = 26.25\%$). Successive percentage decreases must be calculated multiplicatively. ### Final Answer Therefore, the correct answer is **60%**.