1963 AMC 12 Problem 19

Attempt Problem 19 of the 1963 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 1963 AMC 12 solutions, or check the answer key.

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

19.

In counting

n

colored balls, some red and some black, it was found that

49

of the first

50

counted were red. Thereafter,

7

out of every

8

counted were red. If, in all,

90\%

or more of the balls counted were red, the maximum value of

n

is:

225

210

200

180

175

Write the number of red balls as

49+\frac78(n-50)

Require that quantity to be at least

0.9n

The condition is

49+\frac78(n-50)\geq\frac9{10}n.

Simplifying gives

\frac{21}{4}\geq \frac{n}{40},

n\leq210.

The maximum is

210.

Thus, the correct answer is B.

Problem 19 in other years