Paying the rent

Task

A checking account is set up with an initial balance of $4800, and $400 is removed from the account each month for rent (no other transactions occur on the account).

1. Write an equation whose solution is the number of months, $m$, it takes for the account balance to reach $2000.
2. Make a plot of the balance after $m$ months for for $m=1,3,5,7,9,11$ and indicate on the plot the solution to your equation in part (a).

IM Commentary

This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.

Solution

1. Since the account starts with $4,800 and decreases by $400 each month, there will be $4800-400m$ dollars left in the account after $m$ months have passed. The question asks us to set up an equation representing the number of months that pass before this quantity equals 2,000, and so the desired equation is $$4800-400m=2000.$$

2. The following table collects the relevant data points.

   Balance in an account 1–11 months after its establishment

   $m$ (months)	1	3	5	7	9	11
   Balance $4800-400m$	4400	3600	2800	2000	1200	400

   As is found in the table or by solving the equation in part (a), the balance in the account is $2,000 after 7 months. The point $(7,2000)$, and a dashed line plotting the more general relationship, is indicated in the graph below: