Let's go through each question on the page to understand and answer them:

Question 19

The equation given is $10 = \frac{a}{b}$. This represents an inverse variation because as $b$ increases, $a$ must decrease to keep the ratio constant.

• Answer: b. inverse

Question 20

The equation given is $xy = 12$. This also represents inverse variation since the product of $x$ and $y$ is constant. If $x$ increases, $y$ must decrease, and vice versa.

• Answer: b. inverse

Question 21

The question asks for an equation where the volume $V$ varies directly with the temperature $T$ and inversely with the pressure $P$. This translates to $V = \frac{kT}{P}$.

• Answer: a. $V = \frac{kT}{P}$

Question 22

$y$ is inversely proportional to $x^2$. So, $y = \frac{k}{x^2}$. Given $y = \frac{4}{9}$ when $x = -3$:

1. Find $k$: $\frac{4}{9} = \frac{k}{(-3)^2} \Rightarrow k = 4$.
2. Now, when $x = -6$: $y = \frac{4}{(-6)^2} = \frac{4}{36} = \frac{1}{9}$.

• Answer: d. $\frac{1}{9}$

Question 23

If $m$ is directly proportional to $n^2$, then $m = kn^2$. If $n$ increases by 25%, $n$ becomes $1.25n$:

1. New $m$ becomes $k(1.25n)^2 = k \cdot 1.5625 \cdot n^2$.
2. So $m$ increases by 56.25%.

• Answer: c. $m$ increases by 56.25%

Question 24

If $y$ varies directly with $x$, $y = kx$. Given $x = 10$ when $y = 50$:

1. $k = \frac{y}{x} = \frac{50}{10} = 5$.
2. To find $x$ when $y = 60$: $60 = 5x \Rightarrow x = 12$.

• Answer: b. 12

Question 25

If $V$ is inversely proportional to $X$ and $V = 7$ when $X = 2$, then $V \cdot X = k$.

1. $k = 7 \times 2 = 14$.
2. Let's analyze each statement:
   • I. True, as $k = 14$.
   • II. False, as $V = \frac{k}{X}$, not $V = kX$.
   • III. True, if $V = 14$, $X = \frac{k}{V} = \frac{14}{14} = 1$.
   • IV. True, $X$ is inversely proportional to $V$.

• Answer: d. I, III, and IV

Question 26

The area $A$ of a circle is directly proportional to the square of its radius $r$, which means $A = kr^2$. Given $A \approx 314$ when $r = 10$:

1. $k = \frac{A}{r^2} = \frac{314}{10^2} = 3.14$.

• Answer: d. $A = 3.14r^2$

Would you like detailed explanations for any of these answers?

Here are 5 questions that build on this material:

1. How would you determine if a variation is direct or inverse from a given equation?
2. How does the concept of proportionality apply to real-world situations like gas laws?
3. If an inverse relationship is given, what happens to one variable when the other is doubled?
4. What steps are involved in calculating the constant of proportionality for different types of variation?
5. Can you provide examples of equations representing joint variation?

Tip: When working with variation problems, identify if the relationship is direct or inverse by checking if the variables multiply to a constant (inverse) or if one is a multiple of the other (direct).