A Web music store offers two versions of a popular song. The size of the standard version is 2.4 megabytes (MB). The size of the high-quality version is 4.4 MB. Yesterday, the high-quality version was downloaded four times as often as the standard version. The total size downloaded for the two versions was 4400 MB. How many downloads of the standard version were there?
The correct answer and explanation is :
Let the number of downloads of the standard version be denoted as ( x ).
Given that the high-quality version was downloaded four times as often as the standard version, the number of downloads of the high-quality version will be ( 4x ).
The size of the standard version is 2.4 MB, and the size of the high-quality version is 4.4 MB. We are told that the total size of the downloads for both versions is 4400 MB. This can be expressed as:
[
(2.4 \times x) + (4.4 \times 4x) = 4400
]
Now, simplify the equation:
[
2.4x + 17.6x = 4400
]
[
20x = 4400
]
Solve for ( x ):
[
x = \frac{4400}{20} = 220
]
Thus, there were 220 downloads of the standard version.
Explanation:
To solve this problem, we set up an equation based on the information provided in the question. We start by introducing a variable, ( x ), to represent the number of standard version downloads. From the problem statement, we know that the high-quality version was downloaded four times as often as the standard version, so the number of high-quality downloads is ( 4x ).
Next, we use the sizes of the downloads: the standard version is 2.4 MB and the high-quality version is 4.4 MB. We know that the total size of all downloads is 4400 MB, so we write an equation to represent the total download size:
[
(2.4 \times x) + (4.4 \times 4x) = 4400
]
By simplifying this equation, we can solve for ( x ), which represents the number of downloads of the standard version. The result tells us that there were 220 downloads of the standard version.
This approach is an example of setting up a system of equations based on a real-world scenario and then solving it step by step.