AMC 10 · 2007 · #6

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
linear-equations-one-varmulti-digit-arithmetic work-backwards ↑ Prerequisites: linear-equations-one-var
📏 Medium solution 💡 2 insights
Problem
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.

Pick an answer.

(A)
13
(B)
14
(C)
15
(D)
16
(E)
17

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Work Backwards

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.

1STEP 1

Lock in the guaranteed points

The 3 blanks pay 1.5 points each no matter what, so Sarah banks 4.5 points before answering anything.

3 × 1.5 = 4.5
2STEP 2

Work backwards from the target

Peel that off the 100-point goal: the correct answers alone must supply at least 95.5 points.

100 - 4.5 = 95.5
3STEP 3

Divide to count the correct answers

Each correct answer is worth 6 points, so divide: 95.5 ÷ 6 is about 15.9 correct answers.

95.5 ÷ 6 = 15.916
4STEP 4

Round up to a whole answer

A question can't be part right, so round up: 15 correct scores 94.5, but 16 scores 100.5 — answer (D).

15 · 6 + 4.5 = 94.5 ＜ 100 16 · 6 + 4.5 = 100.5 ≥ 100
Answer
16
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.
💡Key takeaway

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

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