If the selling price of an article is five times the discount offered and if the percentage of discount is equal to the percentage profit, find the ratio of the discount offered to the cost price.

### Concept & Relationship between SP, MP, CP, and Discount The core relationship connects the variables: $\text{Marked Price (MP)} = \text{Selling Price (SP)} + \text{Discount (D)}$ and $\text{Selling Price (SP)} = \text{Cost Price (CP)} + \text{Profit (P)}$. $$ \text{Discount \%} = \frac{D}{\text{MP}} \times 100 $$ $$ \text{Profit \%} = \frac{P}{\text{CP}} \times 100 $$ ### Step-by-Step Solution * Given: $\text{SP} = 5D$. * Since $\text{MP} = \text{SP} + D$, we have $\text{MP} = 5D + D = 6D$. * Calculate Discount Percentage: $\text{Discount \%} = \frac{D}{6D} \times 100 = \frac{100}{6}\% = 16.66\%$. * We are given that $\text{Profit \%} = \text{Discount \%}$. So, $\text{Profit \%} = \frac{100}{6}\% = \frac{1}{6}$ as a fraction. * This means Profit $P = \frac{1}{6} \text{ of CP}$. * Since $\text{SP} = \text{CP} + P$, we substitute: $\text{SP} = \text{CP} + \frac{\text{CP}}{6} = \frac{7\text{CP}}{6}$. * We know from the beginning that $\text{SP} = 5D$. * Equating both expressions for SP: $5D = \frac{7\text{CP}}{6}$. * Rearranging to find the ratio of Discount to Cost Price: $\frac{D}{\text{CP}} = \frac{7}{5 \times 6} = \frac{7}{30}$. ### Exam Strategy & Shortcut Express everything in terms of fractions based on $D$. SP = $5D$, MP = $6D$, so discount fraction is $1/6$. Since profit fraction is also $1/6$, CP must be scaled such that adding $1/6$ of itself yields $5D$. $\text{CP} \times (7/6) = 5D \Rightarrow D/\text{CP} = 7/30$. ### Common Pitfall Confusing the base for the percentages. Discount is always calculated on the Marked Price (MP), not the Selling Price (SP), while Profit is always calculated on the Cost Price (CP). ### Final Answer Therefore, the correct answer is **7 : 30**.