AMC 10 · 2005 · #4
Grade 6 arithmeticPick an answer.
The question asks for below-A quizzes, but the goal is stated in A's, so I flip focus with Tool #16: first pin down how many A's she must still earn, then the below-A quizzes are simply whatever is left over among the remaining quizzes. To get that A-count I break the problem into small pieces with Tool #7 (Identify Subproblems): total A's needed, A's already banked, A's still owed, and finally the leftover. Each piece is one short arithmetic step, so no algebra is required.
Turn 80% into a count
The percent becomes a target of 40 quizzes.
A percent of a total is just a fraction of it, so 80% of 50 is a concrete count of quizzes.
Eighty percent of fifty quizzes is a concrete count of quizzes, not a rate.
▸ Why?
A percent of a total is that many hundredths of it, which for a whole number of quizzes is another whole number.
▸ Why?
The quizzes already taken and those still to come make up the whole term, so the goal splits between them.
Subtract the A's already earned
Subtracting what she has leaves 18 still needed.
The A's she already banked shrink the goal by exactly that many.
4.OA.A.3Identify SubproblemsCount the quizzes still left
Only 20 quizzes remain.
Only the quizzes she has not taken yet can still change her total.
4.OA.A.3Identify SubproblemsThe leftover is the below-A cap
The leftover is the allowance, so the answer is 2, choice (B).
Every remaining quiz is either a needed A or a spare, and the spares are what may drop below an A.
4.OA.A.3Change Focus Count The ComplementWhen a goal is set in one kind of thing, count that thing first — the answer you want is often just whatever is left over.
- Turn 80% into a count
- Subtract the A's already earned
- Count the quizzes still left
- The leftover is the below-A cap