Problem 18

To solve the problem, we start by defining variables for the prices of the fruits per kilogram. Let:
- $x$ represent the price of $1$ kilogram of apples in yuan,
- $y$ represent the price of $1$ kilogram of watermelons in yuan,
- $z$ represent the price of $1$ kilogram of oranges in yuan.

Given the purchases made by Xiaoming and Xiaohui, we can set up the following system of equations based on the total cost of their purchases:

1. For Xiaoming's purchase:
$$x + 4y + 2z = 27.2 \quad \text{(1)}$$

2. For Xiaohui's purchase:
$$2x + 6y + 2z = 32.4 \quad \text{(2)}$$

To find the total cost of buying $1$ kilogram of apples and $2$ kilograms of watermelons, we need to eliminate $z$ from the equations. We do this by subtracting equation (1) from equation (2), which eliminates $z$ and gives us an equation in terms of $x$ and $y$ only:

$$\begin{align*} (2x + 6y + 2z) - (x + 4y + 2z) &= 32.4 - 27.2 \\ 2x + 6y - x - 4y &= 5.2 \\ x + 2y &= 5.2 \quad \text{(3)} \end{align*}$$

Equation (3) directly gives us the total cost of buying $1$ kilogram of apples and $2$ kilograms of watermelons, which is the sum of $x$ and $2y$. Therefore, the total cost is:

$$\boxed{5.2 \text{ yuan}}$$