**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

$n-\frac{1}{4}n=\frac{1}{2}c,or\frac{3}{2}n=c$

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

The answer is (B).

**Method II (Plugging in):**

Suppose there are 100 marbles in the bowl—choice (A). Removing 1/4 of them would leave 75 marbles in the bowl. Since this is 1/2 the capacity of the bowl, the capacity of the bowl is 150. But if we add 100 marbles to the original 100, we get 200 marbles, not 150. This eliminates (A).

Next, suppose there are 200 marbles in the bowl—choice (B). Removing 1/4 of them would leave 150 marbles in the bowl. Since this is 1/2 the capacity of the bowl, the capacity of the bowl is 300. Now, if we add 100 marbles to the original 200, we get 300 marbles—the capacity of the bowl. The answer is (B).