Let’s solve the problems step by step. Since $y$ varies directly as $x$, the relation can be expressed as:

$y = kx$

where $k$ is the constant of proportionality.

1. $y = 12$ when $x = 4$, find $y$ when $x = 12$:

First, find $k$: $k = \frac{y}{x} = \frac{12}{4} = 3$

Now use $k$ to find $y$ when $x = 12$: $y = kx = 3 \times 12 = 36$

Answer: $y = 36$

2. $y = -18$ when $x = 9$, find $y$ when $x = 7$:

Find $k$: $k = \frac{y}{x} = \frac{-18}{9} = -2$

Now use $k$ to find $y$ when $x = 7$: $y = kx = -2 \times 7 = -14$

Answer: $y = -14$

3. $y = -3$ when $x = 4$, find $x$ when $y = 2$:

Find $k$: $k = \frac{y}{x} = \frac{-3}{4}$

Now use $k$ to find $x$ when $y = 2$: $y = kx \implies 2 = \frac{-3}{4}x \implies x = \frac{2 \times 4}{-3} = \frac{8}{-3} = -\frac{8}{3}$

Answer: $x = -\frac{8}{3}$

4. $y = 3$ when $x = 10$, find $x$ when $y = 1.2$:

Find $k$: $k = \frac{y}{x} = \frac{3}{10} = 0.3$

Now use $k$ to find $x$ when $y = 1.2$: $y = kx \implies 1.2 = 0.3x \implies x = \frac{1.2}{0.3} = 4$

Answer: $x = 4$

5. $y = 2.5$ when $x = 0.25$, find $y$ when $x = 0.75$:

Find $k$: $k = \frac{y}{x} = \frac{2.5}{0.25} = 10$

Now use $k$ to find $y$ when $x = 0.75$: $y = kx = 10 \times 0.75 = 7.5$

Answer: $y = 7.5$

Summary of Answers:

1. $y = 36$
2. $y = -14$
3. $x = -\frac{8}{3}$
4. $x = 4$
5. $y = 7.5$

Would you like me to explain any of these steps in more detail or solve similar problems?

Here are 5 related questions to consider:

1. What is the difference between direct and inverse variation?
2. Can you explain how to calculate the constant of proportionality in detail?
3. How does direct variation differ when the constant is negative?
4. Can you provide more examples of direct variation problems?
5. How would this problem change if $y$ varied inversely with $x$?

Tip: In direct variation problems, always start by finding $k$ (the constant) using the given values—it simplifies everything else!