Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?
- A)
- B)
- C)
- D)
- E)
Answer
A
Key insight
A total ending in 5 forces the number of pennies to be a multiple of 5, hence zero; then four coins of at least 5 cents total at least 20.
Solution
Every answer choice is a multiple of . Nickels, dimes and quarters are all worth multiples of cents, so the pennies alone must contribute a multiple of cents. With only four coins in hand, the number of pennies is at most , so it must be .
Therefore all four coins are worth at least cents each, and the total is at least cents. A total of cents is impossible.
The other totals are all achievable:
-
-
-
-
The answer is .
Why this works
A divisibility observation (everything but pennies is a multiple of ) eliminates a whole coin type, and then a simple lower bound finishes. For "which cannot happen" questions, look for a structural reason one choice fails, then confirm the others with explicit examples.
The trap
Trying to assemble each total by trial and error without noticing that four coins with no pennies cannot total less than 20 cents.
Common mistakes
- Trying to assemble each total by trial and error without noticing that four coins with no pennies cannot total less than 20 cents.
- Forgetting that the four coins may repeat a denomination, and wrongly declaring or impossible.
Techniques
Use the answer choices (mod checks, size, form) to eliminate or select · Bound the quantity above/below or estimate to pin it down