Geometric Series and Loan Repayments (1 Viewer)

[FDownes]

I could never really get my head around these problems in class, and now I'm stuck on one the questions in my textbook... I was hoping someone here could help me out by going through the question step-by-step. Here's the question;

A farmer borrows $50000 for farm machinery at 18% p.a. over 5 years and makes equal yearly repayments on the loan at the end of each year.

a) How much does he owe at the end of the first year, just before he makes the first repayment?

b)How much is each yearly repayment?

Part a) is pretty easy, the answer is obviously $59000. It's the second one that's giving me difficulties.

 

[Trebla]

I could never really get my head around these problems in class, and now I'm stuck on one the questions in my textbook... I was hoping someone here could help me out by going through the question step-by-step. Here's the question;

A farmer borrows $50000 for farm machinery at 18% p.a. over 5 years and makes equal yearly repayments on the loan at the end of each year.

a) How much does he owe at the end of the first year, just before he makes the first repayment?

b)How much is each yearly repayment?

Part a) is pretty easy, the answer is obviously $59000. It's the second one that's giving me difficulties.

b) Let A be amount owing and X be yearly repayments.
After first repayment: A1 = 50 000.(1.18) - X
In the next year, the interest applies over the amount owing so:
After second repayment: A2 = 1.18 [50 000.(1.18) - X] - X
= 50 000.(1.18)² - X.(1.18) - X
and so on till fifth repayment: A5 = 50 000.(1.18)⁵ - X.(1.18)⁴ - X(1.18)³ - X.(1.18)² - X.(1.18) - X
But amount owing after 5 years is nothing so A5 = 0
50.000.(1.18)⁵ = X.(1.18)⁴ + X(1.18)³ + X.(1.18)² + X.(1.18) + X
The RHS is a geometric series hence:
50 000.(1.18)⁵ = X.(1.18⁵ - 1)/(1.18 - 1)
.: X ~ $15 988.89
Check my working to see if it makes sense....I haven't done these for a long time lol

 

[FDownes]

Ack! I figured out why I couldn't get the answer for the question! For some bizzare reason I was calculating the interest at 12% p.a., not 18%... Man, I feel so stupid right now...

Thanks for your help Trebla!

 

[FDownes]

Hmm... I've run in to problems with a similar question... Could someone run this one past me as well? Here it is;

A store offers a hire-purchase loan on a lounge suite costing $5000. If you pay $200 deposit, the loan on the balance is paid over 3 years, at 18.5% p.a. interest. You do not have to make a payment for the first two months. What are the monthly repayments, and how much money will you pay for the lounge suite?

I seem to keep getting an answer of $2600 for the monthly repayments, instead of $188.12 like it should be.

 

[namburger]

Hmm... I've run in to problems with a similar question... Could someone run this one past me as well? Here it is;

A store offers a hire-purchase loan on a lounge suite costing $5000. If you pay $200 deposit, the loan on the balance is paid over 3 years, at 18.5% p.a. interest. You do not have to make a payment for the first two months. What are the monthly repayments, and how much money will you pay for the lounge suite?

I seem to keep getting an answer of $2600 for the monthly repayments, instead of $188.12 like it should be.

Make sure you change the 18.5% p.a. into months

18.5/12 = 1.54 % per month
Note: there are 34 months

A1 = 4800(1.0154) - X
A2 = [4800(1.0154) - X ](1.0154) - X
= 4800(1.0154)^2 -X (1+1.0154)
.
.
.
A34 = 4800(1.0154)^34 - X(1+1.0154 ......... + 1.0154^33)

To find X : let A34 = 0

X(1+1.0154 ......... + 1.0154^33) = 4800(1.0154)^34
X [(1.0154^34 - 1)/(1.0154-1)] = 4800(1.0154)^34
X = $182.40


To find how much you are paying it = 34 x 182.40 =$6201.85

 

[FDownes]

Got it, thanks.