AMC 10 · 2010 · #8
Grade 4 number-theoryPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks how many prices work, so the plan is to nail down the rule every valid price must obey, then list all numbers that obey it and count them. A price times a whole number of tickets equals a total, so the price must divide each total exactly. Listing the numbers that divide both totals turns the puzzle into a short, checkable list.
Price divides $48
Let t be how many tickets the 9th graders bought. Then 48 = t × x, and t is a whole number, so x must divide 48 exactly.
If the money splits into whole tickets, the price must go into the total evenly.
If the money splits into whole tickets, the price must go into the total evenly.
▸ Why?
Anything else leaves a remainder, which would be money that buys no whole ticket.
▸ Why?
Divisors come in pairs that multiply back to the total, so the candidate prices form a short closed list.
Price divides $64
The same reasoning for the 10th graders: 64 = s × x with s a whole number, so x must divide 64 too.
The one price has to fit both totals, so it must divide each of them.
4.OA.B.4Introduce A VariableList common factors
So x is a common factor of 48 and 64. Write out the factors of each, then keep the numbers on both lists.
A price that works must live in both factor lists at once.
4.OA.B.4Make A Systematic ListCount the values
Both totals are divided by 1, 2, 4, 8, and 16 — that is 5 possible prices, choice (E).
Counting the shared factors directly answers how many prices are possible.
4.OA.B.4Make A Systematic ListA ticket price has to divide every group's total exactly, so just count the numbers that go evenly into both totals.
- Price divides $48
- Price divides $64
- List common factors
- Count the values