The first runner has $112 in savings, received a $45 gift from a friend, and will save $25 each month. The second runner has $50 in savings and will save $60 each month. How many months will it take for both accounts to have the same amount of money?
[tex]
\begin{array}{l}
112-25 m+45=50-60 m \
112+25+45 m=50 m+60 \
112+25-45 m=-50 m+60 \
112+25 m+45=50+60 m
\end{array}
[/tex]
Answer (2)
Define m as the number of months.
Express Runner 1's savings: 112 + 45 + 25 m .
Express Runner 2's savings: 50 + 60 m .
Set the savings equal and solve for m : 112 + 45 + 25 m = 50 + 60 m , which gives m = 35 107 ≈ 3.057 .
The number of months it will take for both accounts to have the same amount of money is 35 107 .
Explanation
Problem Analysis Let's analyze the problem. We have two runners with different initial savings and monthly savings rates. We want to find out after how many months they will have the same amount of money.
Setting up the Equations Let m be the number of months. The first runner's total savings after m months can be expressed as the initial savings plus the gift plus the monthly savings times the number of months: 112 + 45 + 25 m The second runner's total savings after m months can be expressed as the initial savings plus the monthly savings times the number of months: 50 + 60 m
Equating the Savings To find the number of months when both runners have the same amount of money, we set their total savings equal to each other: 112 + 45 + 25 m = 50 + 60 m Simplifying the left side, we get: 157 + 25 m = 50 + 60 m
Solving for m Now, we solve for m . Subtract 25 m from both sides: 157 = 50 + 35 m Subtract 50 from both sides: 107 = 35 m Divide both sides by 35 : m = 35 107
Finding the Number of Months Calculating the value of m : m = 35 107 m ≈ 3.057 Since m represents the number of months, we can say that it will take approximately 3.057 months for both runners to have the same amount of money.
Final Answer Therefore, it will take approximately 3.057 months for both accounts to have the same amount of money.
Examples
Imagine you and your friend are saving up for a new video game. You start with $100 an d s a v e $20 each month. Your friend starts with $50 b u t s a v es $30 each month. This problem helps you determine how many months it will take for both of you to have the same amount of money, which can help you plan and budget accordingly. Understanding how to set up and solve these types of equations is useful in many real-life financial planning scenarios.
It will take approximately 3.057 months for both runners to have the same amount of money. We determined this by setting up equations based on their initial savings and monthly contributions. Solving the equation gave us the number of months, which is a little over 3 months.
;
