Compound Interest

[ayman123]

Hi i need help in this exercise i didn't understand it
Une personne a emprunté 15000 euro à intérêts composés. Au lieu de
rembourser le capital et les intérêts 5 ans après, comme convenu, elle
propose de rembourser à cette date 8000 euro et le reste est versé 5 ans plus
tard par un montant de 29110.90 euro.
Quel est le taux d'intérêts composés ?

thanks

 

[Otis]

Hi ayman. From translate.google, I've interpreted the exercise statement as:

A person borrowed 15,000 euro at compound interest. Instead of repaying the principal and interest 5 years later, as agreed, they propose to repay only 8,000 euro on that date and to repay the rest (including additional interest) 5 years after the original due date. If the second payment is 29,110.90 euro, then what is the compound interest rate?

I hope that's correct.

Did they provide a compounding period? If not, then are you assuming continuous compounding?

Please tell us also what formula(s) you've learned for compounding interest, as well as any work you may have tried. Thanks!

My initial thought is to write an expression for the original amount due, in terms of your rate variable, and then use that expression-8000 as the principle for a second 5-year loan.

?

 

[Otis]

Ayman may have finished the exercise, so for other readers here's an example of what I had in mind.

A person borrowed €4,000 for 3 years at continuously-compounded interest rate r. Instead of repaying the principal and interest 3 years later, as agreed, they propose to repay only €2,500 on that date and to repay the balance (including additional interest) 3 years after the original due date. If the second payment is €3,100.29, then what is r rounded to two decimal places?

We use the formula for continuously-compounded interest. The original amount due is:

4000 e^3r

After paying €2,500, they start a second 3-year loan at the same rate, and the amount due will be €3,100.29:

(4000 e^3r - 2500) e^3r = 3100.29

Distributing and moving all terms to one side yields an equation in quadratic form:

4000 (e^3r)² - 2500 (e^3r) - 3100.29 = 0

Let u = e^3r

4000u² - 2500u - 3100.29 = 0

The Quadratic Formula gives one positive solution for u: 1.2467 (rounded)

1.2467 = e^3r

We solve for r, by taking one-third of the natural logarithm of each side.

r = 0.0735

The interest rate r is 7.35%

?