Write And Solve An Equation To Represent The Given Situation. Be Sure To Define Your Variable.Samantha Currently Has $1600 In The Bank And Is Spending $95 Per Week. How Many Weeks Will It Take Until Her Account Is Worth Only $1220?
Understanding the Problem
Samantha currently has $1600 in her bank account and is spending $95 per week. We need to determine how many weeks it will take until her account is worth only $1220. To solve this problem, we will use algebraic equations to represent the given situation.
Defining the Variable
Let's define the variable 'x' as the number of weeks it will take for Samantha's account to be worth $1220. We will use this variable to represent the unknown quantity in our equation.
Writing the Equation
The initial amount in Samantha's account is $1600. Each week, she spends $95, which means her account balance decreases by $95. After 'x' weeks, her account balance will be $1600 - $95x. We want to find the number of weeks 'x' when her account balance is $1220. Therefore, we can write the equation:
$1600 - $95x = $1220
Simplifying the Equation
To simplify the equation, we can subtract $1600 from both sides:
-$95x = -$380
Dividing Both Sides
Next, we can divide both sides of the equation by -$95 to solve for 'x':
x = -$380 / -$95
Solving for 'x'
x = 4
Interpreting the Result
The result 'x = 4' means that it will take 4 weeks for Samantha's account to be worth only $1220.
Conclusion
In this problem, we used algebraic equations to represent the given situation and solve for the unknown quantity 'x'. By defining the variable, writing the equation, simplifying the equation, dividing both sides, and solving for 'x', we were able to determine that it will take 4 weeks for Samantha's account to be worth only $1220.
Real-World Applications
This problem has real-world applications in finance and accounting. For example, it can be used to calculate the time it will take for a savings account to reach a certain balance, or to determine the number of weeks it will take for a business to pay off a loan.
Tips and Variations
- To make the problem more challenging, you can add more variables or constraints, such as a fixed interest rate or a minimum balance requirement.
- To make the problem easier, you can use a calculator or a computer program to solve the equation.
- You can also use this problem as a starting point to explore more advanced topics in algebra, such as quadratic equations or systems of equations.
Common Mistakes
- Failing to define the variable and write the equation correctly.
- Not simplifying the equation properly.
- Not dividing both sides of the equation correctly.
- Not interpreting the result correctly.
Practice Problems
- A person has $2000 in their savings account and is spending $50 per week. How many weeks will it take for their account to be worth only $1500?
- A business has $5000 in their checking account and is spending $200 per week. How many weeks will it take for their account to be worth only $3000?
- A person has $1000 in their savings account and is earning 5% interest per year. How many years will it take for their account to be worth $1500?
Answer Key
- 10 weeks
- 10 weeks
- 2 years
Frequently Asked Questions (FAQs) =====================================
Q: What is the main concept of the problem?
A: The main concept of the problem is to use algebraic equations to represent a real-world situation and solve for the unknown quantity.
Q: What is the variable 'x' in the problem?
A: The variable 'x' represents the number of weeks it will take for Samantha's account to be worth $1220.
Q: How do I define the variable in the problem?
A: To define the variable, you need to identify the unknown quantity in the problem and assign a letter to it. In this case, we defined the variable 'x' as the number of weeks it will take for Samantha's account to be worth $1220.
Q: What is the equation in the problem?
A: The equation in the problem is $1600 - $95x = $1220.
Q: How do I simplify the equation?
A: To simplify the equation, you need to perform algebraic operations such as adding, subtracting, multiplying, or dividing both sides of the equation.
Q: What is the result of the problem?
A: The result of the problem is x = 4, which means that it will take 4 weeks for Samantha's account to be worth only $1220.
Q: What are some real-world applications of the problem?
A: Some real-world applications of the problem include calculating the time it will take for a savings account to reach a certain balance, or determining the number of weeks it will take for a business to pay off a loan.
Q: What are some tips and variations for the problem?
A: Some tips and variations for the problem include adding more variables or constraints, such as a fixed interest rate or a minimum balance requirement, or using a calculator or a computer program to solve the equation.
Q: What are some common mistakes to avoid in the problem?
A: Some common mistakes to avoid in the problem include failing to define the variable and write the equation correctly, not simplifying the equation properly, not dividing both sides of the equation correctly, and not interpreting the result correctly.
Q: How can I practice solving similar problems?
A: You can practice solving similar problems by using the practice problems provided at the end of the article, or by creating your own problems that involve algebraic equations and real-world situations.
Q: What are some advanced topics in algebra that I can explore?
A: Some advanced topics in algebra that you can explore include quadratic equations, systems of equations, and functions.
Q: How can I apply the concepts learned in this problem to real-world situations?
A: You can apply the concepts learned in this problem to real-world situations by using algebraic equations to represent real-world situations and solve for the unknown quantity. This can be useful in finance, accounting, business, and other fields.
Q: What are some resources that I can use to learn more about algebra and problem-solving?
A: Some resources that you can use to learn more about algebra and problem-solving include textbooks, online tutorials, and practice problems. You can also seek help from a teacher, tutor, or mentor.