A positive integer n not exceeding 100 is chosen in such a way that if n≤50, then the probability of choosing n is p, and if n>50, then the probability of choosing n is 3p. The probability that a perfect square is chosen is

1978 AMC 12 Problem 19

Attempt Problem 19 of the 1978 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 1978 AMC 12 solutions, or check the answer key.

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

19.

A positive integer

n

not exceeding

100

is chosen in such a way that if

n\le50,

then the probability of choosing

n

p,

and if

n\gt50,

then the probability of choosing

n

3p.

The probability that a perfect square is chosen is

0.05

0.065

0.08

0.09

0.1

First use the total probability to determine

p

Count the perfect squares at most

50

and those from

51

through

100

separately

The total probability is

50p+50(3p)=200p=1,

p=0.005.

There are seven perfect squares at most

50

and three more,

64,

81,

100,

above

50.

Hence the desired probability is

7p+3(3p)=16p=0.08.

Therefore, the correct answer is C.