Question

A shopkeeper charges sales tax of \(x \%\) up to Rs. 2,000 and above it he charges \(y \%\) A customer pays total tax of Rs 320 , when he purchases the goods worth Rs. 6,000 and he pay's the total tax of Rs. 680 for the goods worth Rs. 12,000 . The value of \((x-y)\) is : (a) 0 (b) \(-2\) (c) \(-4\) (d) 5

Short answer

The customer pays a total tax of Rs. 320 for a purchase of goods worth Rs. 6,000 and a total tax of Rs. 680 for goods worth Rs. 12,000. Answer: (x-y) = -2%

Step-by-step solution

Determine the expressions for the total tax

For a purchase amount of exactly Rs. 2,000, the total tax paid is \(2000 \cdot x\%\). For any purchase amount above Rs. 2,000, there is a fixed tax amount of \(2000 \cdot x\% = 20x\) (for the first 2,000) plus an additional tax amount for the money above 2,000. So, the total tax paid can be expressed as follows: - If the purchase amount (A) is less than or equal to 2,000, then the total tax paid(T) = \(A \cdot x\%\). - If the purchase amount (A) is more than 2,000, then the total tax paid(T) = \(20x + (A - 2000) \cdot y\%\).

Set up the system of equations

We will use the given information to create a system of equations. The customer pays a total tax of Rs. 320 for a purchase of goods worth Rs. 6,000, and a total tax of Rs. 680 for goods worth Rs. 12,000. For the Rs. 6,000 purchase amount, the total tax equation becomes: \(T = 20x + (6000 - 2000) \cdot y\% = 320\) For the Rs. 12,000 purchase amount, the total tax equation becomes: \(T = 20x + (12000 - 2000) \cdot y\% = 680\) Now our system of equations is: 1) \(20x + 4000y\% = 320\) 2) \(20x + 10000y\% = 680\)

Solve the system of equations

To solve the system, we will use the subtraction method. We will subtract equation 1 from equation 2: \((20x + 10000y\%) - (20x + 4000y\%) = 680 - 320\) Simplifying, we get: \(6000y\% = 360\) Now, we can solve for y by dividing by 6000: \(y\% =\frac{360}{6000} = 0.06\) To find the value of x, plug y% value back into equation 1: \(20x + 4000 \cdot 0.06 = 320\) Solve for x: \(20x + 240 = 320\) \(20x = 80\) \(x\% = 4\) Finally, to find \((x-y)\), we can simply subtract y% from x%: \((x-y) = 4\%-0.06\% = 4\% - 6\% = -2\%\) So, the value of \((x-y)\) is -2, which is option (b).

Key concepts

Quantitative Aptitude

When tackling sales tax problems, quantitative aptitude plays a crucial role. This aspect of problem-solving involves the skill to understand and use mathematical concepts to conclude from the provided data. Specifically, it deals with numerical reasoning, calculation speed, and accuracy, which are essential when processing taxation problems similar to the one provided in our exercise.

For instance, the ability to discern between different tax rates applied to different price ranges demonstrates quantitative aptitude. It requires you to not only perform accurate calculations but also to understand mathematical relationships and how changes in one quantity affect another. This skill is honed through practice and familiarity with a range of quantitative problems, solidifying one's mathematical base.

System of Equations

Sales tax problem-solving often requires the formation of a system of equations, a fundamental tool when there is more than one unknown quantity. In the given exercise, we have two different tax rates, one up to Rs. 2,000 and one above it. This necessitates creating two separate equations linking these rates to the total tax paid.

To successfully solve a system of equations, it's imperative to understand methods like substitution, elimination, or graphical representation. In our case, we employed the idea of elimination to determine the values of the two unknown tax rates.

By subtracting the first equation from the second, we eliminate one of the variables, thereby simplifying the problem to a single variable equation, which is more straightforward to solve. This step-by-step methodical approach is vital in breaking down complex multi-variable problems into simpler solitary equations.

Percentage Calculation

Every sales tax problem hinges upon percentage calculations, as tax rates are typically presented in percentages. To calculate these percentages, you need a strong grasp of basic arithmetic, particularly multiplication and division, and an understanding of how to work with percentages.

In our exercise, we had to convert the percentage of the tax rate into a decimal form to accurately calculate the total tax paid on different amounts. This concept underscores the importance of recognizing that 'percent' means 'per hundred,' so a tax rate of, for example, 4% is equivalent to 0.04 in decimal form when performing calculations.

Understanding how to manipulate and calculate percentages is integral to solving sales tax problems, whether it’s calculating the tax on a single item or finding the total tax paid on a larger scale purchase.