Problem 18 - Algebra 2 and Trigonometry

Question

In \(18-23,\) write and solve an equation or an inequality to solve the problem. Peter had 156 cents in coins. After he bought 3 packs of gum he had no more than 9 cents left. What is the minimum cost of a pack of gum?

Step-by-Step Solution

Answer

The minimum cost of a pack of gum is 49 cents.

Step 1: Understand the Problem

Peter starts with 156 cents. After buying 3 packs of gum, he has no more than 9 cents left. We need to find the minimum cost of each pack of gum.

Step 2: Set Up the Inequality

Let the cost of one pack of gum be represented by \( x \) cents. The inequality capturing the scenario is: \[ 156 - 3x \leq 9 \] This represents that after buying the gum, Peter has no more than 9 cents left.

Step 3: Isolate the Variable

First, subtract 156 from both sides of the inequality: \[ -3x \leq 9 - 156 \] which simplifies to: \[ -3x \leq -147 \]

Step 4: Solve for x

Divide each side by -3, and since we are dividing by a negative number, we reverse the inequality sign: \[ x \geq 49 \] This means that each pack of gum must cost at least 49 cents.

Key Concepts

Cost calculation

The concept of cost calculation is straightforward. It's about adding up expenses and determining how much something costs overall. In our problem about Peter and his purchase of gum, we are interested in calculating how much each pack of gum costs, given a budget constraint. Peter starts with a total of 156 cents, and after purchasing the gum packs, the statement reveals that he is left with no more than 9 cents. This situation sets the stage for calculating the minimum price per pack.
The calculation proceeds by assessing the total monetary departure from starting balance, focusing on how much money he uses to buy the gum. Once we know how much his coins were reduced, we can deduce the minimum cost per gum pack by distributing this expense evenly across them.
This approach involves understanding both the initial and final amount of money and systematically using this information to determine expenses, which is the backbone of any basic cost calculation.

Variables in algebra

Variables are crucial in algebra. They act as placeholders for numerical values that we aim to find. In the context of the problem with Peter, we used a variable, denoted as \( x \), to stand for the unknown cost of a single gum pack.

• The variable allows us to construct equations or inequalities to model real-life situations mathematically.
• It provides flexibility. Instead of having a fixed number, we can adjust and solve for it, depending on varying circumstances.

In our problem, the variable \( x \) fits into the inequality \( 156 - 3x \leq 9 \). Here, it captures Peter's expenses (total expenses on gum = \( 3x \)). Once you have this inequality set, solving for \( x \) gives the minimal cost of each gum pack, bringing the power of algebraic manipulation to life in practical scenarios.

Inequality reversal

An essential part of solving inequalities involves understanding and accurately applying inequality reversal. This concept comes into play when you multiply or divide both sides of an inequality by a negative number.

• In our problem, when we got to the inequality \( -3x \leq -147 \), a division by -3 was necessary.
• Because we divided by a negative, we reversed the inequality sign, changing \( \leq \) to \( \geq \). This step ensures that the inequality rightly reflects the conditions of the problem post transformation.

Such a reversal might seem simple, but it underpins a foundational principle in algebraic operations with inequalities. Ignoring it could result in incorrect solutions. Remember, whenever you multiply or divide an inequality by a negative value, flip the inequality sign. This step is crucial to accurately solving and interpreting inequalities.