Problem 38 - Elementary Algebra

Question

Set up an equation and solve each problem. (Objectives 2 and 3) A retailer has some skirts that cost her \(\$ 24\). If she sells them for \(\$ 31.20\), find her rate of profit based on the cost.

Step-by-Step Solution

Answer

The rate of profit is 30\%.

Step 1: Identify the Cost and Selling Price

The cost to the retailer for each skirt is given as \(\\( 24\), and the selling price is \(\\) 31.20\).

Step 2: Calculate the Profit Per Skirt

Profit is calculated as the selling price minus the cost price. \[\text{Profit} = \text{Selling Price} - \text{Cost Price} = 31.20 - 24 = \$ 7.20\]

Step 3: Calculate the Rate of Profit

The rate of profit is calculated on the cost price and is given by the formula:\[\text{Rate of Profit} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\%\]Substitute the known values into the formula:\[\text{Rate of Profit} = \left( \frac{7.20}{24} \right) \times 100\% = 30\%\]

Key Concepts

Equation Setup

Starting with the equation setup is crucial to understanding how to solve profit-related problems systematically. In our exercise, we are given two values: the cost price of the skirts and the selling price. The objective is to determine the rate of profit. The first step is to express the profit in terms of these two values. We set up our basic equation for profit as:

• Profit = Selling Price - Cost Price

This simple equation allows us to calculate the actual monetary profit from selling one skirt. Once that's known, we can proceed to find the rate of profit. Remember, setting up the right equation helps to logically arrange the problem before diving into the calculation.

Cost Price

The cost price plays a pivotal role in profit calculations. In our scenario, the cost price of a skirt is given as \( \$24 \). This value is the baseline for all profit calculations since it represents what the retailer paid for each item. Knowing the cost price:

• Allows us to determine how much extra is being made when the item is sold at a higher price.
• Helps in setting both the selling price strategy and profit targets.

It's beneficial to always keep track of cost prices in any retail business because it directly influences profit margins.

Selling Price

The selling price of an item is what the customer pays when they purchase it. For the skirts in this exercise, the selling price is \( \$31.20 \). The selling price is crucial for determining how profitable a sale is in comparison to the cost price. Key points about selling price:

• A higher selling price than the cost price results in profit.
• Setting an attractive selling price can influence customer buying decisions.

In this exercise, the difference between the selling price and the cost price gives us the numerical profit, which is required to calculate the rate of profit.

Rate of Profit

The rate of profit is an essential metric as it shows profitability in percentage terms, making it easier to compare across different products or time periods. In our example, once we know the profit is \( \$7.20 \), we can calculate the rate of profit using this formula:\[ \text{Rate of Profit} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100\% \]Why it matters:

• The rate provides insight into how efficiently a business is generating profit relative to its costs.
• A higher rate indicates better profitability.

With a \(30\%\) rate of profit, as calculated, the retailer knows that for every dollar spent on cost, \(30\%\) is being earned as profit.