Comparing cell phone plans over time
A common comparison problem is choosing between two options whose cost increases over time as a linear function. Ex: People commonly must choose between cell phone plans. One plan might be costlier per month but includes a phone, while another plan is cheaper per month but requires buying a phone. A person can graph each option to visualize the cost over time, and see the breakeven point.
PARTICIPATION ACTIVITY
2.4.1: Graphing to help choose a cell phone plan.
Start
2x speed
2
4
6
8
10
12
14
100
200
300
400
16
18
20
500
22
24
Months
Cost (dollars)
600
700
Cell contract, with phone
Cell contract, no phone
$50 per month
y = 50x
$30 per month
Phone: $300
y = 300 + 30x
800
900
1000
1100
1200
Breakeven point:
15 months
2.4.2: Comparing cell phone plans.
1)
What was the equation for the total cost of the with-phone contract?
CheckShow answer
2)
Shay chose to plot points for x = 10 and x = 20 because those values were round numbers and so a bit easier to calculate and draw. For the no-phone contract, what did Shay determine as the y value when x was 10?
$
CheckShow answer
3)
The graph of both contracts shows the with-phone contract to be how much cheaper at the start, when months is 0?
$
CheckShow answer
4)
The breakeven point, where the no-phone contract’s total cost becomes cheaper, is at how many months?
months
CheckShow answer
5)
The graphs not only show the breakeven point (15 months), but give a person a quick way to see the cost difference for any number of months. At 24 months, a person might visually see the blue line at about 1200 and the purple line at about 1000. What is the overall savings for the no-phone contract after 24 months?
CheckShow answer
Refinancing a home
A home loan monthly payment is often a person’s largest expense, perhaps consuming 30% or more of monthly earnings. A home loan has an interest rate, like 6%, that lasts for the loan’s duration, which might be 30 years. If rates go down, a newer loan might have a lower interest rate, like 4.5%. Replacing a home loan with a new lower-interest loan is called refinancing. Refinancing may cost thousands of dollars, so a task is determining whether refinancing is worth that cost.
2.4.3: Graphing the linear functions for existing and new home loans can help determine when refinancing is worth the initial cost.
Start
2x speed
1
2
3
2000
4000
6000
8000
4
5
10000
Months
12000
14000
Total cost ($)
6
7
8
9
10
Current payment: $2000
y = 2000x
Payment after refi: $1500
Refi cost: $3000
y = 1500x + 3000
Breakeven point
keyboard_arrow_downCaptions
2.4.4: Refinancing a home loan.
1)
Initially (months = 0), what is the cost difference between the current loan versus refinancing? Answer by looking where each line touches the y axis.
0
3000
2)
At 2 months, what is the difference in cost of the current loan versus refinancing? Answer by looking at each line’s height when x is 2.
2000
4000
3)
At 6 months, the two options cost the same (the breakeven point). What is that cost?
0
12000
4)
Which option will be cheaper after 1 year?
Current loan
Refinanced loan
5)
The graphs below show the difference between the current loan and the refinanced loan over 10 years (120 months). What is the savings from the refinance after 10 years?
$5000
$57000
6)
In the refinancing example above, rates must have dropped a lot since the original loan, causing a refi to yield a substantially lower monthly payment. The refi decision is easy for that case. Sometimes the decision is harder. The graphs below show another refinancing situation, where the refi still costs $3,000, but the payment only drops from $2,000 to $1,900 per month. About where is the breakeven point?
6 months
30 months
