AMC 10 · 2007 · #10
Grade 6 algebraPick an answer.
The trick is that the total head count never changes, so it is the natural quantity to name. Introduce a variable for the total (Tool #4), then translate the two percent facts into expressions for the girl count before and after (Tool #13). Because a swap of two-for-two changes the girls by 2 but not the total, one clean equation pops out and solves in a line. Tool #3 (Eliminate Possibilities) is kept in reserve to sanity-check the result against the five answer choices.
Name the total and the girls
The swap leaves the total unchanged.
Give the quantity that stays fixed a name, and everything else can be measured against it.
6.EE.B.6Introduce A VariableTranslate the after-swap fact
So the after-picture is one equation in that total.
The same number of girls written two ways must be equal, which turns the words into an equation.
The same number of girls written two different ways must be equal, which turns the words into an equation.
▸ Why?
Two expressions naming the same quantity name the same number, so they can be set equal.
▸ Why?
A percent of the group is that many hundredths of it, so each fraction is a concrete head count.
Solve and read off the girls
Solving and taking the fraction gives 8, choice (B).
Gathering the variable on one side leaves a simple px=q that hands you the total directly.
6.EE.B.7Convert To AlgebraWhen a swap keeps the total the same, name that total, write the girl count two ways, and set them equal.
- Name the total and the girls
- Translate the after-swap fact
- Solve and read off the girls