1. **State the problem:** Agent Hunt transferred files at a rate of 4.4 megabytes per second onto a flash drive that already had some files. After 32 seconds, the drive contained 384 megabytes. The drive's maximum capacity is 1000 megabytes. We need to find: - How full the drive was before the transfer started. - How long it takes for the drive to be completely full from the start of the transfer. 2. **Define variables and formula:** Let $x$ be the initial amount of data on the drive in megabytes. The transfer rate is $4.4$ megabytes per second. The time elapsed is $t$ seconds. The total data on the drive after time $t$ is given by: $$\text{Total data} = x + (4.4 \times t)$$ 3. **Find the initial amount $x$:** Given after 32 seconds, total data is 384 megabytes: $$x + 4.4 \times 32 = 384$$ Calculate $4.4 \times 32$: $$4.4 \times 32 = 140.8$$ So: $$x + 140.8 = 384$$ Subtract 140.8 from both sides: $$x = 384 - 140.8$$ $$x = 243.2$$ 4. **Find the time to fill the drive completely:** The drive capacity is 1000 megabytes, so: $$x + 4.4 \times t = 1000$$ Substitute $x = 243.2$: $$243.2 + 4.4t = 1000$$ Subtract 243.2 from both sides: $$4.4t = 1000 - 243.2$$ $$4.4t = 756.8$$ Divide both sides by 4.4: $$t = \frac{756.8}{4.4}$$ Show cancellation: $$t = \frac{\cancel{4.4}172}{\cancel{4.4}1} = 172$$ 5. **Final answers:** - Initial amount on the drive: **243.2 megabytes** - Time to fill the drive completely: **172 seconds**