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Difficulty: MediumLinear Equations in One Variable

A digital archive charges an annual subscription fee of $120\$120, which includes 5050 free document downloads per year. For each additional document downloaded beyond the first 5050, the archive charges a flat rate of $1.50\$1.50. If a research group paid a total of $231\$231 to the archive last year, how many total documents did the research group download?

  1. A
    24
  2. B
    74
  3. C
    114
  4. 124Answer
  5. E
    204

Answer

The total number of documents downloaded by the research group last year was 124.
Subtracting the base subscription fee of 120fromthetotalpaymentof120 from the total payment of 231 leaves 111spentstrictlyonextradownloads.Dividing111 spent strictly on extra downloads. Dividing 111 by the $1.50 per-document rate yields 74 extra downloads. Adding the 50 included free downloads gives a total of 124 documents downloaded.

Step-by-Step Solution

1
Define the linear equation representing total cost.
120+1.50(x−50)=231120 + 1.50(x - 50) = 231, where xx represents the total number of documents downloaded.
The base subscription fee is 120,andtheper−documentrateof120, and the per-document rate of 1.50 applies only to documents in excess of the initial 50 free documents.
2
Isolate the variable term by subtracting the base subscription fee from both sides.
1.50(x−50)=231−120=1111.50(x - 50) = 231 - 120 = 111
This determines the portion of the total cost that resulted strictly from additional downloads.
3
Solve for the number of additional documents (x−50)(x - 50).
x−50=1111.50=74x - 50 = \frac{111}{1.50} = 74
Dividing the extra cost of 111bytheper−documentrateof111 by the per-document rate of 1.50 gives the exact count of extra downloads.
4
Add the initial 50 free documents to find the total document count xx.
x=74+50=124x = 74 + 50 = 124
The question asks for the total documents downloaded, which combines the 50 free downloads and the 74 additional paid downloads.

Key Concept

Linear Equations in One Variable
Estimated Time:1m 30s
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Topics

Linear Equations in One Variable

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