AMC 10 · 2005 · #1
Grade 6 arithmeticPick an answer.
Each English sentence hides one multiplication, so turning the words into two equations makes the problem mechanical. Each equation gives a part and asks for the whole, which is a backwards question: undo the multiplication. The request x - y splits cleanly into three small jobs — find x, find y, subtract — so solve them in that order.
Turn the first sentence into an equation
The first sentence says a whole was shrunk to a tenth.
A percent is a rate per 100, so "10% of" is nothing more than multiplying by 1/10.
6.RP.A.3Convert To AlgebraUndo the tenth to recover x
Undoing that shrinking recovers 20.
If one tenth of the whole is 2, the whole must be ten copies of 2.
If one tenth of a number is 2, the number is ten copies of 2.
▸ Why?
A percent is a rate out of a hundred, so ten percent is exactly one tenth of the whole.
▸ Why?
Multiplying by ten undoes dividing by ten, so the whole is recovered from the part in one move.
Repeat the move for y
The same move on a fifth recovers 10.
Same reversal, a different fraction — a bigger percent needs a smaller whole to produce the same part.
6.EE.B.7Work BackwardsSubtract to finish
Subtracting gives 10, choice (D).
The two subproblems were solved separately; the last step just combines their answers.
6.EE.A.2Identify SubproblemsA percent tells you how the part was made from the whole, so running that one step backwards hands you the whole.
- Turn the first sentence into an equation
- Undo the tenth to recover x
- Repeat the move for y
- Subtract to finish