2007 AMC 12B Problem 5

Attempt Problem 5 of the 2007 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

5.

The 20072007 AMC 1212 contests will be scored by awarding 66 points for each correct response, 00 points for each incorrect response, and 1.51.5 points for each problem left unanswered. After looking over the 2525 problems, Sarah has decided to attempt the first 2222 and leave the last 33 unanswered. How many of the first 2222 problems must she solve correctly in order to score at least 100100 points?
1313
1414
1515
1616
1717
Answer: D
Concepts:inequality
Small hint:
The 33 unanswered problems already contribute 3⋅1.53\cdot1.5 points
Big hint:
Solve 6x≥100−4.56x\ge100-4.5 and round the number correct xx up to a whole number
Solution:
The three unanswered problems give 3⋅1.5=4.53\cdot1.5=4.5 points.
So Sarah needs at least 100−4.5=95.5100-4.5=95.5 points from correct answers. Since 15<95.56<16, 15\lt\dfrac{95.5}{6}\lt16, she must solve at least 1616 problems correctly, which would give her 16⋅6+4.5=100.516\cdot6+4.5=100.5 points.
Thus, the correct answer is D.
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