AMC 10 · 2010 · #3
Grade 4 number-theoryPick an answer.
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
A whole ticket count makes the price a divisor.
If the money splits into whole tickets, the price must go into the total evenly.
If the money splits into whole tickets, the price has to go into the total evenly.
▸ Why?
A count of tickets is a whole number exactly when the division leaves no remainder.
▸ Why?
Each price pairs with the ticket count it produces, so the possible prices are exactly the divisors.
Price divides $64
The same must hold for the other total.
The one price has to fit both totals, so it must divide each of them.
4.OA.B.4Introduce A VariableList common factors
So only the common divisors survive.
A price that works must live in both factor lists at once.
4.OA.B.4Make A Systematic ListCount the values
There are 5 of them, choice (A).
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