1994 AMC 12 Problem 16

Attempt Problem 16 of the 1994 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 1994 AMC 12 solutions, or check the answer key.

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

16.

Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally?

8

22

36

57

71

Answer: B

Concepts:systems of equations proportions

Difficulty rating: 1510

Small hint:

Let

R

be the red count and

T

the total count

Big hint:

Translate the two experiments into

\frac{R-1}{T-1}=\frac{1}{7}

and

\frac{R}{T-2}=\frac{1}{5}

Solution:

The two conditions give

\frac{R-1}{T-1}=\frac17

and

\frac{R}{T-2}=\frac15.

Thus

T=7R-6

and

T=5R+2.

Hence

R=4

and

T=22.

Thus the correct answer is B.