# Task 4.

In Donut's pantry and in Syrup's pantry, there is a total of 100 kg of jam. Each dwarf took the same amount of time to eat their own supplies, 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?
# 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.