Allison has 5 times as many baseball cards as football cards. In all, she has 120 baseball and football cards. How many baseball cards does Allison have?
The correct answer and explanation is :
To determine how many baseball cards Allison has, let’s define variables and set up an equation.
Step 1: Define Variables
Let:
- ( x ) be the number of football cards Allison has.
- Since she has 5 times as many baseball cards as football cards, the number of baseball cards she has is ( 5x ).
Step 2: Set Up the Equation
The problem states that the total number of baseball and football cards is 120. This gives the equation:
[
x + 5x = 120
]
Step 3: Solve for ( x )
[
6x = 120
]
[
x = \frac{120}{6} = 20
]
So, Allison has 20 football cards.
Step 4: Find the Number of Baseball Cards
Since she has 5 times as many baseball cards:
[
5 \times 20 = 100
]
Thus, Allison has 100 baseball cards.
Step 5: Verify the Answer
Checking the total:
[
100 + 20 = 120
]
which matches the given total, confirming our answer is correct.
Explanation (300 words)
This problem is an example of an algebraic word problem that requires setting up and solving an equation. We started by defining variables: the number of football cards as ( x ) and the number of baseball cards as ( 5x ). Since the total number of both cards is given as 120, we wrote the equation:
[
x + 5x = 120
]
This equation expresses that the sum of football and baseball cards must equal 120. By simplifying, we combined like terms:
[
6x = 120
]
Next, we solved for ( x ) by dividing both sides by 6:
[
x = 20
]
This means Allison has 20 football cards. Since the number of baseball cards is 5 times that, we computed:
[
5 \times 20 = 100
]
Thus, she has 100 baseball cards.
To verify, we checked that ( 100 + 20 = 120 ), confirming that our calculations are correct.
This problem demonstrates how algebra simplifies complex relationships into solvable equations. By translating the word problem into an equation, solving for ( x ), and verifying the solution, we ensured accuracy. This method is widely used in real-world applications, such as finance and engineering, where relationships between quantities must be determined systematically.