Problem 7 - Prealgebra

Question

Koto delivers pizzas on weekends. Her average tip is \(\$ 1.50\) for each pizza that she delivers. How many pizzas must she deliver to earn at least \(\$ 20\) in tips? $$\mathbf{A} 10$$ $$\mathbf{B} 13$$ $$\mathbf{C} 14$$ $$\mathbf{D} 20$$

Step-by-Step Solution

Answer

Koto must deliver at least 14 pizzas to earn at least $20 in tips.

Step 1: Understanding the Problem

Koto earns an average tip of $1.50 per pizza delivered. Our goal is to find out how many pizzas she needs to deliver to earn at least $20 in tips.

Step 2: Set Up the Equation

To determine the number of pizzas, we use the equation for total tips earned: number of pizzas × average tip per pizza ≥ required total tips. This becomes: \( 1.50 \, \times \, x \geq 20 \).

Step 3: Solve the Equation for x

Rearrange our equation to solve for x: first divide both sides by 1.5 to isolate x. Thus, we have \( x \geq \frac{20}{1.5} \).

Step 4: Calculate the Division

Divide 20 by 1.5 to find the minimum number of pizzas: \( \frac{20}{1.5} = 13.33 \).

Step 5: Determine the Minimum Integer Solution

Since Koto cannot deliver a fraction of a pizza, we round 13.33 up to the nearest whole number, which means Koto must deliver at least 14 pizzas.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They are often represented by symbols such as \(<\), \(>\), \(\leq\), and \(\geq\). In the context of Koto's pizza delivery problem, the inequality \(1.50 \times x \geq 20\) is used to determine the minimum number of pizzas Koto needs to deliver to earn at least \(\$20\) in tips. This inequality tells us that the product of 1.50 (the average tip per pizza) and \(x\) (the number of pizzas) must be greater than or equal to 20.

To solve this type of inequality, you need to isolate the variable \(x\) on one side. In Koto's example, this is achieved by dividing both sides of the inequality by 1.50, resulting in \(x \geq \frac{20}{1.5}\). From here, we calculate \(\frac{20}{1.5}\), which simplifies to approximately 13.33. Because it doesn't make sense to deliver a fraction of a pizza, we round up to the nearest whole number, which is 14.

Problem Solving

Problem solving in math often involves a series of logical steps to find a solution, which can be both analytical and creative. The process begins with understanding the problem. Here, Koto wants to earn at least \(\$20\) delivering pizzas.

After understanding the problem, the next step is to set up a mathematical model, which is an equation or inequality that represents the situation. In Koto's case, this involved creating the inequality \(1.50 \times x \geq 20\). Setting up this model helps to visualize what calculations are necessary to find the solution.

The final step involves solving the inequality for \(x\), which includes performing mathematical operations such as division and rounding up to ensure the number meets the real-world constraints of the problem. This approach of breaking down a problem into smaller, manageable steps is a central skill in algebra and beyond, as it allows for tackling more complex problems efficiently.

Real-World Applications

Algebra and inequalities are not just numbers on a page; they model real-world scenarios. In Koto's situation, we're applying basic algebra to a practical problem: optimizing efforts to reach financial goals. This showcases how algebra helps in budgeting and planning.

With inequalities, you learn to deal with situations where exact matches aren't necessary; rather, finding a satisfactory range is the key. For instance, while calculating areas such as minimum wage calculations, budgeting, and even in project management, setting up inequalities helps define limits within which a solution must fall.

Real-world applications make these concepts relevant and relatable. By practicing with problems like Koto's pizza delivery, students can see how algebra helps in decision-making and planning, emphasizing its value beyond classroom exercises. It also encourages logical thinking, ensuring that the solutions fit into realistic scenarios, like rounding 13.33 up to 14 pizzas because delivering a fraction of a pizza is impractical.