Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem C. 977. (February 2009)

C. 977. An entrepreneur has taken out a subsidized loan for 12 million forints (HUF, Hungarian currency) at a fixed annual interest rate of 8%. How much will he owe in 10 years if he pays back the loan in annual instalments of 1.2 million forints?

(5 pont)

Deadline expired on March 16, 2009.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Az első év végén 12.1,08-1,2 millió forint lesz a tartozása, a második év végén (12.1,08-1,2).1,08-1,2 millió forint, és így tovább, a 10. év végén a tartozás (milló forintban):

x=(((12\cdot1,08-1,2)\cdot1,08-1,2)\cdot1,08\ldots)\cdot1,08-1,2,

ahol összesen 10-szer szerepel az 1,08-as szorzó.

Innen:

x=12\cdot1,08^{10}-1,2\cdot1,08^9-1,2\cdot1,08^8-\ldots-1,2\cdot1,08-1,2=

=12\cdot1,08^{10}-1,2\cdot(1+1,08+1,08^2+\ldots+1,08^9)=

=12\cdot1,08^{10}-1,2\cdot1\cdot\frac{1,08^{10}-1}{1,08-1}\approx8,523225.

Vagyis 8 523 225 Ft lesz a tartozása 10 év múlva.


Statistics:

285 students sent a solution.
5 points:210 students.
4 points:21 students.
3 points:18 students.
2 points:2 students.
1 point:2 students.
0 point:23 students.
Unfair, not evaluated:9 solutions.

Problems in Mathematics of KöMaL, February 2009