Question

Pepsi and Coke, there are two companies, selling the packs of cold-drinks. For the same selling price Pepsi gives two successive discounts of \(10 \%\) and \(25 \%\). While Coke sells it by giving two successive discounts of \(15 \%\) and \(20 \%\). What is the ratio of their marked price? (a) \(143: 144\) (b) \(19: 11\) (c) \(136: 135\) (d) \(73: 77\)

Short answer

Answer: The ratio of marked prices of Pepsi and Coke packs is approximately \(136: 135\).

Step-by-step solution

Calculate the effective discount for Pepsi

Since Pepsi offers two successive discounts of \(10\%\) and \(25\%\), we can use the formula for successive discounts: $$ \text{Effective Discount} = \text{First Discount} + \text{Second Discount} - (\text{First Discount} \times \text{Second Discount}) $$ Substitute the given values: $$ \text{Effective Discount for Pepsi} = 10\% + 25\% - (10\% \times 25\%) $$ Calculate the effective discount for Pepsi: $$ \text{Effective Discount for Pepsi} = 10\% + 25\% - 2.5\% = 32.5\% $$

Calculate the effective discount for Coke

Similarly, since Coke offers two successive discounts of \(15\%\) and \(20\%\), we can use the formula for successive discounts: $$ \text{Effective Discount for Coke} = 15\% + 20\% - (15\% \times 20\%) $$ Calculate the effective discount for Coke: $$ \text{Effective Discount for Coke} = 15\% + 20\% - 3\% = 32\% $$

Calculate the ratio of marked prices

Let the marked price of Pepsi be \(P\) and that of Coke be \(C\). Since both have the same selling price after their respective discounts, we can write the equation: $$ P(1 - 0.325) = C(1 - 0.32) $$ Now, we need to find the ratio of their marked prices, i.e., \(\frac{P}{C}\). Divide both sides by \(C(1 - 0.32)\): $$ \frac{P}{C} = \frac{1 - 0.32}{1 - 0.325} $$ Calculate the ratio: $$ \frac{P}{C} = \frac{0.68}{0.675} $$ Approximating the values, we get the ratio: $$ \frac{P}{C} \approx \frac{136}{135} $$ Therefore, the correct answer is (c) \(136: 135\).

Key concepts

Effective Discount Calculation

Understanding how to calculate effective discount is crucial for determining the final price after multiple discounts are applied. When a product or service is offered with successive discounts, simply subtracting each discount percentage from the original price does not yield the correct final price. Instead, we need a way to combine these discounts to find the total, or 'effective,' discount.

To find the effective discount when two discounts are offered successively, we use a specific formula: \[\text{Effective Discount} = \text{First Discount} + \text{Second Discount} - (\text{First Discount} \times \text{Second Discount})\]
This formula accounts for the fact that the second discount is applied to the already reduced price after the first discount. The minus part of the formula corrects for the overlap of the two discounts. It's important to convert percentages to decimals when using this formula to calculate. For instance, a 10% discount is represented as 0.10 in the formula.

Marked Price Ratio

The concept of marked price ratio emerges when comparing the initial prices, or marked prices, of two or more items before any discounts are applied. Understanding this ratio is especially useful in comparing deals or discounts from different retailers or manufacturers.

When two products have the same selling price after discounts but had different original marked prices, the marked price ratio can tell us how those original prices relate to each other. The formula used to find the marked price ratio when dealing with successive discounts is derived from setting up a proportion of the net selling prices, considering the effective discounts on each product:

Marked Price Ratio and Effective Discounts

\[\frac{\text{Marked Price of First Item}}{\text{Marked Price of Second Item}} = \frac{1 - \text{Effective Discount of Second Item}}{1 - \text{Effective Discount of First Item}}\]
This proportion equates the reduced prices after all discounts are applied, allowing us to solve for the marked price ratio. The result gives us a direct comparison of the original prices before any discounts, and it can often lead to insights about the pricing strategies of different sellers.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are fundamental in representing real-world situations and in solving equations systematically. In the context of discounts and pricing, algebraic expressions allow us to create formulas and equations that represent the relationships between different financial quantities.

For example, if we denote the marked price for Pepsi as 'P' and for Coca-Cola as 'C,' and we know the selling prices are equal after discounts, we can form an algebraic expression to represent this situation. This expression can be manipulated to find the marked price ratio. In the exercise provided, we use algebra to equate the discounted prices, which leads to the development of a ratio:\[\frac{P}{C} = \frac{1 - \text{Effective Discount for Coke}}{1 - \text{Effective Discount for Pepsi}}\]
By using algebraic expressions, we transform a word problem into an equation that can be solved using algebraic methods. This makes finding solutions not just possible, but also systematic and reliable.