Morgan started the year with $615 in the bank and is saving $25 per week. Kendall started with $975 and is spending $15 per week. Write an equation to represent Morgan's bank accou... Morgan started the year with $615 in the bank and is saving $25 per week. Kendall started with $975 and is spending $15 per week. Write an equation to represent Morgan's bank account and an equation to represent Kendall's bank account. They will have the same amount of money after a certain number of weeks. How many weeks will it take for them to have the same amount?
Understand the Problem
The question involves writing equations to represent the bank accounts of two individuals, Morgan and Kendall, based on their starting amounts, savings, and spending. It also requires finding the point at which both accounts will have the same amount.
Answer
Morgan and Kendall will both have $840 after 9 weeks.
Answer for screen readers
Morgan and Kendall will have the same amount after 9 weeks. In that week, they will both have $840 in their bank accounts.
Steps to Solve
- Equation for Morgan's Bank Account
Morgan starts with $615 and saves $25 per week. The equation for Morgan's bank account can be expressed as:
$$ y = 615 + 25x $$
where $y$ is the total amount in Morgan's bank account after $x$ weeks.
- Equation for Kendall's Bank Account
Kendall starts with $975 and spends $15 per week. The equation for Kendall's bank account can be expressed as:
$$ y = 975 - 15x $$
where $y$ is the total amount in Kendall's bank account after $x$ weeks.
- Setting the Equations Equal
To find when both accounts are equal, set the equations from Morgan and Kendall equal to each other:
$$ 615 + 25x = 975 - 15x $$
- Solving for x
Combine like terms and solve for $x$:
First, add $15x$ to both sides:
$$ 615 + 25x + 15x = 975 $$
Simplify to:
$$ 615 + 40x = 975 $$
Now subtract 615 from both sides:
$$ 40x = 975 - 615 $$
This simplifies to:
$$ 40x = 360 $$
Finally, divide both sides by 40:
$$ x = \frac{360}{40} = 9 $$
This means Morgan and Kendall will have the same amount after 9 weeks.
- Finding the Amount in Their Accounts
Substitute $x = 9$ back into either equation to find the total amount. Using Morgan's equation:
$$ y = 615 + 25(9) = 615 + 225 = 840 $$
So, in 9 weeks, both will have $840.
Morgan and Kendall will have the same amount after 9 weeks. In that week, they will both have $840 in their bank accounts.
More Information
The situation represents a real-world application of linear equations where two people's savings and expenses can be modeled mathematically. The calculations show how linear growth and decline can lead to equality in amounts over time.
Tips
- Forgetting to keep track of whether money is being saved or spent, leading to incorrect signs (positive for savings, negative for spending).
- Failing to simplify or combine like terms correctly when solving the equation, which can lead to wrong values for $x$.
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