The instructions on a -gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?
- A)
- B)
- C)
- D)
- E)
Answer
B
Key insight
Each mug needs a full 20 grams. Seventeen mugs use 340 grams, and the 10 grams left over cannot make another proper mug.
Solution
- Each proper mug takes grams. Count how many full groups of grams fit into grams.
- Try a round number first: mugs use grams, which leaves grams.
- Seven more mugs use grams, which leaves grams. So far that is mugs using grams.
- The last grams are only half of what a mug needs. An eighteenth mug would not be properly brewed, so it does not count.
So the most mugs is .
The answer is .
Why this works
Dividing gives , but you can only make whole mugs, so you drop the half. Whenever a question asks how many complete things you can make from a fixed supply, divide and then round down, even if the decimal part is or more.
The trap
Working out 350 ÷ 20 = 17.5 and rounding up to 18, choice (C). The eighteenth mug would be short of beans.
Concise solution
Every properly brewed mug uses exactly grams, so mugs use grams and the requirement is . Divide:
Seventeen mugs use grams and leave grams, which is not enough for an eighteenth mug brewed properly. Hence the greatest number 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)
Techniques
Set up the equation/formula and compute; no special trick needed