Problem 185

To solve the problem, we start by defining variables for the prices of the notebooks and pencil cases. Let's denote:

- The price of each notebook as $x$ yuan.
- Consequently, the price of each pencil case will be $x + 3$ yuan, since it's given that each pencil case costs $3$ yuan more than each notebook.

The problem states that the quantity of notebooks that can be bought with $200$ yuan is the same as the quantity of pencil cases that can be bought with $350$ yuan. This can be translated into an equation based on the concept that the total cost divided by the price per item gives the quantity of items that can be bought. Therefore, we have:

$$\frac{200}{x} = \frac{350}{x + 3}$$

To solve for $x$, we cross-multiply:

$$200(x + 3) = 350x$$

Expanding the left side:

$$200x + 600 = 350x$$

Rearranging the equation to solve for $x$:

$$350x - 200x = 600$$

$$150x = 600$$

$$x = \frac{600}{150}$$

$$x = 4$$

Thus, the price of each notebook is $4$ yuan. To find the price of each pencil case, we add $3$ to the price of a notebook:

$$x + 3 = 4 + 3 = 7$$

Therefore, the prices of the pencil cases and notebooks are $7$ yuan and $4$ yuan, respectively. Encapsulating the final answer:

$$\boxed{\text{The prices of the pencil cases and notebooks are 7 yuan and 4 yuan, respectively.}}$$