6. To reduce heating costs in a residential building, the tenants decided to change the facade, and they were granted non-repayable funds from the Environmental Protection and Energy Efficiency Fund, which cover 60% of all total costs. The total cost for the new facade is 1,200,000 kn. The new facade guarantees that the average annual heating costs will be reduced by 35%. If the average annual heating cost for the building was 168,000 kn, how many years will it take for this investment to pay off?
6. The cost for the facade is 1200000 kn. The non-refundable portion of funds is $60\%$, which means that the residents have to cover $40\%$ of the costs, amounting to $480000 \mathrm{kn}$.

1 POINT

The average annual heating cost was 168000 kn, and with a savings of 35%, it will amount to 109200 kn.

1 POINT

Let $n$ be the number of years needed for the investment to pay off. Then the total cost for $n$ years of heating before the new facade is $168000 n$,

and the total cost for $n$ years of heating after the new facade is $109200 n$.

1 POINT

We need to determine the smallest number of years for which the heating cost before the new facade will be greater than the sum of the total heating costs after the new facade and the costs for the facade paid by the residents.

Determine the smallest natural number $n$ for which the following holds:

$168000 n > 109200 n + 480000$.

1 POINT

$168000 n - 109200 n > 480000$

$58800 n > 480000$

1 POINT

$n > \frac{480000}{58800}$

1 POINT

Since $\frac{480000}{58800} \approx 8.2$,

1 POINT

the smallest natural number that satisfies this condition is 9, meaning the investment will pay off in 9 years.