RUMM - Comparing Models Lesson
Comparing Models
Derrick is trying to save money for the down payment on a used car. His parents have said that, in an effort to help him put aside money, they will pay him 10% interest on the money Derrick accumulates each month. At the moment, he has saved $200.
Suppose Derrick does not add any money to the savings. Write a recursive rule and an explicit function rule that model the money Derrick will accumulate with only the addition of the interest his parents pay.
The Recursive Rule Model
Let n=the number of months and d=the dollars saved
d0=200
n = 1, 2, 3, ....
dn +1 = next term
dn = current term
1.10 = percentage rate (x1 there would be no increase in value, 10% = 0.10, so the percentage rate increase = 1.10)
The Functional Rule Model
Using this same example, a functional rule model would be:
where n = 0, 1, 2, 3...
How long will it take Derrick to save at least $2,000 for the down payment if the only additions to his savings account are his parents' interest payments? Hint - You can figure this out using a table, scatterplot, or by solving for n using the function rule.
Using the Functional Rule Model
d = $2,000, the amount of money that Derrick wants
n = the number of months, you are solving for this exponent
With the addition of only the interest paid by his parents, it will take Derrick 24.16, or 25, months (just over two years) to earn enough for the $2,000 down payment.
Using the Recursive Rule Model
In an effort to speed up the time needed to save $2,000, Derrick decides to take on some jobs in his community. Suppose he commits to adding $50 per month to his savings, starting with the initial deposit from his parents. Fill in the table, showing the amount of money Derrick will have over several months.
The following is a scatterplot of the data you generated in the table and the graph to the function rule you found for Question 1. How does adding $50 per month to Derrick's savings change the way in which his money grows?
The scatterplot shows the savings with the added $50 per month, while the graph of the function rule shows the savings with only the interest paid by the parents. The graph shows that the scatterplot increases at a faster rate than the function rule. Adding $50 per month greatly affects the overall time needed to save $2,000.
How long will it take Derrick to save $2,000 for the down payment if he continues to add $50 every month?
The recursive rule in this situation says to multiply the prior answer by 1.10 and add 50. According to this routine, it will take Derrick only 14 months to save up the money for the down payment.
How would you write a recursive routine to model this situation?
A recursive rule describes this situation, in which $50 is added to the previous amount and that sum is multiplied by 1.10.
dn +1 =dn •1.10+50,where d0 =200 and n =1,2,3,...
Suppose Derrick changes the amount of money he adds to his savings each month to $100. How does this affect the time it takes to save $2,000? How much does he have to add to the savings each month to have enough money for the down payment on his car in six months?
By adding $100 to the savings each month, it will take Derrick 10 months to save $2,000. Using a table of values supports this conclusion. After testing several amounts greater than $100, it seems that Derrick needs to save $200 per month to have $2,000 in six months.
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