Question

Prep Exercise 5 One month a business earns \(\$ 40,000\) in revenue. The next month the business increases the revenue55000. a. Is \(\$ 40,000\) an initial amount, a final amount, or an increase amount? b. Is \(\$ 5000\) an initial amount, a final amount, or an increase amount? c. With the information given, to find the percent of increase, do you need to subtract prior to setting up the proportion? Why or why not? d. Write a proportion that can be used to find the percent of increase. Melliah purchases a tennis racquet that is priced \(\$ 54.95 .\) She notes that the tax on her receipt is \(\$ 3.85\) What is the sales tax rate?

Short answer

Answer: The percent of increase in business revenue is \(37.5\%\) and the sales tax rate for Melliah's tennis racquet purchase is approximately \(7\%\).

Step-by-step solution

Identifying \(40000\)

The revenue for the first month is \(\$ 40000\). Since this amount represents the starting revenue value, it is an initial amount. #b. Identifying Initial Amount, Final Amount, or Increase Amount for 5000#

Identifying \(\$ 5000\)

The business revenue increased from \(\$ 40000\) to \(\$ 55000\). The increase in revenue is \(\$ 15000\), which is different from the given \(\$ 5000\). As such, the \(\$ 5000\) does not represent an initial, a final, or an increase amount in the context of the given problem. #c. Check Whether to Subtract Prior to Setting up the Proportion#

Subtraction Requirement

Yes, it is necessary to subtract the initial amount from the final amount to determine the increase amount. This increase amount will then be used to set up the proportion to find the percent of increase. #d. Proportion to Find Percent of Increase#

Finding Percent of Increase

First, let's find the increase amount: Increase Amount = Final Amount - Initial Amount Increase Amount = \(\$ 55000\) - \(\$ 40000\) Increase Amount = \(\$ 15000\) Now let's set up the proportion to find the percent of increase:

Percent of Increase Proportion

Percent of Increase = \(\dfrac{Increase \: Amount}{Initial \: Amount}\) x 100 Percent of Increase = \(\dfrac{\$ 15000}{\$ 40000}\) x 100 Percent of Increase = \(0.375\) x 100 Percent of Increase = \(37.5\%\) #Finding Sales Tax Rate for Melliah's Purchase#

Calculating Sales Tax Rate

To find the sales tax rate, we can use the following formula: Sales Tax Rate = \(\dfrac{Sales \: Tax \: Amount}{Purchased \: Item \: Price}\) x 100 Melliah's purchase has given us the sales tax amount as \(\$ 3.85\) and the purchased item price as \(\$ 54.95\). Using these values, we can calculate the sales tax rate as follows: Sales Tax Rate = \(\dfrac{\$ 3.85}{\$ 54.95}\) x 100 Sales Tax Rate ≈ \(0.0700\) x 100 Sales Tax Rate ≈ \(7\%\) Hence, the sales tax rate for Melliah's tennis racquet purchase is approximately \(7\%\).

Key concepts

Initial Amount

Understanding the concept of the initial amount is crucial when calculating percentages, particularly the percent of increase. An initial amount refers to the starting value, before any changes, like increases or decreases, take place. In real-world scenarios, this could be the original price before a discount or the original revenue before growth.

For instance, when a company's revenue jumps from \(\$40,000\) to \(\$55,000\), the initial amount is \(\$40,000\). This is the benchmark against which any increase is measured. It’s essential to correctly identify this number, as it will be the denominator when setting up proportions to find percent changes.

Final Amount

The term final amount refers to the value of an item or total after an increase or decrease has taken place. It's essential when calculating the percentage change, as it represents the new state of affairs.

To illustrate this with a business example, if the business's revenue was \(\$40,000\) in the first month and then grew to \(\$55,000\) the following month, \(\$55,000\) represents the final amount. It's also important to note that the difference between the final and initial amounts represents the actual increase, which, in this case, is \(\$15,000\), not the initially mentioned \(\$5,000\).

Setting Up a Proportion

To uncover the percent of increase, we set up a proportion. A proportion is an equation that states that two ratios are equal. It’s a central tool in finding out how one quantity relates to another in terms of percentage.

For example, to find the percent of an increased revenue, we compare the increase amount (numerator) to the initial amount (denominator) and then multiply by 100. Mathematically, this looks like \(\frac{Increase \: Amount}{Initial \: Amount} \times 100\). Always remember, to keep your ratios consistent, and ensure that both terms of the proportion relate to the same thing, in this case, the revenue amount.

Sales Tax Rate Calculation

Understanding how to calculate the sales tax rate plays a vital role in everyday financial literacy. The sales tax rate gives you the percentage of the tax charged on a purchase.

It's calculated by dividing the sales tax amount by the price of the item before tax, and then multiplying by 100 to get the percentage. Here's the formula: \(Sales \: Tax \: Rate = \frac{Sales \: Tax \: Amount}{Purchased \: Item \: Price} \times 100\). This formula is demonstrated by Melliah’s tennis racquet purchase, where the purchase price was \(\$54.95\), and the tax was \(\$3.85\). The calculated sales tax rate of approximately 7% reveals the additional cost incurred due to taxation on the item.