# Task 4.

In Donut's pantry and in Syrup's pantry, a total of 100 kg of jam is stored. Each dwarf took the same amount of time to eat their own supply, despite having different appetites. "If my supply were equal to yours, I would have eaten it in 45 days" - Donut told his friend. "And if my supply were equal to yours, I would have eaten it in just 20 days" - Syrup replied. How much jam did each dwarf eat and at what rate? (Don't forget to include the units of measurement.)
# Solution.

Let Donut have $x$ kg of jam, and Syrup have $n$ times more, i.e., $n x$ kg. Then Donut's consumption rate is $\frac{n x}{45}$, and Syrup's consumption rate is $\frac{x}{20}$. The equality of the time it takes to consume their reserves gives the equation

$$
\frac{x}{n x / 45}=\frac{n x}{x / 20}
$$

from which

$$
n^{2}=\frac{45}{20}=\frac{9}{4}
$$

thus, $n=\frac{3}{2}$.

From the equation

$$
x+\frac{3}{2} \cdot x=100
$$

we find $x=40$. Therefore, Donut had 40 kg of jam, and Syrup had 60 kg.

Their consumption rates are $\frac{60}{45}=\frac{4}{3}$ kg/day and $\frac{40}{20}=2$ kg/day, respectively.

## Answer:

Donut: 40 kg of jam, $\frac{4}{3}$ kg/day;

Syrup: 60 kg of jam, and 2 kg/day.