Problem 87 - Algebra 1

Question

You have 32 coins in a jar. Each coin is either copper or silver. You have 8 more copper coins than silver coins. Let \(c\) be the number of copper coins. Which equation correctly models the situation? (Review 1.5) A. \((c-8)+c=32\) B. \(c+(c+8)=32\)

Step-by-Step Solution

Answer

The equation that correctly models the situation is \((c-8) + c = 32\), which corresponds to option A

Step 1: Identify the variables

The problem involves two types of coins: copper and silver coins. Let's represent the number of copper coins as \(c\) and the number of silver coins as \(c-8\), since there are 8 fewer silver coins than copper coins.

Step 2: Create the equation

According to the problem, the total number of coins equals the number of silver coins plus the number of copper coins. This allows us to create the equation: \((c-8) + c = 32\). This equation reflects both the difference in quantity between the two types of coin and the total number of coins.

Step 3: Match the equation

The equation \((c-8) + c = 32\), obtained from the problem correctly matches with option A from the given alternatives. This equation accurately represents the problem's constraints.

Key Concepts

Coin Problems

Coin problems are a common topic in algebra, often used to teach students the concepts of variable representation and equation formulation. These problems typically involve a collection of coins of different types and a set of conditions concerning their numbers or values.
Solving a coin problem typically involves:

• Defining variables to represent the unknown quantities of coins.
• Understanding the relationships between these quantities (e.g., more of one type than another).
• Formulating equations that mirror these relationships, based on given totals.

Coin problems not only help students practice setting up equations but also improve their ability to translate real-world scenarios into mathematical terms. This skill is foundational for tackling more complex algebraic problems.

Variable Identification

In algebra, identifying the right variables is crucial to solving any problem effectively. When tackling a coin problem, begin by focusing on what you need to find: the quantities of different types of coins.
In the example problem, since we need to find the number of copper coins, we define the variable:

• Let \( c \) represent the number of copper coins.
• Then, establish a relationship for other types of coins; for instance, if there are 8 more copper coins than silver coins, the number of silver coins can be expressed as \( c - 8 \).

By properly identifying these variables, we lay the foundation for forming an accurate equation that reflects the problem's conditions. This identification process requires careful reading and interpreting of the problem's text to spot crucial information about the quantities and their relationships.

Equation Formulation

Formulating equations is a vital step in solving algebra problems. Once variables are defined, you turn the relationships and conditions described in the problem into mathematical equations.
In this problem, the total number of coins is 32. You know:

• Copper coins: \( c \)
• Silver coins: \( c - 8 \)

Combine these to express the total number of coins:\[(c - 8) + c = 32\]This equation effectively represents the scenario where the number of copper coins plus the number of silver coins equals 32, considering there are 8 more copper coins than silver ones.
By setting up this equation correctly, you're able to find the solution to how many of each type of coin you have. The ability to write and solve such equations is a fundamental skill in algebra, as it translates a verbal problem description into a solvable mathematical representation.