Certainly! Let's solve the problem step-by-step.
### Part (i):
We know that the computer can download [tex]\(\frac{1}{3}\)[/tex] of the file in 40 seconds.
To find out how much time it would take to download the entire file, we need to consider the whole file as 1 part.
Since [tex]\(\frac{1}{3}\)[/tex] of the file takes 40 seconds, the whole file will take:
[tex]\[
\text{Total time} = 40 \text{ seconds} \times 3 = 120 \text{ seconds}
\][/tex]
Next, we convert this time from seconds to minutes. Since 1 minute = 60 seconds:
[tex]\[
\text{Total time in minutes} = \frac{120}{60} = 2 \text{ minutes}
\][/tex]
### Part (ii):
To find out how much of the file the computer can download in 1 minute, we need to calculate the download speed in terms of file parts per second.
Firstly, the speed, in terms of file parts per second, is:
[tex]\[
\text{Speed (parts per second)} = \frac{1/3 \text{ part}}{40 \text{ seconds}} = \frac{1}{120} \text{ parts per second}
\][/tex]
Now convert this speed to parts per minute:
[tex]\[
\text{Speed (parts per minute)} = \left(\frac{1}{120} \text{ parts/second}\right) \times 60 = \frac{60}{120} = \frac{1}{2} \text{ parts per minute}
\][/tex]
Thus, the computer can download:
[tex]\[
0.5 \text{ parts of the file in 1 minute}
\][/tex]
### Part (iii):
We need to determine how much of the file is left to download after downloading for 90 seconds.
First, we calculate how much of the file is downloaded in 90 seconds. We already know the download speed per second is [tex]\(\frac{1}{120}\)[/tex] parts per second. Hence:
[tex]\[
\text{Downloaded in 90 seconds} = 90 \text{ seconds} \times \frac{1}{120} \text{ parts/second} = \frac{90}{120} = \frac{3}{4}
\][/tex]
Therefore, after 90 seconds, the amount of the file that remains to be downloaded is:
[tex]\[
\text{Remaining file} = 1 \text{ (whole file)} - \frac{3}{4} = \frac{1}{4}
\][/tex]
Summarizing the results:
1. It will take 2 minutes to download the whole file.
2. The computer downloads 0.5 parts of the file in 1 minute.
3. After 90 seconds, 0.25 parts of the file remain to be downloaded.
