2005 AMC 10B Problem 5
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
Show solution without answering
Key insight
If one third of the CDs costs one fifth of her money, then all the CDs cost three times as much: three fifths. Two fifths is left.
Solution
1. All the CDs cost the same, so buying three times as many CDs costs three times as much money.
2. One third of the CDs costs one fifth of her money. All of the CDs is three times one third, so they cost
3. The question asks what is left. She started with all of it, whole, so
The answer is .
Why this works
The money spent grows in step with the number of CDs bought, so tripling one amount triples the other. You never need her actual amount of money. Before choosing an answer, reread the question: here it asks for what is left, not what is spent.
Alternative approach
Make up numbers. Say she has dollars and wants CDs. One fifth of her money is dollars, and that buys one CD (one third of the CDs). So each CD costs dollars, and all three cost dollars. She has dollars left, and .
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
One third of the CDs cost one fifth of her money. Since the CDs are equally priced, the full set costs three times as much:
What remains is
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. Also appeared as 2005 12B #3.