2011 AMC 10A Problem 1
A cell phone plan costs each month, plus ¢ per text message sent, plus ¢ for each minute used over hours. In January Michelle sent text messages and talked for hours. How much did she have to pay?
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Key insight
Split the bill into three parts. Only the extra half hour costs money, and half an hour is 30 minutes at 10 cents each.
Solution
The bill has three parts, so work out each part on its own and add them at the end.
- The monthly fee. This is , no matter what.
- The texts. Michelle sent texts at cents each. That is cents, which is .
- The extra talking time. The first hours are included. She talked hours, so only hour is extra.
- The charge is per minute, so change the half hour into minutes: hour is minutes. At cents a minute, that is cents, which is .
- Add the parts. dollars.
The answer is .
Why this works
A bill like this is a fixed fee plus a few "number times price" pieces. The only danger is mixing units: cents with dollars, or hours with minutes. Put every piece into the same unit before you add. And when a plan says "over 30 hours", you pay only for the part above 30 hours.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
Add the three parts of the bill separately.
Fixed charge: .
Texts: messages at cents each is cents, or .
Overage: Michelle talked hours beyond the free allowance, and hours is minutes. At cents per minute that is cents, or .
Total: dollars.
The answer is .
Review in the AoPS books
- Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
- Introduction to Algebra, Chapter 6: Ratios and Percents
- Introduction to Algebra, Chapter 7: Proportion (§7.4)
- Prealgebra, Chapter 13: Data and Statistics (§13.1)
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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.