2000 AMC 8 Problem 20
Attempt Problem 20 of the 2000 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
20.
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of with at least one coin of each type. How many dimes must you have?
Answer: A
Small hint:
Use cents modulo to find the number of pennies
Big hint:
After the pennies, solve for nickels, dimes, and quarters
Solution:
The number of pennies must have the same remainder as modulo so there are either or pennies. Seven pennies would leave only two coins for nickels, dimes, and quarters, impossible because at least one of each type is needed.
So there are pennies. The remaining coins are worth cents. If and are the numbers of nickels, dimes, and quarters, then and
Dividing the value equation by and subtracting the coin-count equation gives The only positive solution is and
Thus there must be dime.
Thus, A is the correct answer.
Problem 20 in other years
- 1985 AMC 8
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