Problem C. 916. (November 2007)

C. 916. In 2006, the income of an insurance company increased by 25% and its expenses increased by 15% relative to the previous year. The profit ( $\hbox{income}-\hbox{expenses}$ ) of the company increased by 40%. What percentage of the income was taken by the expenses in 2006?

(5 pont)

Deadline expired on December 17, 2007.

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

Megoldás. Jelölje B₂₀₀₅ a 2005. évi, B₂₀₀₆ a 2006. évi bevételt, K₂₀₀₅ a 2005. évi, K₂₀₀₆ a 2006. évi kiadást. Tudjuk, hogy:

B₂₀₀₆=1,25B₂₀₀₅, K₂₀₀₆=1,15K₂₀₀₅, B₂₀₀₆-K₂₀₀₆=1,4(B₂₀₀₅-K₂₀₀₅).

Az első két egyenletből:

$B_{2005}=0,8B_{2006},~~~K_{2005}=\frac{20}{23}\cdot K_{2006}.$

Ezeket a harmadiik egyenletbe behelyettesítve:

$B_{2006}-K_{2006}=1,4(0,8\cdot B_{2006}-\frac{20}{23}\cdot K_{2006}).$

Innen

$\frac{5}{23}\cdot K_{2006}=0,12B_{2006},$

tehát

$\frac{K_{2006}}{B_{2006}}=\frac{0,12\cdot23}{5}=0,552,$

vagyis 55,2 %-a volt 2006-ban a kiadások összege a bevételeknek.

Statistics:

385 students sent a solution.

5 points: 211 students.

4 points: 77 students.

3 points: 62 students.

2 points: 24 students.

1 point: 4 students.

0 point: 3 students.

Unfair, not evaluated: 4 solutionss.

Problems in Mathematics of KöMaL, November 2007

385 students sent a solution.
5 points:	211 students.
4 points:	77 students.
3 points:	62 students.
2 points:	24 students.
1 point:	4 students.
0 point:	3 students.
Unfair, not evaluated:	4 solutionss.

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