series help- Time payments (1 Viewer)

[speedo]

Hi ppl,

could you'll help me with this question:
A man borrows $2000 at an int. rate of 12% p.a. The loan is to be repayed after 3 years. However, the first 3 months r interest free.Calculate:
a) The amt of each monthly installment.
b) the amount owing after 3 months

 

[insert-username]

Hi ppl,

could you'll help me with this question:
A man borrows $2000 at an int. rate of 12% p.a. The loan is to be repayed after 3 years. However, the first 3 months r interest free.Calculate:
a) The amt of each monthly installment.
b) the amount owing after 3 months

Hey there, welcome to BoS.

Firstly, we need to sort out the important parts of the question:

P = $2000
R = 1% per month (I assume interest is calculated monthly)
n = 36 months

Now we write out the repayment, remembering that the first 3 months are interest free (M is the monthly payment):

Owe₁ = 2000 - M

Owe₂ = 2000 - 2M

Owe₃ = 2000 - 3M

Owe₄ = 2000 x 1.01 - 4M

Owe₅ = 2000 x 1.01² - 4M x 1.01 - M [Note that 4M is multiplied by the interest rate here]

all the way to

Owe₃₆ = 2000 x 1.01³³ - 4M x 1.01³² - .... - M x 1.01 - M

But since the loan is paid off after 36 months, Owe₃₆ is 0, and we can then rewrite the equation with M as the subject. First, take it out as a common factor:

0 = 2000 x 1.01³³ - M(4 x 1.01³² + 1.01³¹ + .... + 1.01 + 1)

Add it to both sides:

M(4 x 1.01³² + 1.01³¹ + .... + 1.01 + 1) = 2000 x 1.01³³

Divide by the geometric series:

M = [2000 x 1.01³³] ÷ [[(1.01³² - 1)/.01] + 4 x 1.01³³] [When we work out the sum of the geometric series, we need to take the 4 x 1.01³³ out and add it on separately]

= $64.52


Now for part b, we need to go back to our first equations. We now know M, so we just substitute in the values:

Owe₃ = 2000 - 3M

Owe₃ = 2000 - 193.56

= $1806.44


I hope that helps you out.

 

[kido_1]

I got for part:
a> $64.52
b> $1806.44

 

[nallask8r]

is there any unoffical forumla/equation to work time payments out? or maybe a simplified way. we just started that topic and, as confirmed by your working out, it looks unnessesarily tedious...

 

[pLuvia]

There are

A_n = PRⁿ

Where P is the initial amount, A_n is the amount that P grows after n periods of time. R = 1+^r/₁₀₀

Some of the Annuities questions require you to actually find out the formulae of the specific question using geometric progression. So not all formulae work for all questions.

PR(Rⁿ-1) / (R-1)

R=1+^r/₁₀₀

If a person invests $P at the beginning at r% per period compound interest. This formula gives you the sum of the investments at the end of n periods of time

 

[SoulSearcher]

Yeah the general formula would be

A_n = PRⁿ - M[ (Rⁿ - 1) / (R - 1) ]

where P is the amount of the loan at the beginning of the loan, R is the interest rate as a percentage plus 1 (remember that the interest rate has to be in the same time frame as the repayments), M is the regular repayment, and n is the amount of payments that the loan asks for. You would be required to manipulate the formula to get the answers that the question asks for.

 

[adgala]

You won't get full marks unless you do the full working out as shown by insert-username, the formula's pretty easy to get when you learn how to set it all up (use letters[forgot what they were called...pronumerals?] and write it out like insert-username instead of putting in numbers directly [gets a bit confusing])

 

[SoulSearcher]

Yes true, but the only steps that you would needd to do before subbing in the values would be the ones to prove that the question is dealing with loan repayments, i.e. the first 6 steps of insert_usernames working out, or more simply, that the first 3 repayments take the form of a loan repayment equation, and thus n amount of repayments leads to the formula for loan repayments.