AMC 10 · 2004 · #2

Grade 7 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
linear-equations-one-vardecimal-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: mental-arithmetic
📏 Medium solution 💡 1 insight
Problem
A test has 25 problems: a correct answer gives 6 points, a wrong answer 0, and a blank 2.5 points. Charlyn leaves 8 problems blank and answers the other 17. Find the smallest number of correct answers that pushes her total to 100 or more.

Pick an answer.

(A)
11
(B)
13
(C)
14
(D)
16
(E)
17
How to solve
Strategy Convert to Algebra

The score splits cleanly into two independent pieces (Tool #7): points from the 8 blanks, which are already fixed, and points from the answered problems, which depend on how many are correct. Wrong answers are worth 0, so they contribute nothing and can be ignored entirely. That makes the whole problem one inequality in one unknown (Tool #13): fixed points plus 6 per correct answer must reach 100. Because the question asks how many she must get right, the target is the smallest whole number satisfying that inequality (Tool #14) — solving the inequality gives a fraction, and the boundary has to be handled carefully.

1STEP 1

Bank the points from the blanks

The blanks are guaranteed points, banking 20 before anything else.

8×2.5 = 20
2STEP 2

Write the scoring inequality

Wrong answers add nothing, so the total is 6c + 20, needing at least 100.

6c + 20 ≥ 100
3STEP 3

Solve for the gap

Solving gives at least 13 and a third correct answers.

6c ≥ 80 ⟹ c ≥ 80/6 = 13 1/3
4STEP 4

Round up and check both sides

Rounding up and checking both sides gives 14, choice (C).

6(13)+20 = 98 ＜ 100, 6(14)+20 = 104 ≥ 100 → (C)
Answer
14
With 8 blanks her ceiling is 17×6+20 = 122 points, so 100 is reachable but demanding — she needs most of the 17 right, and 14 out of 17 fits that picture. The check at the boundary is the real test: 13 gives 98, just 2 points short, and 14 gives 104. Note that (D) 16 and (E) 17 do produce scores above 100 (116 and 122), but they are not answers to this question, because "must" asks for the least number that suffices. Choices (A) 11 and (B) 13 give 86 and 98, both below 100.
💡Key takeaway

Bank the points you already have, then divide the gap by what one problem is worth — 80 points of gap needs 14 six-point problems, because 13 leaves you 2 points short.

  • Bank the points from the blanks
  • Write the scoring inequality
  • Solve for the gap
  • Round up and check both sides

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