Question

Given a percent of decrease and an initial amount, explain how to find the final amount after the decrease. Rachel paid a total of \(\$ 45.10\) for a pair of shoes. She knows that the sales tax rate in her state is \(5 \% .\) What was the initial price of the shoes?

Short answer

Answer: The initial price of the shoes was $47.47.

Step-by-step solution

Convert the percentage of decrease to a decimal

To convert the given percentage of decrease (5%) to a decimal, we divide the percentage by 100. So we have: \(5\%/100 = 0.05\)

Calculate the decimal equivalent of the tax amount

Since the 5% sales tax is added to the initial price, we must find the decimal equivalent with which we need to decrease the final amount (45.10). We subtract the decimal equivalent of the sales tax (0.05) from 1: \(1 - 0.05 = 0.95\)

Find the factor

We now have the decimal factor (0.95) that will help us find the initial price of the shoes. This factor represents the proportion of the initial price after considering the sales tax.

Calculate the initial price of the shoes

Now, we can divide the final amount by the factor (0.95) to get the initial price of the shoes: Initial price = \(\$45.10/0.95 = \$47.47\) So, the initial price of the shoes was \(\$47.47\).

Key concepts

Percent of Decrease

When we talk about a 'percent of decrease', we are referring to how much a certain quantity is reduced in terms of a percentage. This concept is a cornerstone in everyday transactions, especially in calculating discounts or sales. Imagine you see an item with a sign saying '10% off' - that is a percent of decrease in action. To calculate this effect on a price, you simply take the original amount and reduce it by the given percentage.

For instance, if a sweater originally costs \(50 and the store advertises a 20% off sale, you would calculate the amount of the discount as follows: Convert the percentage decrease to a decimal by dividing it by 100 (20% / 100 = 0.20), and then multiply that decimal by the original price (\)50 * 0.20 = \(10). So, the sweater's price decreases by \)10, making the final amount \(40.

Sales Tax Calculation

Sales tax is an additional amount of money, calculated as a percentage, that is added to the cost of goods or services when purchased. Sales tax rates vary depending on the region. For example, if the sales tax rate is 8%, that means for every dollar you spend, an additional 8 cents is added as tax.

To calculate the total amount including sales tax, you add the tax to the initial price of the item. If an item costs \)100 with an 8% sales tax, you convert 8% to a decimal (0.08) and then multiply the price of the item by this decimal to get the tax amount (\(100 * 0.08 = \)8). You then add that to the initial price (\(100 + \)8) for a total of $108.

Converting Percentage to Decimal

Converting a percentage to a decimal is an essential step in various financial calculations, such as determining sales tax or discounts. The process is straightforward: simply divide the percentage by 100. This moves the decimal point two places to the left.

For example, a 25% discount rate or sales tax would be converted to a decimal as follows: 25% divided by 100 equals 0.25. It’s crucial to make this conversion because you use the decimal form to multiply against a total or principal amount to find the portion of decrease or the additional sales tax.

Determining Final Amount After Decrease

To find the final amount after a decrease - be it a discount, depreciation, or after subtracting tax - you first convert the percentage decrease to a decimal. Then, subtract this decimal from 1 to find what portion of the original amount remains. Finally, multiply the original amount by this new decimal to determine the reduced final amount.

Using the shoe example, to find the price before tax when the final amount with tax is known, you reverse the operation. After converting the tax rate to a decimal (0.05), you'd subtract it from 1 to find the remaining percentage of the initial price (1 - 0.05 = 0.95). Then, divide the final amount including tax by 0.95 to arrive at the price before tax. The calculation reveals that the initial price, before adding the tax, was lower than the final amount paid.