We are downloading a 1.5 MB file from the internet to our computer. During the process, the program continuously estimates the remaining time based on the average download speed so far. Looking at the screen, we see that the program has already downloaded exactly half of the file, and at this point, it estimates the remaining time for the operation to be exactly 2 minutes. After any $t$ time has elapsed, we observe that (due to network load) the program still displays 2 minutes as the remaining time for the file download. Provide the size of the already downloaded part of the file as a function of $t$.
Solution. The program calculates the remaining time based on the average speed. If after downloading half of the file, the program estimates the remaining time to be 2 minutes, this means that it downloaded the initial part in 2 minutes. Therefore, the average speed so far is $\frac{0.75}{2} \frac{\text{MB}}{\text{min}}$. If we then download an additional $x$ MB in $t$ time, the average download speed becomes $\frac{0.75+x}{2+t} \frac{\text{MB}}{\text{min}}$. If we were to download the remaining $(0.75-x)$ MB at this speed in 2 minutes, the average speed would be $\frac{0.75-x}{2} \frac{\text{MB}}{\text{min}}$. Due to the equality of the two speeds, we can write the following equation:

$$
\frac{0.75+x}{2+t}=\frac{0.75-x}{2}
$$

Let's introduce a new variable, let $0.75+x=y$, which is the size of the downloaded part. Express $y$ as a function of $t$ from equation (1). After substitution,

$$
\frac{y}{2+t}=\frac{1.5-y}{2} \quad \text{and hence} \quad y=\frac{3+1.5 t}{4+t}
$$

The size of the downloaded part is therefore $\frac{3+1.5 t}{4+t} \text{MB}$ as a function of $t$.