Problem 48 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Jody has a collection of 116 coins consisting of dimes, quarters, and silver dollars. The number of quarters is 5 less than three-fourths of the number of dimes. The number of silver dollars is 7 more than five-eighths of the number of dimes. How many coins of each kind are in her collection?

Step-by-Step Solution

Answer

Jody has 48 dimes, 31 quarters, and 37 silver dollars.

Step 1: Define the Variables

Let's define the variables for the problem. Let \( d \) be the number of dimes, \( q \) be the number of quarters, and \( s \) be the number of silver dollars.

Step 2: Set Up the Equations

We will set up equations based on the information given:1. The total coins are 116: \( d + q + s = 116 \).2. The quarters are 5 less than three-fourths of the number of dimes: \( q = \frac{3}{4}d - 5 \).3. The silver dollars are 7 more than five-eighths of the number of dimes: \( s = \frac{5}{8}d + 7 \).

Step 3: Substitute and Solve for \( d \)

Substitute \( q \) and \( s \) from the second and third equations into the first equation:\[ d + \left( \frac{3}{4}d - 5 \right) + \left( \frac{5}{8}d + 7 \right) = 116 \]Combine like terms:\[ d + \frac{3}{4}d + \frac{5}{8}d - 5 + 7 = 116 \]Convert \( d \) terms to a common denominator:\[ \frac{8}{8}d + \frac{6}{8}d + \frac{5}{8}d = \frac{19}{8}d \]So:\[ \frac{19}{8}d + 2 = 116 \]Subtract 2 from both sides:\[ \frac{19}{8}d = 114 \]Multiply both sides by \( \frac{8}{19} \):\[ d = \frac{8 imes 114}{19} = 48 \]

Step 4: Find \( q \) and \( s \) Using \( d \)

Now that we know \( d = 48 \), we can find \( q \) and \( s \):1. \( q = \frac{3}{4}(48) - 5 = 36 - 5 = 31 \)2. \( s = \frac{5}{8}(48) + 7 = 30 + 7 = 37 \)

Step 5: Verify the Solution

Finally, verify that these values satisfy the original condition that the total is 116 coins:\[ 48 + 31 + 37 = 116 \]This confirms that the values are correct.

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations that involve several variables. In the case of Jody's coin collection, we've identified a system with three variables: dimes (\( d \)), quarters (\( q \)), and silver dollars (\( s \)). Each equation in a system works together, providing different pieces of information about the same variables. To solve a system of equations, the goal is to find values for those variables that satisfy all the given conditions.

In the problem above, the system is represented by three equations:

• The total number of coins, given by \( d + q + s = 116 \).
• The relationship between quarters and dimes, \( q = \frac{3}{4}d - 5 \).
• The relationship between silver dollars and dimes, \( s = \frac{5}{8}d + 7 \).

By carefully setting up these equations, it's possible to use methods like substitution (which we'll cover in detail!) to find the solution, which reveals how many of each type of coin Jody has. Solving systems of equations is a fundamental skill in algebra, useful in many real-life situations where multiple conditions have to be met at once.

Substitution Method

The substitution method is an algebraic technique used to solve systems of equations. It involves expressing one variable in terms of another using one of the equations and then substituting this expression into another equation. This method is particularly useful when one of the equations is easy to manipulate to express one variable in terms of others.

For Jody's coin problem, the substitution method is applied as follows:

• First, express \( q \) and \( s \) in terms of \( d \) using the given relationships: \( q = \frac{3}{4}d - 5 \) and \( s = \frac{5}{8}d + 7 \).
• Then, substitute these expressions into the total coin equation \( d + q + s = 116 \). This simplifies the system into a single equation with one variable.
• Solve this simplified equation for \( d \), leading to the solution \( d = 48 \).
• Finally, use the value of \( d \) to find \( q \) and \( s \) by plugging it back into the expressions derived from the original substitutions.

By using substitution, the problem becomes more straightforward, allowing you to solve complex systems with multiple variables step-by-step.

Variables in Algebra

Variables are fundamental components of algebra. They are symbols, often denoted by letters, that represent unknown or changeable values. Understanding how to work with variables is crucial for solving algebraic equations and systems of equations.

In the exercise about Jody's coins, variables \( d \), \( q \), and \( s \) are introduced to stand for the number of dimes, quarters, and silver dollars, respectively. These variables allow us to write equations that capture the relationships and constraints given in the problem:

• \( d + q + s = 116 \) summarizes the total coin count.
• \( q = \frac{3}{4}d - 5 \) expresses how quarters depend on the number of dimes.
• \( s = \frac{5}{8}d + 7 \) relates silver dollars to dimes.

Handling variables, interpreting their roles in equations, and manipulating them mathematically are essential skills. These are essential to transform and solve equations effectively. By learning to identify and work with variables, you unlock the potential to explore and solve a wide variety of mathematical problems.