AMC 10 · 2004 · #15
Grade 4 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Chasing the two unknown counts head-on invites algebra, but the clean move is Tool #16 (Change Focus): stop staring at the totals and watch what one swap does. Turning a nickel into a dime adds exactly 5 cents, and turning a dime into a nickel loses exactly 5 cents. So the whole 70-cent jump is built out of 5-cent steps, and dividing tells you how many more nickels than dimes there are. Tool #8 (Analyze the Units) keeps every quantity in cents so nothing gets misread, and Tool #6 (Guess and Check) lets you confirm the split lands on 20 coins.
See what one swap does
A nickel is 5 cents and a dime is 10 cents, so swapping any single coin moves the total by exactly 5 cents, up or down.
Nickels and dimes sit 5 cents apart, so swapping any one coin nudges the total by that same 5 cents.
The two coin values sit a fixed amount apart, so swapping any one coin nudges the total by exactly that amount.
▸ Why?
Each coin is worth a fixed number of cents, so a swap replaces one fixed value with another.
▸ Why?
Everything else in the pile stays put, so the total moves by exactly the difference of those two values.
Turn the 70-cent jump into a count
The total rose 70 cents in 5-cent steps, so Patty has 14 more nickels than dimes.
A net rise of 70 cents in 5-cent pieces can only come from 14 extra nickels over dimes.
3.OA.C.7Change Focus Count The ComplementSplit the 20 coins
Set the 14 extra nickels aside and the other 6 coins split evenly, giving 17 nickels and 3 dimes.
After pulling out the 14 extra nickels, the rest splits down the middle into equal counts.
4.OA.A.3Guess And CheckAdd up the value
17 × 5 + 3 × 10 = 85 + 30 = 115 cents, which is $1.15 — choice (A).
Counting each pile of coins in cents and adding gives the money on the table.
4.MD.A.2Analyze The UnitsWatch what one swap changes instead of the whole pile: since each swapped coin shifts the total by 5 cents, a 70-cent jump means 14 more nickels than dimes, which forces 17 nickels and 3 dimes worth $1.15.
- See what one swap does
- Turn the 70-cent jump into a count
- Split the 20 coins
- Add up the value