# Miscellaneous Problems

*The language Q has the following properties:*

*(1) ABC is the base word.
(2) If C immediately follows B, then C can be moved to the front of the code word to generate another word.*

*Which one of the following is a code word in language Q?
A. CAB
B. BCA
C. AAA
D. ABA
E. CCC*

From (1), ABC is a code word.

From (2), the C in the code word ABC can be moved to the front of the word: CAB.

Hence, CAB is a code word and the answer is (A).

*Bowl S contains only marbles. If 1/4 of the marbles were removed, the bowl would be filled to 1/2 of its capacity. If 100 marbles were added, the bowl would be full. How many marbles are in bowl S?*

*A. 100
B. 200
C. 250
D. 300
E. 400*

Let *n* be the number of marbles in the bowl, and let *c* be the capacity of the bowl.

Then translating *“if *1/4* of the marbles were removed, the bowl would be filled to *1/2* of its capacity”* into an equation yields

, or

Next, translating *“if 100 marbles were added, the bowl would be full”* into an equation yields

100 + *n* = c

Hence, we have the system:

100 + *n* = c

Combining the two above equations yields

3*n* = 200 + 2*n*

*n* = 200

The answer is (B).

**Method II (Plugging in):**