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1968 AMC 12 Problem 34

Attempt Problem 34 of the 1968 AMC 12 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 1968 AMC 12 solutions, or check the answer key.

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

34.

With 400400 members voting, the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the re-vote was 1211\dfrac{12}{11} of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time?

7575

6060

5050

4545

2020

Answer: B
Concepts:system of equationsratio and proportion
Difficulty rating: 1900
Small Hint:

Let the original and second numbers voting for be FF and GG

Big Hint:

Use G=1211(400−F)G=\frac{12}{11}(400-F) and 2G−400=2(400−2F)2G-400=2(400-2F)

Solution:

Let F,F, GG be the original and second numbers voting for the bill. The ratio condition gives G=1211(400−F).G=\frac{12}{11}(400-F). The passage margin is twice the original defeat margin, so 2G−400=2(400−2F),2G-400=2(400-2F), or G+2F=600.G+2F=600. Solving gives F=180,F=180, G=240,G=240, so 6060 more members voted for it.

Therefore, the correct answer is B.

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