Question

The list price of an article is \(50 \%\) more than its original cost price. The shopkeeper allowed a discount of \(20 \%\) and charged a sales tax of \(20 \%\) on it. Finally the buyer paid \(\mathrm{Rs} 2880\). What is the cost price of the article? (1) \(\mathrm{Rs} 2500\) (2) \(\operatorname{Rs} 3500\) (3) \(\operatorname{Rs} 3000\) (4) \(\mathrm{Rs} 2000\)

Short answer

Answer: Rs 2000.

Step-by-step solution

Calculate the list price given the cost price

Let the cost price of the article be \(x\). The list price is 50% more than the cost price. So, the list price can be given as \(1.50x\).

Calculate the discounted price

As the shopkeeper allowed a discount of 20%, the discounted price will be 20% less than the list price, so we calculate 80% of the list price: \(0.80 \times 1.50x = 1.20x\).

Calculate the final price after sales tax

Now, the shopkeeper charged a 20% sales tax on the discounted price. So, we calculate 120% of the discounted price which is \(1.20 \times 1.20x=1.44x\). The final price paid by the buyer is Rs 2880.

Solve for the cost price

We know that the final price is Rs 2880. Therefore, we can set up the equation: \(1.44x = 2880\). Solving for \(x\), we find the cost price: \[x = \frac{2880}{1.44} = 2000\] So, the cost price of the article is \(\text{Rs} \ 2000\). The correct answer is (4).

Key concepts

Percentage Calculation

Understanding how percentage calculation works is crucial when dealing with financial transactions, such as sales and taxation. A percentage represents a part per hundred. To calculate the percentage of a number, you multiply the number by the percentage expressed as a decimal. For example, if you have a list price that is 50% more than its cost price, the calculation is straightforward: you take the original cost price and add half of its value to it (50% as a decimal is 0.50).

In the context of the given exercise, if the cost price is denoted by 'x', the list price becomes '1.50x', indicating a 50% increase. Similarly, when applying a discount, you subtract a percentage from 100% to find what proportion of the list price you pay. For a 20% discount, you actually pay 80% of the list price, or '0.80' when expressed as a decimal. It's important for students to practice converting percentages to decimals to adeptly handle these equations in financial dealings.

Discount and Sales Tax Impact

When shopping, the amount you pay for an item can be influenced by discounts and sales tax. A discount reduces the list price of an item, which makes the item more affordable. However, sales tax is an additional charge on the sale of goods and services and increases the final price. To understand the combined effect of both, you must first apply the percentage of discount to the list price to find the discounted price, and then apply the sales tax percentage to the discounted price to find the final amount payable.

In the exercise we're discussing, a discount of 20% and a sales tax of 20% are applied sequentially, not simultaneously. This is a key concept because the sales tax is calculated on the discounted price, not the original list price. It is a common mistake to apply the tax on the list price without considering the discount. The problem illustrates the discount reduces the price to '1.20x', and then sales tax increases it to '1.44x'. These two operations have mutually opposing effects, but their sequence significantly impacts the final cost.

Algebraic Problem-Solving

Algebraic problem-solving involves using symbols and letters to represent numbers in equations, thus allowing us to solve for unknown values. In algebra, the goal is to isolate the variable you are solving for by performing inverse operations. When you're working with equations that have compounded percentage changes like discount followed by tax, setting up the equation correctly is half the battle.

In our example, the final equation '1.44x = 2880' represents the culmination of such steps. The unknown value, 'x', represents the cost price, and the other part of the equation represents the sequence of percentage adjustments that lead to the final price. To solve for 'x', you use the inverse operation of multiplication, which is division. Dividing both sides of the equation by '1.44' isolates 'x' and allows you to find the cost price. It's essential for students to get comfortable with these algebraic manipulations, as they serve as the foundation for various advanced mathematical concepts.