AMC 10 · 2007 · #5
Grade 6 arithmeticPick an answer.
The target score is fixed at 100, and the unknown is how many correct answers reach it, so it is cleanest to start from 100 and peel back. First remove the points that are already guaranteed by the 3 blanks, then see how many 6-point correct answers are needed to cover what is left. Because a partial correct answer is impossible, the last move is to round up to a whole number.
Lock in the guaranteed points
The blanks bank 4.5 points before anything is answered.
The blanks are settled, so treat their points as money already in the bank.
5.NBT.B.7Analyze The UnitsWork backwards from the target
Subtracting from the target leaves what is still needed.
Take the guaranteed points off the goal, and what remains is the job the correct answers must do.
5.NBT.B.7Work BackwardsDivide to count the correct answers
Dividing gives a fractional count of answers.
Sharing the needed points into 6-point chunks tells you how many correct answers those chunks require.
6.NS.B.3Analyze The UnitsRound up to a whole answer
Rounding up and checking both neighbours gives 16, choice (C).
When a decimal count of whole things falls short, you always round up to reach the goal.
When a decimal count of whole things falls short, it always has to be rounded up to reach the goal.
▸ Why?
Dividing leaves a remainder, and a partial answer cannot be handed in as a whole one.
▸ Why?
One fewer correct answer would leave the total below the target, so the next whole number is the smallest that works.
Bank the points you are guaranteed, subtract them from the goal, divide by what each correct answer is worth, then round up because you can't get part of a question right.
- Lock in the guaranteed points
- Work backwards from the target
- Divide to count the correct answers
- Round up to a whole answer