Miscellaneous Problems

Example 1:

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).

Example 2:

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

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

100 + n = c

Hence, we have the system:

$\frac{3}{2}n=c$

100 + n = c

Combining the two above equations yields

$\frac{3}{2}n=100+n$

3n = 200 + 2n

n = 200