Problem 19 - Precalculus

Question

Carl's Sasquatch Attack! Game Card Collection is a mixture of common and rare cards. Each common card is worth $$\$ 0.25$$ while each rare card is worth $$\$ 0.75 .$$ If his entire 117 card collection is worth $$\$ 48.75,$$ how many of each kind of card does he own?

Step-by-Step Solution

Answer

Carl has 78 common cards and 39 rare cards.

Step 1: Define Variables

Let \( c \) represent the number of common cards and \( r \) represent the number of rare cards in Carl's collection.

Step 2: Set Up the Equations

Write two equations based on the given information. The first equation is the total number of cards equation: \[ c + r = 117 \] The second equation represents the total value of the cards: \[ 0.25c + 0.75r = 48.75 \]

Step 3: Solve One Equation for a Variable

From the first equation, solve for \( c \) in terms of \( r \): \[ c = 117 - r \]

Step 4: Substitute the Expression

Substitute \( c = 117 - r \) into the second equation: \[ 0.25(117 - r) + 0.75r = 48.75 \]

Step 5: Simplify the Equation

Expand and simplify the equation: \[ 29.25 - 0.25r + 0.75r = 48.75 \] Combine like terms:\[ 29.25 + 0.5r = 48.75 \]

Step 6: Solve for r

Isolate \( r \) by subtracting 29.25 from both sides:\[ 0.5r = 19.50 \]Divide by 0.5 to find \( r \):\[ r = 39 \]

Step 7: Solve for c

Use the expression for \( c \) from Step 3:\[ c = 117 - r \]Substitute \( r = 39 \): \[ c = 117 - 39 = 78 \]

Step 8: Verify the Solution

Verify the number of cards and the total value. The total number should be \( 39 + 78 = 117 \). Calculate the value:\[ 0.25 \times 78 + 0.75 \times 39 \]\[ = 19.50 + 29.25 \]\[ = 48.75 \]The values match the given total, confirming our solution.

Key Concepts

Linear Equations

Linear equations are the backbone of algebra. They represent relationships between variables using simple algebraic expressions. In this exercise, we are dealing with two linear equations:

• The first equation represents the total count of cards: \( c + r = 117 \)
• The second equation captures their total value: \( 0.25c + 0.75r = 48.75 \)

These equations are linear because they involve constant terms and variables raised only to the first power. Understanding how to express real-world situations through these simple relationships can help in solving various problems by finding unknowns.

Substitution Method

The substitution method is a powerful technique used to solve systems of linear equations. In this exercise, we used substitution to find the number of common and rare cards:

• First, solve one of the equations for one variable. We found \( c \) in terms of \( r \) from the equation \( c + r = 117 \), rewriting it as \( c = 117 - r \).
• Next, substitute \( c = 117 - r \) into the second equation, \( 0.25c + 0.75r = 48.75 \).
• This substitution yields a single equation in one variable, which can be solved to find \( r \).

Once \( r \) is known, substitute back to find \( c \). This systematic approach simplifies a two-variable problem into simpler steps, making it easier to handle.

Word Problems

Word problems require interpreting real-life scenarios into mathematical models. This requires identifying relevant information and determining which unknowns (variables) to solve for.Finding Carl's card collection count involves:

• Defining variables: Let \( c \) and \( r \) represent the number of common and rare cards.
• Setting up equations based on details given: number of cards and their value.
• Using these equations to form a solvable system.

Understanding what the problem is detailing and the relationships between quantities is key. As with Carl's cards, once we've represented the problem mathematically, the solution is within reach.

Problem Solving

Effective problem solving in math involves a clear, strategic approach:

• Read the problem carefully to understand what is required and which variables are involved.
• Translate words into equations that model these relationships practically.
• Solve the system using a method like substitution, as demonstrated in this exercise.
• Verify your results by checking if they satisfy the original problem.

These steps help demystify word problems by breaking them down. By systematically following these principles, you can tackle various mathematical challenges efficiently.