Problem 51 - Algebra 1

Question

Write and solve an equation to answer the question. You ate three of the eight pizza slices and you paid \(\$ 3.30\) as your share of the cost. How much did the whole pizza cost?

Step-by-Step Solution

Answer

The cost of the whole pizza is approximately $7.27.

Step 1: Setting up the Proportion

Firstly, form a proportion relationship between the number of slices and the cost of each slice. This can be denoted as \(\frac{Cost}{Slices}\). The given problem states that 3 slices cost $3.30. So we can write the proportion as \(\frac{3.30}{3} = \frac{x}{8}\), where x denotes the cost of the whole pizza.

Step 2: Solving the equation

To get the cost of 8 slices of pizza, simply cross multiply and solve for x. Therefore, \(3^*8 = 3.30^*x\). Which simplifies to \(24 = 3.3^*x\).

Step 3: Find the total cost

Finally solve for x by dividing both sides by 3.3, i.e. \(x = \frac{24}{3.3}\), which is approximately $7.27 when rounded to the nearest hundredth.

Key Concepts

Proportion Problems

When encountering a proportion problem, it's like piecing together a puzzle of parts and wholes. Imagine you are breaking down an item into equal parts, such as slices of a pizza, and you aim to find the cost of the entire item based on knowing the price of a single part. In the given exercise,
you ate three slices of an eight-slice pizza and paid \(3.30. The question then becomes a quest to determine the whole based on these parts. This forms a proportion, which is a statement that two ratios are equal. The key to solving these puzzles is to set up the proportion correctly, which involves establishing a ratio that represents the known parts (3 slices for \)3.30) and an equivalent ratio that represents the entire item (the whole pizza cost in relation to 8 slices).

This skill is valuable, as proportion problems are common in real-world situations—be it in cooking recipes, shopping discounts, or even converting units in scientific experiments.

Writing Equations

Equations are the language of mathematics—they help us translate scenarios into algebraic expressions that can be solved to find unknown values. Writing an equation involves identifying the quantities involved and the relationships between them. In the context of our pizza problem, the slices and the cost are our essential quantities.
The relationship is the cost per slice, which remains constant—that is the crux of a proportion. Hence, the equation is built around this relationship. By equating the ratio of the cost to slices already consumed, \(\frac{3.30}{3}\), with the cost to total slices, \(\frac{x}{8}\), we create an equation that, when solved, will reveal the total cost. With practice, writing equations becomes intuitive, opening up a world of problem-solving possibilities.

Cross Multiplication

Cross multiplication is a deft maneuver in the mathematician's toolkit. It's a shortcut that simplifies the process of solving proportions. The 'cross' refers to the action of multiplying diagonally across the equals sign in a proportion. For instance, with the equation \(\frac{3.30}{3} = \frac{x}{8}\), we 'cross multiply' to get \(3.30 \times 8 = 3 \times x\).

There's a dance to it: one value from each ratio multiplies with the opposing value in the other, creating a bridge that connects the unknown with the known. This step converts our proportion into an ordinary equation: \(24 = 3.3 \times x\), leaving us just one step away from unveiling the total pizza cost. That final step is simple algebraic manipulation—another key concept on its own.

Algebraic Problem Solving

Where cross multiplication leaves off, algebraic problem-solving picks up the baton. Our problem now resembles a straightforward algebraic equation, with the cost \(x\) being our variable. The equation \(24 = 3.3 \times x\) is begging to be balanced, and the balancing act is performed through division.
By dividing both sides of the equation by 3.3, we isolate \(x\), the star of our show, revealing that the whole pizza costs approximately $7.27.

Why Divide?

We divide because it undoes the multiplication that has ensnared our variable, granting us a clear view of its value. It is through algebraic problem-solving that we extract meaning from abstract equations, turning symbols into solutions with real-world significance.