Question

The list price of a T.V. is Rs 15000 and the shopkeeper allows a discount of \(20 \%\) and \(10 \%\) successively on list price. On the remaining amount, he charges \(20 \%\) as sales tax. If buyer paid Rs \(\mathrm{x}\), then by how much amount will the list price exceed \(\operatorname{Rs} x\) ? (1) \(\mathrm{Rs} 3000\) (2) \(\operatorname{Rs} 4200\) (3) \(\mathrm{Rs} 2040\) (4) \(\mathrm{Rs} 5000\)

Short answer

Answer: Rs 2040

Step-by-step solution

Calculate the first discount

A discount of 20% is applied to the list price. To calculate the price after the discount, first find 20% of the list price and then subtract it from the initial price. The formula to find a percentage is: \(Discount = \frac{Percentage}{100} \times List\ Price\) In this case, the list price is Rs 15,000, and the percentage is 20: \(Discount = \frac{20}{100} \times 15000 = 3000\) Now, to find the price after the first discount, subtract the discount from the list price: \(New\) \(Price = List\ Price - Discount = 15000 - 3000 = 12000\).

Calculate the second discount

A second discount of 10% is applied to the new price. To calculate the price after this discount, first find 10% of the new price, and then subtract it from the new price: \(Discount_2 = \frac{10}{100} \times New\ Price = \frac{10}{100} \times 12000 = 1200\) Now, calculate the price after the second discount: \(Final\ Price_{discounted} = New\ Price - Discount_2 = 12000 - 1200 = 10800\)

Calculate sales tax

A 20% sales tax is applied to the final discounted price. To calculate the sales tax, find 20% of the final discounted price: \(Sales Tax = \frac{20}{100} \times Final\ Price_{discounted} = \frac{20}{100} \times 10800 = 2160\)

Calculate the final price including sales tax

Now, to find the final price, add the sales tax to the final discounted price: \(Final\ Price_{total} = Final\ Price_{discounted} + Sales\ Tax = 10800 + 2160 = 12960\)

Calculate the difference between the list price and final price

Now that we have the final price, we can find the difference between the list price and the final price: \(Difference = List\ Price - Final\ Price_{total} = 15000 - 12960 = 2040\) The list price exceeds Rs x (the amount the buyer paid) by Rs 2040. The correct answer is (3) Rs 2040.

Key concepts

Percentage Calculations

Understanding how to calculate percentages is vital for determining successive discounts or any form of percentage reduction. A percentage represents a fraction of 100, and calculating it involves multiplying the original value by the percentage expressed as a fraction. For instance, to understand what 20% of Rs 15,000 is, you use the formula: \[ \text{Discount} = \frac{20}{100} \times 15000 \]By calculating, we find the discount to be Rs 3,000. After any percentage calculation, recalculating the base amount is essential. Subtract the discount from the original price to get the new reduced amount. This methodology applies similarly when calculating successive discounts, as each discount is a percentage of the new reduced price, not the original price. Grasping this core math skill is pivotal for just about any financial transaction that involves percentage discounts.

Sales Tax Calculation

After discounts have been applied, the sales tax calculation could play a significant role in determining the final price a buyer has to pay. Sales tax is typically calculated as a percentage of the final selling price. First, just like with discounts, change the percentage into a fraction by dividing by 100. Then, multiply it by the final price after discounts. For example, when a 20% sales tax is applied on a final discounted price of Rs 10,800, we calculate:\[ \text{Sales Tax} = \frac{20}{100} \times 10800 \]Resulting in a Rs 2,160 tax amount. This tax is then added to the already discounted price to find the total amount inclusive of tax. Thus, clear understanding of sales tax calculations is essential, as it affects the final amount a consumer is obliged to pay. Without it, a buyer might miscalculate budget or a seller might misinform a customer about the true cost.

Mathematical Problem Solving

Solving mathematical word problems can be daunting, but breaking them down into smaller parts makes them manageable. Begin by comprehending the problem requirements: what is given and what needs to be found? Here, the problem requires calculating how much the original list price of a TV (Rs 15,000) exceeds the final price after discounts and taxes (Rs 12,960). To solve, translate each word detail into a mathematical operation or formula, such as finding successive percentages and final price, step-by-step. It involves calculating sequential reductions and additions which involve logical thinking and organization. Properly employing mathematical symbols and formulas helps in visualizing each part of the process, thereby avoiding overwhelming situations. Mastering such problem-solving strategies enhances analytical thinking and strengthens overall math comprehension.

Step-by-Step Algebraic Solutions

Implementing step-by-step analysis for algebra-related problems helps clear the complexity surrounding them. Each successive step is simplification of the previous calculation, leading towards the final solution. It's much like following a recipe step by step to avoid missing any ingredient. Begin with calculating the first discount, followed by the second on the reduced price. Next, calculate the sales tax on this newly reduced price. Finally, by adding sales tax to this price, the total or complete price is determined. This structured approach is paramount in reducing errors and ensuring accuracy at each stage. The systematic solving of each calculation step by step ensures clarity and precision, making algebra manageable even for individuals new to the subject. By carefully reviewing each step, learners gain confidence in solving these kinds of problems independently.