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2009 AMC 12A Problem 3

Below is the professionally curated solution for Problem 3 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:fractionratio and proportion

Difficulty rating: 1050

3.

What number is one third of the way from 14\dfrac{1}{4} to 34?\dfrac{3}{4}?

13\dfrac{1}{3}

512\dfrac{5}{12}

12\dfrac{1}{2}

712\dfrac{7}{12}

23\dfrac{2}{3}

Solution:

The gap is 34−14=12.\dfrac{3}{4} - \dfrac{1}{4} = \dfrac{1}{2}. One third of the way adds 13⋅12=16.\dfrac{1}{3} \cdot \dfrac{1}{2} = \dfrac{1}{6}.

So the number is 14+16=312+212=512.\dfrac{1}{4} + \dfrac{1}{6} = \dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}.

Thus, the correct answer is B.

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