Christina and Frieda want to buy the same book. Christina has $\frac{3}{4}$ of the money needed to buy the book and Frieda has half of the money needed to buy the book. If the book was $\$ 3$ cheaper, then together they would have exactly enough money to buy 2 copies of the book. What is the original price of the book?
(A) $\$ 4$
(B) $\$ 16$
(C) $\$ 12$
(D) $\$ 10$
(E) $\$ 8$

## Part C: Each correct answer is worth 8.
Solution 1

Suppose that the cost of one book, in dollars, is $C$.

Then Christina has $\frac{3}{4} C$ and Frieda has $\frac{1}{2} C$.

Combining their money, together Christina and Frieda have $\frac{3}{4} C+\frac{1}{2} C=\frac{3}{4} C+\frac{2}{4} C=\frac{5}{4} C$.

If the book was $\$ 3$ cheaper, then the cost to buy one book would be $C-3$.

If the cost of one book was $C-3$, then the cost to buy two at this price would be $2(C-3)$ or $2 C-6$.

Combined, Christina and Frieda would have enough money to buy exactly two books at this reduced price.

Thus, $2 C-6=\frac{5}{4} C$.

Solving,

$$
\begin{aligned}
2 C-6 & =\frac{5}{4} C \\
2 C-\frac{5}{4} C & =6 \\
\frac{8}{4} C-\frac{5}{4} C & =6 \\
\frac{3}{4} C & =6
\end{aligned}
$$

Since $\frac{3}{4}$ of 8 is 6 , then $C=8$.

Therefore, the original price of the book is $\$ 8$.

## Solution 2

We proceed by systematically trying the five multiple choice answers given.

| Initial cost  of the book | Combined money for  Christina and Frieda | Reduced cost  of the book | Number of books  they may buy |
| :---: | :---: | :---: | :---: |
| $\$ 4$ | $\frac{3}{4}$ of $\$ 4+\frac{1}{2}$ of $\$ 4=\$ 3+\$ 2=\$ 5$ | $\$ 1$ | $5 \div 1=5$ |
| $\$ 16$ | $\frac{3}{4}$ of $\$ 16+\frac{1}{2}$ of $\$ 16=\$ 12+\$ 8=\$ 20$ | $\$ 13$ | $20 \div 13=1.53 \ldots$ |
| $\$ 12$ | $\frac{3}{4}$ of $\$ 12+\frac{1}{2}$ of $\$ 12=\$ 9+\$ 6=\$ 15$ | $\$ 9$ | $15 \div 9=1.66 \ldots$ |
| $\$ 10$ | $\frac{3}{4}$ of $\$ 10+\frac{1}{2}$ of $\$ 10=\$ 7.50+\$ 5=\$ 12.50$ | $\$ 7$ | $12.50 \div 7=1.78 \ldots$ |
| $\$ 8$ | $\frac{3}{4}$ of $\$ 8+\frac{1}{2}$ of $\$ 8=\$ 6+\$ 4=\$ 10$ | $\$ 5$ | $10 \div 5=2$ |

We see that if the original price of the book is $\$ 8$, Christina and Frieda are able to buy exactly two copies of the book at the reduced price.

Thus, the initial cost of the book is $\$ 8$.

ANSWER: (E)