Problem 62 - Algebra 1

Question

Use the following information. A person has quarters, dimes, and nickels with a total value of 500 cents ( 5.00 dollar). The number of nickels is twice the number of quarters. The number of dimes is four less than the number of quarters. Explain why the expression \(5(2 q)\) represents the value of the nickels if \(q\) represents the number of quarters. How can you simplify the expression?

Step-by-Step Solution

Answer

In this problem, the expression \(5(2q)\) represents the total value of the nickels in cents. This is because the number of nickels is twice the number of quarters (\(2q\)) and each nickel is worth 5 cents. So, the value of the nickels is the value of one nickel (5 cents) times the number of nickels (\(2q\)). The expression can be simplified as \(10q\), where 10 is the result of the product of 5 and 2.

Step 1: Understanding the Coin Values

Each type of the coins has a specific value. A nickel is worth 5 cents, a dime is worth 10 cents and a quarter is worth 25 cents. So when calculating the total value of a number of the same coin type, you would multiply the number of coins by the coin's value in cent.

Step 2: Explaining the Expression

In the absence, the number of nickels is given as twice the number of quarters, represented as \(2q\). Since each nickel is worth 5 cents, to find the value of all these nickels, you multiply \(2q\) by 5. The expression \(5(2q)\) therefore correctly represents the total value of the nickels, in cents.

Step 3: Simplify the Expression

To simplify the expression \(5(2q)\), you just have to follow the multiplication operation in it. The multiplication of 5 and 2 gives 10, so the simplified expression is \(10q\), meaning the value of the nickels is 10 times the number of quarters.

Key Concepts

Equation Solving

Equation solving is a fundamental concept in algebra. It involves finding the value of the unknown variable that satisfies a given equation. In the context of coin problems, you're often dealing with variables representing quantities, like the number of different types of coins.
Whenever you encounter a word problem, it's essential to turn the words into equations. This means identifying known quantities and relationships and expressing them using mathematical language. For example, if you know the number of nickels is twice the number of quarters, you can express this as the equation:

• Number of nickels = 2 × (Number of quarters)

Appending value to these coins allows you to form expressions, as shown in the original exercise. This builds a literal equation which can be further analyzed.
In our exercise, besides using numbers, letters like "q" for quarters help convert scenarios into solvable algebraic formats. Understanding which operations to perform means identifying relationships and ensuring you include all necessary components (like coin value). Thus, solving these equations step by step helps find out the quantities in question.

Coin Word Problems

Coin word problems are a classic type of algebra problem. They involve figuring out the quantities of different coins based on certain conditions or relationships given in the problem.
To approach these problems effectively, it's helpful to:

• Identify what you are solving for: Understand the relationship between different coins. In this case, nickels, dimes, and quarters.
• Know the value: Comprehend the value of each type of coin as this affects your calculations. Recall that quarters are 25 cents, dimes 10 cents, and nickels 5 cents each.
• Set up equations: Use the relationships given, like the number of nickels being twice the number of quarters, to create useful equations.
• Express total value: Look at the total value of all coins together to guide further equation formation.

By laying out these elements, the word problem transforms from just statements to actionable equations that can be solved through algebra.

Simplification of Expressions

Simplification of expressions is about making a complex expression easier to understand and use. In algebra, this process often involves combining like terms or performing arithmetic operations across the expression.
In our context, you might have an expression like \(5(2q)\). To simplify, you need to understand and perform multiplication, which significantly reduces the complexity. The operation here would be:

• First, multiply the numbers outside and inside the parentheses: \(5 \times 2\).
• This results in \(10q\), an expression that clearly indicates you multiply 10 by the number of quarters to find the total value in cents of the nickels.

Such simplifications help in quickly comprehending the quantity without losing key relationships described in the original problem. The goal is to arrive at a form that is both accurate and easy to interpret.