Assume that you have a balance of $3000 on your Discover credit card and that you make no more charges. Assume that Discover charges 15% APR and that each month you make only the minimum payment of 2% of the balance.

 

Find a formula for the remaining balance B after t monthly payments.

 

Assume that you have a balance of $3400 on your Discover credit card and that you make no more charges. Assume that Discover charges 18% APR and that each month you make only the minimum payment of 2.5% of the balance.

 

On what balance do you begin making payments of $80 or less?

$_______

 

Assume that you have a balance of $4600 on your Discover credit card and that you make no more charges. Assume that Discover charges 21% APR and that each month you make only the minimum payment of 2% of the balance.

 

Find how many months it will take to bring the remaining balance down to $2500. (Round your answer to the nearest whole number.)

______ months

1. B = $3,000(0.99225)t

 

2. $3,152.71

 

3. 214 months

 

 

Step-by-step explanation

1.

Every month, the formula to get the remaining balance is

 

Remaining Balance = Starting Balance + Finance Charge - Minimum Payment 

 

Where:

 

Finance Charge = Staring Balance x APR/12 (APR means annual percentage rate, that is why we should divide the APR by 12 to get the monthly percentage rate since all other information are on monthly basis.)

 

The Minimum Payment = (Starting Balance + Finance Charge) x rate of payment (it is assumed that the payment is done every end of the month that is why the finance charge should be added to the starting balance first before computing for the payment)

 

So we can further expound the formula through

 

Remaining Balance = Starting Balance + (Staring Balance x APR/12) - (Starting Balance + Finance Charge) x payment rate

 

But since the finance charge is also Staring Balance x APR/12 the formula will be

 

Remaining Balance = Starting Balance + (Staring Balance x APR/12) - (Starting Balance + Staring Balance x APR/12 ) xpayment rate

 

For simplicity, let us use variables

B = remaining balance

S = starting balance

t = months

 

So we can now also express the formula above as

 

B = S + (S x APR/12) - (S+ S x APR/12 ) x payment rate

 

Now, we can use the formula above to make another formula using the given in the problem.

 

B = S + (S x 15%/12) - (S+ S x 15%/12 ) x 2%

 

B = S + (S x 1.25%) - (S + S x 1.25%) x 2%

 

B = S + 0.0125S - (S + 0.0125S) x 2%

 

B = S + 0.0125S - 1.0125S x 2%

 

B = S + 0.0125S - 0.02025S

 

B = S - 0.00775S

 

B = S(0.99225)

 

Every month, the formula to get the remaining balance will be B = S(0.99225). Which means there is a recurring of formula to be used.

1st month B = staring balance of $3,000(0.99225)

2nd month B = ending balance of the 1st month(0.99225)

3rd month B = ending balance of the 2nd month(0.99225)

 

              month 1     month 2     month 3

Or it can simply be starting balance of $3,000 x 0.99225 x 0.99225 x 0.99225 and so on....

 

Since it is as if you are multiplying 0.99225 by itself, then we can further simplify the formula

 

B = $3,000(0.99225)t

 

Checking:

Let us try to use the formula for the 2nd month

 

B = 3,000(0.99225)2

 

B = 3,000(0.9845600625)

 

B = $2,953.68

 

Let us compute the remaining balance of the 2nd month using the individual balance of the 1st and 2nd month.

 

1st month

B = S(0.99225)

 

B = 3,000(0.99225)

 

B = 2,976.75

 

2nd month

B = S(0.99225)

 

B = 3,025.25(0.99225)

 

B = $2,953.68

 

2.

Since the only info being asked in this question is the balance you will start making a payment of $80 or less then the only relevant part of the formula is the formula for minimum payment which is

 

Minimum Payment = (Starting Balance + Finance Charge) x payment rate

 

or 

 

Minimum Payment = (Starting Balance + Staring Balance x APR/12) x payment rate

 

 

Then we can compute for the Starting Balance by substituting the given in the formula above

 

$80 = (S + S x 18%/12) x 2.5%

 

$80 = (S + S x 1.5%) x 2.5%

 

$80 = (S + 0.015S) x 2.5%

 

$80 = 1.015S x 2.5%

 

  $80    = 1.015S

  2.5%

 

$3,200 = 1.015S

 

$3,200 = S

  1.015

 

$3,152.71 = S

 

Checking:

$80 = (S + S x 18%/12) x 2.5%

 

$80 = ($3,152.71 + $3,152.71 x 1.5%) x 2.5%

 

$80 = ($3,152.71 + 47.29) x 2.5%

 

$80 = $3,200 x 2.5%

 

$80 = $80

 

3.

In this problem, we can use again the formula above to get the time being asked.

 

B = S + (S x 21%/12) - (S + S x 21%/12) x 2%

 

B = S + (S x 1.75%) - (S + S x 1.75%) x 2%

 

B = S + 0.0175S - (S + 0.0175S) x 2%

 

B = S + 0.0175S - 1.0175S x 2%

 

B = S + 0.0175S - 0.02035S

 

B = S - 0.00285S

 

B = S(0.99715)

 

Now we can substitute the given in the derived formula above.

 

B = S(0.99715)t

 

$2,500 = $4,600(0.99715)t

 

$2,500 = 0.99715t

$4,600

 

0.54347826086 = 0.99715t

 

Unfortunately, the way to compute for exponent or for the time with this complex problem is through the use of logarithm

 

t = log_b(m)

 

Where: 

 

b is the base

m is the result

t is the exponent

 

And then substitute the given to get the exponent

 

t = log_0.99715(0.54347826086)

 

t = 213.648 or 214 months

 

However, this function is not always available in some calculator but natural logarithm or "ln" is often available in most scientific calculator. This can be used using the formula 

 

t = ln (m)

      ln (b)

 

t = ln (0.54347826086)

            ln (0.99715)

 

t = 213.648 or 214 months

 

Checking:

B = S(0.99715)t

 

$2,500 = $4,600(0.99715)213.648

 

$2,500 = $4,600(0.99715)213.648

 

$2,500 = $4,600(0.5434779574)

 

$2,500 = $2,500