13.039. For the first quarter, the car factory completed $25 \%$ of the annual production plan for cars. The number of cars produced in the second, third, and fourth quarters turned out to be proportional to the numbers $11,25,12$ and 13.5. Determine the overfulfillment of the annual plan in percentages, if in the second quarter the car factory produced 1.08 times more products than in the first.
Solution.

Let $x$ be the annual plan of the car factory. In the first quarter, the factory completed $0.25 x$ of the plan. In the second quarter, the factory produced $11.25 y$ cars, which amounted to $1.08 \cdot 0.25 x = 0.27 x$ of the plan: $11.25 y = 1.08 \cdot 0.25 x = 0.27 x$, hence $y = 0.024 x$. In the third quarter, the factory completed $12 y = 12 \cdot 0.024 x = 0.288 x$ of the plan, and in the fourth quarter, $13.5 y = 0.324 x$ of the plan. Over the entire year, the factory completed $0.25 x + 0.27 x + 0.288 x + 0.324 x = 1.132 x$ of the plan, which means it exceeded the plan by $0.132 = 13.2\%$.

Answer: $13.2\%$.