Problem 315 - Elementary Algebra

Question

In the following exercises, solve. Joni sells kitchen aprons online for \(\$ 32.50\) each. How many aprons must she sell next month if she wants to earn at least \(\$ 1,000 ?\)

Step-by-Step Solution

Answer

Joni must sell at least 31 aprons.

Step 1: Identify the variables

Let the number of aprons Joni needs to sell be represented by the variable \(x\). The price per apron is given as \(\$ 32.50\) and Joni's target earning is \(\$ 1,000\).

Step 2: Set up the inequality

Joni wants to earn at least \(\$ 1,000\). Therefore, set up an inequality where the product of the number of aprons sold and the price per apron is greater than or equal to \(\$ 1,000\). This can be written as: \[ 32.50x \geq 1,000 \]

Step 3: Solve the inequality

To find the value of \(x\), divide both sides of the inequality by \(32.50\): \[ x \geq \frac{1,000}{32.50} \] Compute the division: \[ x \geq 30.77 \]

Step 4: Interpret the result

Since the number of aprons must be a whole number, Joni must sell at least the next whole number that is greater than \(30.77\), which is \(31\).

Key Concepts

variables in algebra

Understanding variables in algebra is key to setting up and solving problems effectively. In this exercise, the variable represents the number of aprons Joni needs to sell. We use the letter \(x\) to stand for this unknown amount. By using variables, we can write mathematical expressions that relate to real-world situations. This approach helps to generalize and solve problems where specific values might change. It's like setting up a placeholder that we can manipulate mathematically to find a solution.

setting up inequalities

Setting up inequalities involves translating a real-world requirement into a mathematical condition. Here's how we did it for Joni's apron business:

• First, identify the key quantities: price per apron and the target earnings.
• Next, relate these quantities using an inequality. Since Joni wants to earn at least \(\$1,000\), we write the inequality as \(32.50x \geq 1,000\).

This inequality shows that the product of the number of aprons (\(x\)) and the price per apron (\(\$32.50\)) must be at least \(\$1,000\). Inequalities help us model conditions like 'at least,' 'at most,' and 'greater than' in a mathematical form.

solving inequalities

Solving inequalities involves finding the range of values for the variable that make the inequality true. Here, we want to determine how many aprons Joni needs to sell. To solve, we:

• Isolate the variable on one side of the inequality. For our problem, we divide both sides by \(\$32.50\): \[ x \geq \frac{1,000}{32.50} \]
• Perform the division to get \( x \geq 30.77 \).

This calculation shows that to meet or exceed \(\$1,000\), Joni needs to sell at least \(30.77\) aprons. Since she can't sell a fractional apron, we round up to the next whole number, which means she needs to sell at least \(31\) aprons.

interpreting results

Interpreting the results is about understanding the solution in the context of the problem. After solving the inequality \( x \geq 30.77 \), we conclude that Joni must sell at least \(31\) aprons to achieve her target. This step is crucial because mathematical solutions need to make practical sense.

• We recognize that the number of aprons must be a whole number.
• Thus, Joni should aim for a minimum of \(31\) aprons to earn at least \(\$1,000\).

This interpretation ensures we apply the mathematical solution appropriately in real-life scenarios. Always check if your solution fits the context of the problem.