Noah has just graduated high school and has some credit card offers! The first credit card (Credit Card A) charges 18% compounded monthly. The second credit card (Credit Card B) charges 18% compounded semi-annually. The third credit card (Credit Card C) charges 17% compounded quarterly.
Think about it: Which credit card do you assume will be the best deal?
Noah isn’t sure which credit card is the best deal, so he wants to do some comparison. You’re going to help him out using the skills you learned in the Exponents Unit to create some equations and graphs to determine the best deal.
Let’s suppose he is going to buy a smartphone for $607.99 and put the entire balance on the credit card. Use the following formulas.
Credit card A:
B = $607.99
Credit card B:
B = $607.99
Credit card C:
B = $607.99
1. Use the equations to help you fill out the table below, assuming there are NO payments made to the credit card companies. Hint: 6 months = 0.5 years (9 pts)
Balance After
Credit Card Equation (copy from above) 0 year 6 months 1
year 2
year 5
years
A $607.99t $607.99 $644.80
B $607.99 $607.99
C $607.99 $607.99
First of all, I am not aware of any credit card that does not have monthly compounding.
anyway.....
18% compounded monthly vs charges 18% compounded semi-annually.
If the nominal rates are the same, the card with the more frequent compounding would have the higher effective rat.
So #1 is worse than #2
compare them:
(1 + .18/12)^12 = 1.19562 ---> effective rate is 19.562 per annum
(1+.18/2)^2 = 1.1881 ----> effective rate is 18.81% per annum
(1 + /17/4)^4 = 1.18114.. --> effective rate is 18.114% per annum
ok, rate them and since I don't see any tables you are on your own.
Btw, you said "assuming there are NO payments made to the credit card companies" . Lol, what do you think would happen if you did that ??
18% monthly is .18/12 each month = 0.015/month
so every month multiply by 1.015
total cost = $607.99 * 1.01^number of months
18% semi annual = .18/2 every six months = .09 /half year
so every half year (6 months) multiply by 1.09
total cost = $607.99 * 1.09^ (number of months/6)
17% quarterly = .17/4 every 3 months = .0425 /quarter year
so every quarter year (3 months) multiply by 1.0425
total cost = $607.99 * 1.025^(number of months/3)
To determine the balance after a certain period of time for each credit card, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = final amount (balance)
P = principal amount (initial balance)
r = annual interest rate (expressed in decimal form)
n = number of times interest is compounded per year
t = number of years
1. For Credit Card A:
To find the balance after a certain time (t) using the equation B = $607.99t, we substitute the values into the equation:
- Balance after 0 years: B = $607.99 * 0 = $0
- Balance after 6 months (0.5 years): B = $607.99 * 0.5 = $303.995
- Balance after 1 year: B = $607.99 * 1 = $607.99
- Balance after 2 years: B = $607.99 * 2 = $1215.98
- Balance after 5 years: B = $607.99 * 5 = $3039.95
2. For Credit Card B:
To find the balance after a certain time (t) using the equation B = $607.99 * (1 + r/2)^(2t), we substitute the values into the equation:
- Balance after 0 years: B = $607.99 * (1 + 0.18/2)^(2 * 0) = $607.99
- Balance after 6 months (0.5 years): B = $607.99 * (1 + 0.18/2)^(2 * 0.5) ≈ $646.41
- Balance after 1 year: B = $607.99 * (1 + 0.18/2)^(2 * 1) ≈ $689.96
- Balance after 2 years: B = $607.99 * (1 + 0.18/2)^(2 * 2) ≈ $779.27
- Balance after 5 years: B = $607.99 * (1 + 0.18/2)^(2 * 5) ≈ $1295.11
3. For Credit Card C:
To find the balance after a certain time (t) using the equation B = $607.99 * (1 + r/4)^(4t), we substitute the values into the equation:
- Balance after 0 years: B = $607.99 * (1 + 0.17/4)^(4 * 0) = $607.99
- Balance after 6 months (0.5 years): B = $607.99 * (1 + 0.17/4)^(4 * 0.5) ≈ $644.80
- Balance after 1 year: B = $607.99 * (1 + 0.17/4)^(4 * 1) ≈ $688.27
- Balance after 2 years: B = $607.99 * (1 + 0.17/4)^(4 * 2) ≈ $776.40
- Balance after 5 years: B = $607.99 * (1 + 0.17/4)^(4 * 5) ≈ $1292.25
Now the table can be filled out as follows:
Balance After
Credit Card Equation 0 year 6 months 1 year 2 years 5 years
A $607.99t $0 $303.995 $607.99 $1215.98 $3039.95
B $607.99 * (1 + 0.18/2)^(2t) $607.99 $646.41 $689.96 $779.27 $1295.11
C $607.99 * (1 + 0.17/4)^(4t) $607.99 $644.80 $688.27 $776.40 $1292.25
By comparing the balances at different time intervals, we can determine which credit card is the best deal.