Problem 65 - Algebra for College Students

Question

Solve each equation and express the solutions in decimal form. Be sure to check your solutions. Use your calculator whenever it seems helpful. A retailer buys an item for \(\$ 90\), resells it for \(\$ 100\), and claims that she is making only a \(10 \%\) profit. Is this claim correct?

Step-by-Step Solution

Answer

The retailer's profit percentage is approximately 11.11%, not 10%.

Step 1: Understand the Problem Statement

The retailer claims a 10% profit on an item bought for $90 and sold for $100. To verify this claim, we'll calculate the actual profit percentage.

Step 2: Determine the Profit

Calculate the profit made by subtracting the cost price (CP) from the selling price (SP). Here, the SP is \(100, and the CP is \)90.\[\text{Profit} = \text{Selling Price} - \text{Cost Price} = 100 - 90 = 10\]

Step 3: Calculate the Profit Percentage

Use the formula for profit percentage, which is\[\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\]Substitute the values obtained:\[\text{Profit Percentage} = \left( \frac{10}{90} \right) \times 100 \approx 11.11\%\]

Step 4: Compare Calculated Profit Percentage With Claim

The calculated profit percentage is \(11.11\%\), while the retailer claims a \(10\%\) profit. Therefore, the retailer's claim is not correct.

Key Concepts

Profit Calculation

Profit calculation is an essential concept in understanding how much gain is earned from a business transaction. When you buy something at one price and sell it at a higher price, the difference is known as a profit. To calculate the profit:

• Identify the Cost Price (CP), which is the initial amount spent to purchase the item.
• Determine the Selling Price (SP), the price at which the item is sold.
• Subtract the Cost Price from the Selling Price: \[\text{Profit} = \text{Selling Price} - \text{Cost Price}\]

Using the example from the original problem, the item was bought for \(90 and sold for \)100, generating a profit of $10.Understanding this concept allows you to analyze business performance by assessing the profitability of selling activities.

Cost Price and Selling Price

The concepts of Cost Price (CP) and Selling Price (SP) are key to comprehending any business's financial transactions.

• Cost Price: This is the amount one spends to acquire a product. It's important to note that costs involved in the acquisition may sometimes include additional expenses such as shipping or taxes.
• Selling Price: This refers to the price at which the product is sold to the customer. Ideally, this should be higher than the Cost Price to realize a profit.

In the retailer scenario, the Cost Price was $90, and the Selling Price was $100. The difference between these two amounts not only determines the profit but is also critical for calculating the profit percentage.

Decimal Form Solutions

Presenting solutions in decimal form is a vital skill, especially when working with percentages and other fractional values. Decimal representation can make it easier to understand and communicate precise values.When calculating the profit percentage, you might start with a fraction:\[\text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\]In our example, the fraction for the profit percentage is \(\frac{10}{90}\). To express this as a decimal, use division to convert the fraction:

• \(\frac{10}{90} \approx 0.1111\), which translates to a profit percentage of \(11.11\%\) when multiplied by 100.

By converting the fractional expression to decimal form, you can easily verify whether a claim made about a percentage is accurate. In this case, the retailer claimed a 10% profit, yet the calculated decimal form revealed an 11.11% profit, proving the claim incorrect.