Question

An article was marked at \(\mathrm{Rs} \mathrm{m}\). A discount of \(\mathrm{x} \%\) was given on it. A sales tax of \(\mathrm{x} \%\) was then charged. A person bought it for \(\mathrm{Rs} \mathrm{s}\) which included the sales tax. Which of the following can be concluded? (1) \(\mathrm{s}<\mathrm{m}\) (2) \(\mathrm{s}=\mathrm{m}\) (3) \(\mathrm{s}>\mathrm{m}\) (4) None of the previous choices

Short answer

The relationship between s and m cannot be definitively concluded from the given information.

Step-by-step solution

Write the discount equation

Let's first write the equation to find the discounted price. Since a discount of x% was given on the original price m, the discounted price will be: $$(1 - \frac{x}{100})m$$

Write the sales tax equation

Now we need to write the equation to calculate the sales tax on the discounted price. Since a sales tax of x% was added to the price after the discount, the final price, including the sales tax, will be: $$s = (1 + \frac{x}{100})((1 - \frac{x}{100})m)$$

Simplify the equation

Now we need to simplify the equation by expanding and combining like terms. $$s = (1 - \frac{x}{100} + \frac{x^2}{10000})(m)$$

Analyzing the equation

Analyzing the expression, we can see that the sign of the term \(\frac{x^2}{10000}m\) will depend on the actual value of x. If x > 0, then this term will be positive, meaning the fraction part of the price will be positive. If x < 0, then this term will be negative, meaning the fraction part of the price will be negative. As we can see, the relationship between s and m depends on the value of x and cannot be definitively concluded from the given information.

Answer

Therefore, the answer is: (4) None of the previous choices.

Key concepts

Discount Calculation

When you see a discount being offered, it means you're getting a reduction from the original price. Suppose an item is marked at a price of \(m\). If a discount of \(x\%\) is offered, you pay only a part of the original amount. To find the price after the discount, you subtract this percentage from 100%.

Here's how the math works:

• Discount rate = \(x\%\)
• Discounted price = \((1 - \frac{x}{100})m\)

Essentially, you calculate by finding the fraction of the original price that remains after the discount, which is \(1 - \frac{x}{100}\). Multiply this with the original price \(m\) to get the new reduced price.

Sales Tax

Adding sales tax works a bit like the reverse of a discount. After figuring out the discounted price, we need to account for any tax added to the item. Sales tax, like discounts, is a percentage of the cost. So, when a sales tax of \(x\%\) is charged, you add this percent to the discounted price.

Calculating this involves these steps:

• Discounted price = \((1 - \frac{x}{100})m\)
• Price including tax = \((1 + \frac{x}{100})\times\text{{discounted price}}\)

The equation becomes \(s = (1 + \frac{x}{100})((1 - \frac{x}{100})m)\). Each percentage step references the previous stages, ensuring that calculations add up to the final price \(s\) that includes both the discount and the sales tax.

Equation Simplification

To make the math more understandable, we will simplify the equation used for finding the final price after discounts and taxes.

Once the equation \(s = (1 + \frac{x}{100})((1 - \frac{x}{100})m)\) is set up, our goal is to expand and simplify:

• Expand the terms: This means multiplying out each part of the equation.
• Equation turns into: \(s = (1 - \frac{x}{100} + \frac{x^2}{10000})m\).
• Combine like terms: Simplifying the expression so it's easier to see how the terms affect each other.

By carefully manipulating this, we reach a simpler form which shows clearly that the final price \(s\) depends intricately on \(x\), the discount and sales tax rate.

Price Comparison

Understanding how prices relate when discounts and taxes are involved can be tricky. We want to see how \(s\) (the final price including tax) relates to \(m\) (the original price).

This comparison is guided by looking at the simplified equation:

• If \(x > 0\), the term \(\frac{x^2}{10000}m\) is positive, indicating that \(s\) could be higher than \(m\).
• If \(x < 0\), then the term means a decrease, implying a reduced price.
• Without specific numbers, we can't definitively determine if \(s\) is greater or less than \(m\).

Hence, it's challenging to make a clear conclusion just from the given information, leading us to see that none of the initial choices can be definitively chosen without further details.