Q:You have borrowed $8000 from the bank. Suppose you want to repay a fixed amount of money for each of the following n years (except possibly the last year), and the annual interest rate r does not change in these n years. For example, if r = 10% and you repay $4000 each year, then you will own the bank $(8000+800-4000) = $4800 next year, $(4800+480-4000) = $1280 two years after, and at the end of the third year you only need to repay $(1280 + 128) = $1408.

Question

Design a visually stimulating concept that symbolically represents a process of bank loan repayment over the course of several years, using a numerical equation as a metaphor. Include a visualization of a pile of currency representing the initial loan amount of $8000, progressively diminishing as annual payments are made. Also, encapsulate the concept of interest rates, represented by additional sums of money added to the pile every year. Show two distinct sets of year markers indicating different repayment schedules, one at a 20% interest rate with $2500 annual repayments and another at a 24% interest rate with $2000 annual repayments.

Q:You have borrowed $8000 from the bank. Suppose you want to repay a fixed amount of money for each of the following n years (except possibly the last year), and the annual interest rate r does not change in these n years. For example, if r = 10% and you repay $4000 each year, then you will own the bank $(8000+800-4000) = $4800 next year, $(4800+480-4000) = $1280 two years after, and at the end of the third year you only need to repay $(1280 + 128) = $1408.

(a) If r = 20% and you want to repay $2500 each year (except possibly the last year). How many years do you need to repay all the money?
(d) If r = 24% and you want to repay $2000 each year (except possibly the last year). How many years do you need to repay all the money.

Answer 1

a)

amount at end of 1st year = 8000 + 1600 - 2500 = 7100
amount at end of 2nd year = 7100+1420-2500=6020
amount at end of 3rd year = 6020+1240-2500=4724
amount at end of 4th year = 4724+944.8-2500=3168.80
amount at end of 5th year = 3168.8 +633.76-2500= 1302.56 , which is less than the 2500 payment

amount owing at end of 6th year = 1302.56 + 260.51 = 1563.07

So you will need 5 full payments of $2500 plus a partial payment of 1563.07 at the end of the 6th year

checking with my formula:
outstanding after 5 years
= 8000(1.2)^5 - 2500(1.2^5 - 1)/.2
= 19906.56 - 18604.00
= 1302.56 <---- amount owing after 5 years, see above

adding one more interest = 1302.56 + 260.51
= 1563.07 <--- the final partial payment I found above

d) follow the same steps as I used in a)
of course since the rate is higher, and the payment is only 2000, it will take considerably longer.

Answer 2

(a) Well, it seems like you're in quite a "percent-age" with this question. But fear not, I'm here to lend a helping hand, or should I say, "lend some humor."

If the annual interest rate is 20% and you want to repay $2500 each year, let's calculate how many years it would take to repay all the money.

In the first year, you would owe the bank $8000. But after repaying $2500, you would still owe $8000 - $2500 = $5500.

Now, multiply $5500 by 20% and you get $1100. This means that at the end of the first year, you would owe $5500 + $1100 = $6600 to the bank.

Let's continue this hilarious cycle. In the second year, you would owe $6600. After repaying $2500, you would still owe $6600 - $2500 = $4100.

Multiply $4100 by 20% and you get $820. So, at the end of the second year, you would owe $4100 + $820 = $4920.

Now, in the third year, after repaying $2500, you would still owe $4920 - $2500 = $2420.

Multiply $2420 by 20% and you get $484. Therefore, at the end of the third year, you would owe $2420 + $484 = $2904.

Seems like we're getting closer to paying off that debt! But hold on, it's not over yet.

In the fourth year, after repaying $2500, you would still owe $2904 - $2500 = $404.

Multiply $404 by 20% and you get $80.80. This means that at the end of the fourth year, you would owe $404 + $80.80 = $484.80.

But hey, we're almost there! In the fifth and final year, after repaying $2500, you would still owe $484.80 - $2500 = -$2015.20.

Wait a minute, that's a negative number! It means you've paid off your debt and have some extra cash to celebrate! Hooray!

So, it would take approximately 5 years for you to repay all the money.

(d) Alrighty, let's dive into another "percent-age" conundrum. If the annual interest rate is a whopping 24% and you want to repay $2000 each year, how many years will it take to repay all the money?

In the first year, you would owe the bank $8000. But after repaying $2000, you would still owe $8000 - $2000 = $6000.

Multiply $6000 by 24% and you get $1440. This means that at the end of the first year, you would owe $6000 + $1440 = $7440 to the bank.

Oh boy, let's keep this going! In the second year, you would owe $7440. After repaying $2000, you would still owe $7440 - $2000 = $5440.

Multiply $5440 by 24% and you get $1305.60. So, at the end of the second year, you would owe $5440 + $1305.60 = $6745.60.

Now, in the third year, after repaying $2000, you would still owe $6745.60 - $2000 = $4745.60.

Multiply $4745.60 by 24% and you get $1138.94. Therefore, at the end of the third year, you would owe $4745.60 + $1138.94 = $5884.54.

Hey, we're making progress! In the fourth year, after repaying $2000, you would still owe $5884.54 - $2000 = $3884.54.

Multiply $3884.54 by 24% and you get $932.29. So, at the end of the fourth year, you would owe $3884.54 + $932.29 = $4816.83.

Okay, almost there! In the fifth year, after repaying $2000, you would still owe $4816.83 - $2000 = $2816.83.

Multiply $2816.83 by 24% and you get $676.04. Therefore, at the end of the fifth year, you would owe $2816.83 + $676.04 = $3492.87.

And finally, in the sixth year, after repaying $2000, you would still owe $3492.87 - $2000 = $1492.87.

Multiply $1492.87 by 24% and you get $358.29. So, at the end of the sixth year, you would owe $1492.87 + $358.29 = $1851.16.

Guess what? It's not over yet! In the seventh year, after repaying $2000, you would owe $1851.16 - $2000 = -$148.84.

Hold your 🤡 horses! A negative number means you've paid off the debt with a little money to spare! Congrats!

So, it would take approximately 7 years for you to repay all the money.

Answer 3

To calculate the number of years needed to repay all the money, we need to use the formula for calculating the future value of an annuity.

The formula for the future value of an annuity is given by:

FV = P * [(1 + r)^n - 1] / r

Where:
FV = Future value (the amount you owe the bank after the last payment)
P = Payment amount (the fixed amount you want to repay each year)
r = Annual interest rate
n = Number of years

(a) If r = 20% and you want to repay $2500 each year (except possibly the last year), we can plug in the given values into the formula and solve for n.

FV = $8000
P = $2500
r = 20%

$8000 = $2500 * [(1 + 0.2)^n - 1] / 0.2

Now, we can solve this equation to find the value of n.

8000 = 2500 * [(1.2)^n - 1] / 0.2

Multiplying both sides of the equation by 0.2:

1600 = 2500 * [(1.2)^n - 1]

Dividing both sides of the equation by 2500:

0.64 = (1.2)^n - 1

Adding 1 to both sides of the equation:

1.64 = (1.2)^n

To solve for n, we need to take the logarithm (base 1.2) of both sides of the equation:

log₁.₂(1.64) = n

Using a calculator, we find that:

n ≈ 3.35

Therefore, you would need approximately 3.35 years to repay all the money.

(d) If r = 24% and you want to repay $2000 each year (except possibly the last year), we can follow a similar process to solve for n.

Using the same formula:

FV = $8000
P = $2000
r = 24%

$8000 = $2000 * [(1 + 0.24)^n - 1] / 0.24

Simplifying the equation:

3200 = 2000 * [(1.24)^n - 1] / 0.24

Multiplying both sides of the equation by 0.24:

768 = 2000 * [(1.24)^n - 1]

Dividing both sides of the equation by 2000:

0.384 = (1.24)^n - 1

Adding 1 to both sides of the equation:

1.384 = (1.24)^n

Taking the logarithm (base 1.24) of both sides of the equation:

log₁.₂₄(1.384) = n

Using a calculator, we find that:

n ≈ 1.1

Therefore, you would need approximately 1.1 years to repay all the money.

Answer 4

To find out how many years it will take to repay all the money in both cases, we can use the formula for calculating the future value of an annuity. The formula is:

FV = P * ((1 + r)^n - 1) / r,

where FV is the future value, P is the payment per period, r is the annual interest rate, and n is the number of years.

(a) If r = 20% and you want to repay $2500 each year (except possibly the last year):

We can set up the equation:

8000 = 2500 * ((1 + 0.20)^n - 1) / 0.20.

First, divide both sides of the equation by 2500:

3.2 = (1.20^n - 1) / 0.20.

Now, multiply both sides by 0.20:

0.64 = (1.20^n - 1).

Next, add 1 to both sides:

1.64 = 1.20^n.

Finally, take the logarithm base 1.20 of both sides to solve for n:

n = log base 1.20 (1.64).

Using a calculator, we find that n ≈ 2.7.

Therefore, you will need approximately 3 years to repay all the money.

(d) If r = 24% and you want to repay $2000 each year (except possibly the last year):

We can set up the equation:

8000 = 2000 * ((1 + 0.24)^n - 1) / 0.24.

First, divide both sides of the equation by 2000:

4 = (1.24^n - 1) / 0.24.

Now, multiply both sides by 0.24:

0.96 = (1.24^n - 1).

Next, add 1 to both sides:

1.96 = 1.24^n.

Finally, take the logarithm base 1.24 of both sides to solve for n:

n = log base 1.24 (1.96).

Using a calculator, we find that n ≈ 3.5.

Therefore, you will need approximately 4 years to repay all the money.