2020 AMC 10A Problem 4
A driver travels for hours at miles per hour, during which her car gets miles per gallon of gasoline. She is paid per mile, and her only expense is gasoline at per gallon. What is her net rate of pay, in dollars per hour, after this expense?
Show solution without answering
Key insight
The answer is per hour, so look at one hour. She drives 60 miles, earns 30 dollars, uses 2 gallons costing 4 dollars, and keeps 26 dollars.
Solution
The question asks for dollars per hour, so it is easiest to look at just one hour of driving.
- In one hour she drives miles.
- She is paid per mile, so she earns .
- Her car goes miles on one gallon, so miles uses gallons.
- Gas costs per gallon, so she spends .
- What she keeps in that hour is .
The answer is .
Why this works
Everything here grows at a steady rate with time: twice the hours means twice the miles, pay and gas. So the hours do not change the rate per hour. Picking the time unit that the question asks about saves a division at the end.
Alternative approach
Do the whole trip. In hours she drives miles and earns . She uses gallons, costing . She keeps over hours, which is per hour.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
Since the answer is a rate per hour, look at a single hour of driving.
In one hour she covers miles and earns dollars.
Those miles use gallons of gasoline, costing dollars.
Net pay per hour: 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)
Similar problems
Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.