Question

On the April 1,2005 my salary increased from Rs. 10,000 to Rs. \(16,000 .\) Simultaneously the rate of income tax decreased by \(37.5 \%\), So the amount of income tax paid by me remains constant what is the value of income tax paid by me : (a) Rs. 3000 (b) Rs. 6000 (c) Rs. 1600 (d) can't be determined

Short answer

(a) 120 (b) 180 (c) 240 (d) can't be determined Answer: (d) can't be determined

Step-by-step solution

Set up the proportion

Let x be the original tax rate (as a decimal). Then the new tax rate will be x*(1-37.5/100) or x*(5/8). Since the amount of income tax paid remains constant, we can set up the following proportion: 10000x = 16000 * (5/8)x

Solve for x

Since both sides of the equation are multiplied by x, we can simplify the equation by dividing both sides by x: 10000 = 16000 * (5/8) Now, divide both sides by (5/8): 10000 * (8/5) = 16000 Divide both sides by 160: 10000 * (1/2) = 100 Multiply both sides by 2: 10000 = 200 So the original tax rate (as a decimal) is 200/10000, or 0.02.

Calculate the tax amount and select the correct option

Now that we found the original tax rate, we can calculate the tax amount by multiplying it with the original salary: tax_amount = original_salary * original_tax_rate tax_amount = 10000 * 0.02 tax_amount = 200 However, none of the given options match our calculated tax amount. It seems the tax amount can't be determined based on the given information since it doesn't match any available options. Therefore, the correct answer is: (d) can't be determined

Key concepts

Tax Rate Calculations

Understanding how tax rates are calculated is essential for anyone dealing with income taxes. Tax rates often dictate the portion of your income that you owe to the government. The problem at hand touches on an aspect of tax rates: the impact of a change in the rate. When your salary increases, any change in the tax rate could either increase or decrease the tax you pay, unless it is adjusted in such a way that your tax payment remains the same.

For calculating the new tax rate after a decrease, a common approach is to apply proportional reasoning, which involves calculating the percentage change. If an income tax rate decreases by 37.5%, you can represent the new rate as a fraction of the original rate. By denoting the original tax rate as \(x\) and applying the decrease, we get the new rate as \(x \times \frac{5}{8}\), because a 37.5% decrease leaves 62.5% of the original rate, which is \frac{5}{8} when expressed as a fraction.

Proportional Reasoning

Proportional reasoning is a mathematical tool that involves using the concept of ratios to make comparisons between quantities. In this scenario, we know that the amount of income tax paid remains constant despite the increase in salary and decrease in income tax rate, meaning that the salary and tax rate are inversely proportional. In simpler terms, if one goes up, the other must come down in such a way that the product of the two remains unchanged.

Using proportional reasoning, we can set up a proportion like so:\[ 10000 \times x = 16000 \times \left(\frac{5}{8}x\right) \]This equation shows that the original salary multiplied by the original tax rate is equal to the new salary multiplied by the new, reduced tax rate. By solving this proportion, we can find the value of \(x\), the original tax rate, before the percentage decrease.

Percentage Decrease

A percentage decrease is a reduction in a quantity by a certain percent of its original value. It's a concept frequently encountered in financial contexts, such as when calculating the effect of a sale on prices or the change in tax rates. In this problem, the rate of income tax decreased by 37.5%, which can be tricky to visualize without converting it into a more manageable form, such as a fraction or a decimal.

To calculate the new tax after a percentage decrease, you subtract the percentage decrease from 100% and then multiply the original tax rate by this resultant percentage. Here's a breakdown using the given tax rate decrease:

• 100% - 37.5% = 62.5%
• Converting 62.5% to a decimal for easy calculation gives us 0.625
• The new tax rate can then be calculated by multiplying the original tax rate by 0.625

Nevertheless, such calculations require that the options provided align with the possible outcome, which was not the case in the initial problem, leading to the conclusion that the answer can't be determined.