2009 AMC 10B Problem 3
Paula the painter had just enough paint for identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for rooms. How many cans of paint did she use for the rooms?
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Key insight
The three lost cans were exactly the paint for the 5 rooms she can no longer do, so 3 cans paint 5 rooms, and 25 rooms need 15 cans.
Solution
The only thing that changed is that cans fell off, so start by asking what those cans were worth.
- Before the accident she could paint rooms. After it, only . So the lost cans were the paint for rooms.
- So cans paint rooms.
- rooms is times as many as rooms, so it needs times as many cans: cans.
- Check: she started with enough for rooms, which is cans. Losing leaves . That matches.
The answer is .
Why this works
When a known change in what you have (3 cans) causes a known change in what you can do (5 rooms), those two numbers give you the rate directly. After that, every other amount is found by scaling. Read the last sentence carefully: it asks for the cans she used, not the cans she started with.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
The three lost cans were exactly the paint for rooms. So cans cover rooms, and one can covers rooms.
To cover rooms she needs cans.
(Originally she had cans; losing leaves , consistent.)
The answer is .
Review in the AoPS books
- Introduction to Algebra, Chapter 6: Ratios and Percents
- Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
- 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.