AMC 10 · 2007 · #14
Grade 6 algebraPick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Let n be the total number of people. The swap leaves n fixed, and the starting girls are 40% of n, or 0.4n.
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
After the swap the total is still n, but the girls drop to 0.4n-2, and the new 30% fact says that same count is 0.3n.
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 ways must be equal, which turns the words into an equation.
▸ Why?
Two expressions naming one quantity name the same value, so they can be set equal.
▸ Why?
Each share is a count out of a hundred of a stated total, so both sides can be written as plain numbers.
Solve and read off the girls
Collecting the n terms gives 0.1n=2, so n=20, and the starting girls are 0.4×20=8, choice (C).
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