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# Data Transfer

Alignments to Content Standards: 6.RP.A.3.b

## Task

The data transfer rate of an Internet connection is the rate in bytes per second that a file can be transmitted across the connection. Data transfer is typically measured in kilobytes (KB) per second, or megabytes (MB) where 1 MB = 210 KB = 1024 KB. Suppose the data transfer rate of your internet connection is 500 KB per second.

1. How long will it take to download a music file that is 5 MB?
2. How long will it take to download a video file that is 100 MB?

## IM Commentary

This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While the computations involved are not particularly complex, the units will be abstract for many students. The first solution relies more on reasoning about the meaning of multiplication and division, while the second solution uses units to help keep track of the steps in the solution process. Students in 6th grade begin to make the transition to using unit analysis that helps them solve complex modeling problems in high school (see N-Q.1). The teacher may need to provide some background information if the students struggle to understand the concepts of bytes, kilobytes, and megabytes.

In computer applications we traditionally work with powers of 2, so $1 \text{ MB} = 2^{10} \mbox{ KB}= 1024 \mbox{ KB}.$ However, since 1024 is so close to $1000=10^3$, it is now customary to just use powers if 10. To make the task slightly easier, we could use $1 \mbox{ MB} = 10^3 \mbox{ KB}$ as our conversion factor. An extension could be to also ask for download times of movies that are several gigabytes in size.

## Solutions

Solution: 1

1. We first convert the music file that we wish to download from megabytes to kilobytes, which is how the speed of our internet connection is measured. Since we have 1024 KB for every 1 MB, we know we can multiply the number of megabytes by 1024 to find the number of kilobytes: $$5 \cdot 1024 = 5120$$ So we need to download 5120 KB.

As our internet connection is 500 KB per second, we know that it takes 1 second to download 500 KB. To find the number of seconds it will take to download 5120 KB, we just divide 5120 by 500. $$5120 \div 500 = 10.24$$ So, it will take 10.24 seconds to download a 5 MB music file at a rate of 500 KB per second.

2. For a 100 MB video file, we follow the same process as in part (a) of this task. First, we convert the video file from MB to KB.

$$100 \cdot 1024 = 102400$$

Next, we find how many seconds this download will take, based on the same 500 KB per second internet connection by dividing the number of KB by 500:

$$102400 \div 500 = 204.8$$

So it takes 204.8 seconds to download 100 MB at 500 KB per second. Let's convert this time into minutes. Since there are 60 seconds in a minute, we can divide the number of seconds by 60 to find the number of minutes: $$204.8 \div 60 \approx 3.41$$ So, it will take approximately 3.41 minutes to download a 100 MB video file at 500 KB per second.

Note that we could have simply multiplied our final answer from part (a), 10.24 seconds, by 20: if a 5 MB file takes 10.24 seconds, then a 100 MB file takes 20 times longer, and $10.24 \cdot 20 = 204.8$ seconds.

Solution: Unit analysis

1. We first convert the music file that we wish to download from megabytes to kilobytes, which is how our internet connection is measured. Since 1 MB is equal to 1024 KB, we have

\begin{align} 5 \text{ MB} \cdot \frac{1024 \text{ KB}}{1 \text{ MB}} &= (5)(1024) \text{ KB} \\ &= 5120 \text{ KB} \end{align}

As our internet connection is 500 KB per second, we can find how many seconds this download will take by asking ourselves:

\begin{align} x \text{ seconds} \cdot \frac{500 \text{ KB}}{1 \mbox{ second}} = 5120\text{ KB}? \end{align} Solving for $x$ we have:
\begin{align} 5120 \text{ KB} \cdot \frac{1 \text{ second}}{500 \text{ KB}} = 10.24 \text{ seconds} \end{align}

So, it will take approximately 10.24 seconds to download a 5 MB music file.

2. For a 100 MB video file, we follow the same process as in part (a) of this task. First, we convert the video file from MB to KB.

\begin{align} 100 \text{ MB} \cdot \frac{1024 \text{ KB}}{1 \text{ MB}} &= (100)(1024) \text{ KB} \\ &= 102400 \text{ KB} \end{align}

Next, we find how many seconds this download will take, based on the same 500 KB per second internet connection.

$$102400 \text{ KB} \cdot \frac{1 \text{ second}}{500 \text{ KB}} = 204.8 \text{ seconds}$$

And we finally convert this time into minutes.

$$204.8 \text{ seconds} \cdot \frac{1 \text{ minute}}{60 \text{ seconds}} \approx 3.41 \text{ minutes}$$

So, it will take approximately 3.41 minutes to download a 100 MB video file.

Note that we could have simply multiplied our final answer from part (a), 10.24 seconds, by 20: if a 5 MB file takes 10.24 seconds, then a 100 MB file takes 20 times longer, and (10.24 seconds)(20) = 204.8 seconds .