AMC 10 · 2007 · #6
Grade 6 arithmeticThe 2007 AMC 10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?
Pick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: On a 25-question test, each correct answer is worth 6 points, each wrong answer 0 points, and each unanswered question 1.5 points. Sarah will answer the first 22 questions and leave the last 3 blank. Find the smallest number of the 22 attempted questions she must get right so her total is at least 100 points.
Givens: A correct answer scores 6 points.; A wrong answer scores 0 points.; An unanswered question scores 1.5 points.; Sarah answers the first 22 questions and leaves the last 3 blank.
Unknowns: The least number of correct answers among the 22 attempted that reaches a total of at least 100 points
Understand
Restated: On a 25-question test, each correct answer is worth 6 points, each wrong answer 0 points, and each unanswered question 1.5 points. Sarah will answer the first 22 questions and leave the last 3 blank. Find the smallest number of the 22 attempted questions she must get right so her total is at least 100 points.
Givens: A correct answer scores 6 points.; A wrong answer scores 0 points.; An unanswered question scores 1.5 points.; Sarah answers the first 22 questions and leaves the last 3 blank.
Plan
Primary tool: #11 Work Backwards
Secondary: #8 Analyze the Units, #3 Eliminate Possibilities
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.
Execute — Answer: D
5.NBT.B.7 Step 1 Lock in the guaranteed points
- The 3 unanswered questions each score 1.5 points no matter what, so they contribute a fixed amount.
- Multiply to find it: 3 times 1.5 equals 4.5 points.
- Sarah walks in with 4.5 points already secured.
💡 The blanks are settled, so treat their points as money already in the bank.
5.NBT.B.7 Step 2 Work backwards from the target
- The goal is at least 100 points, and 4.5 of those are already covered by the blanks.
- Subtract to find how many points the correct answers must supply: 100 minus 4.5 equals 95.5.
- So the correct answers alone need to bring in at least 95.5 points.
💡 Take the guaranteed points off the goal, and what remains is the job the correct answers must do.
6.NS.B.3 Step 3 Divide to count the correct answers
- Each correct answer is worth 6 points, so divide the needed points by 6: 95.5 divided by 6 is about 15.9.
- That is how many correct answers it would take to hit exactly 95.5 points.
💡 Sharing the needed points into 6-point chunks tells you how many correct answers those chunks require.
4.OA.A.3 Step 4 Round up to a whole answer
- Sarah cannot answer 15.9 questions correctly, and 15 correct answers fall short: 15 times 6 plus 4.5 is 94.5, which is below 100.
- The next whole number, 16, works: 16 times 6 plus 4.5 is 100.5, which clears 100.
- So the smallest whole number that reaches at least 100 is 16, and the answer is (D).
💡 When a decimal count of whole things falls short, you always round up to reach the goal.
5.NBT.B.7 The 3 unanswered questions each score 1.5 points no matter what, so they contrib 5.NBT.B.7 The goal is at least 100 points, and 4.5 of those are already covered by the bla 6.NS.B.3 Each correct answer is worth 6 points, so divide the needed points by 6: 95.5 di 4.OA.A.3 Sarah cannot answer 15.9 questions correctly, and 15 correct answers fall short: Review
Reasonableness: Check the boundary directly. With 16 correct: 16 correct at 6 points is 96, plus 4.5 from the blanks is 100.5, which is at least 100. With 15 correct: 90 plus 4.5 is 94.5, which misses. So 16 is exactly the tipping point, matching (D). The jump between them is one correct answer worth 6 points, which is why 94.5 leaps to 100.5 and there is no value strictly between that also works.
Alternative: Set it up as one inequality. Let c be the number correct: 6c + 4.5 \ge 100. Subtract 4.5 to get 6c \ge 95.5, then divide by 6 to get c \ge 15.9\overline{1}. Since c must be a whole number, the least value is 16.
CCSS standards used (min grade 6)
5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Computing the guaranteed 3 x 1.5 = 4.5 points and subtracting to get 100 - 4.5 = 95.5.)6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals (Dividing the 95.5 points still needed by 6 points per correct answer to get about 15.9.)4.OA.A.3Solve multistep word problems, interpreting remainders in context (Recognizing that a fractional count of correct answers must be rounded up to the next whole number to reach at least 100.)
⭐ 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.
⭐ 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.
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