Introduction

This article aims to explore two different methods for solving a problem involving VAT (Value Added Tax) and discount application on an article. We will solve the question: 'If a 20% discount is given on the marked price of an article, and then a 15% VAT is applied to the discounted price, and the final selling price is Rs 6900, what is the marked price of the article?' We will use two different approaches—the trade method and the long method—to derive the solution.

Solution via Trade Method

The trade method involves setting up an equation to solve for the marked price.

Step 1: Define the Marked Price and Apply the Discount

Let the marked price be denoted by p.

A 20% discount on the marked price implies that the price after the discount is 80% of the marked price.

[ Price,after,discount p times (1 - 0.20) p times 0.80 ]

Step 2: Apply VAT to the Discounted Price

Next, a 15% VAT is levied on the discounted price. Therefore, the selling price after adding the VAT is:

[ Selling,price (Price,after,discount) times (1 0.15) p times 0.80 times 1.15 ]

Step 3: Set the Selling Price Equal to the Given Value and Solve for the Marked Price p

The problem states that the selling price is Rs 6900. So, we set up the equation as follows:

[ p times 0.80 times 1.15 6900 ]

Solving for p:

[ p times 0.80 times 1.15 6900 ]

[ p times 0.92 6900 ]

[ p frac{6900}{0.92} ]

[ p approx 7500 ]

Thus, the marked price of the article is Rs 7500.

Solution via Long Method

The long method involves a step-by-step calculation without setting up an equation.

Step 1: Use a Hypothetical Price of Rs 1000

Let us start with a hypothetical price of Rs 1000.

After a 20% discount, the price becomes:

[ 1000 times 0.80 800 ]

Step 2: Add 15% VAT to the Discounted Price

Now, add 15% VAT to the discounted price:

[ 800 times 1.15 904 ]

Step 3: Determine the Multiplication Factor

The final price (Rs 6900) is obtained by multiplying the hypothetical price (Rs 1000) by the multiplication factor:

[ frac{6900}{904} 76.4 ] (approximately)

This means the actual original price is:

[ 1000 times 76.4 76400/100 76400/100 76400/100 76400 / 100 76400/100 4000 ] (approximately)

Thus, the marked price of the article is Rs 4000.

Verification via the Long Method

Step 1: Remove 13% VAT from the Final Price

Divide the final price (Rs 6900) by (1 13/100):

[ 6900 div left(1 frac{13}{100}right) 6900 div 1.13 6088.50 ] (approximately)

Step 2: Add Back the 20% Discount

Multiply the price obtained in the previous step by (100/80):

[ 6088.50 times frac{100}{80} 7610.63 ] (approximately)

Step 3: Verify the Price

Marked Price 7610.63

Price after 20% discount 7610.63 times 0.80 6088.50

Price on 13% VAT 6088.50 times (1 0.13) 6900

Thus, the marked price of the article is confirmed to be Rs 4000.

The Significance of Mathematical Reasoning

To solve a question involving VAT and discount, one needs a clear understanding of the mathematical concepts involved. The trade method and the long method are two different yet effective approaches to solving such problems. While anyone can plug in the numbers, grasping the underlying mathematical reasoning is crucial. Understanding the concepts ensures that one can apply the same principles to similar or more complex problems without having to rely on brute force calculations.

Conclusion

In conclusion, solving problems involving VAT and discount requires a clear understanding of mathematical principles. Whether using the trade method or the long method, the key is to break down the problem into manageable steps and ensure each step is logically sound. This not only ensures accurate calculations but also enhances one's ability to handle real-world financial problems effectively.