Direct and Inverse Variation worksheet - Free Printable

Let's solve each problem step by step and explain the reasoning.

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Part 1: Find the Missing Variable

#### 1) y varies directly with x. If y = -4 when x = 2, find y when x = -6.

- Direct variation means:
$$
y = kx
$$
where $k$ is the constant of variation.

- Given: $y = -4$, $x = 2$
So:
$$
-4 = k(2) \Rightarrow k = \frac{-4}{2} = -2
$$

- Now use $k = -2$ to find $y$ when $x = -6$:
$$
y = (-2)(-6) = 12
$$

✔ Answer: 12

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#### 2) y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

- Inverse variation means:
$$
y = \frac{k}{x} \quad \text{or} \quad xy = k
$$

- Given: $y = 40$, $x = 16$
$$
k = xy = 40 \times 16 = 640
$$

- Now find $x$ when $y = -5$:
$$
-5 = \frac{640}{x} \Rightarrow x = \frac{640}{-5} = -128
$$

✔ Answer: -128

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#### 3) y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

- Inverse variation: $xy = k$

- $k = 7 \times (-4) = -28$

- Now find $y$ when $x = 5$:
$$
y = \frac{-28}{5} = -5.6
$$

✔ Answer: -5.6 (or $-\frac{28}{5}$)

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#### 4) y varies directly with x. If y = 15 when x = -18, find y when x = 1.6.

- Direct variation: $y = kx$

- $k = \frac{y}{x} = \frac{15}{-18} = -\frac{5}{6}$

- Now find $y$ when $x = 1.6$:
$$
y = -\frac{5}{6} \times 1.6 = -\frac{5}{6} \times \frac{8}{5} = -\frac{40}{30} = -\frac{4}{3} \approx -1.333...
$$

✔ Answer: $-\frac{4}{3}$ or $-1.\overline{3}$

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#### 5) y varies directly with x. If y = 75 when x = 25, find x when y = 25.

- $y = kx$, so:
$$
k = \frac{75}{25} = 3
$$

- Now $y = 25 = 3x \Rightarrow x = \frac{25}{3} \approx 8.333...$

✔ Answer: $\frac{25}{3}$ or $8.\overline{3}$

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Part 2: Classify as Direct, Inverse, or Neither

Recall:
- Direct variation: $y = kx$ → one variable is a constant multiple of the other.
- Inverse variation: $y = \frac{k}{x}$ → product is constant.
- Neither: doesn't fit either form.

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#### 6) m = -5p
- This is of the form $m = kp$, with $k = -5$.
✔ a) Direct

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#### 7) r = $\frac{9}{t}$
- This is $r = \frac{k}{t}$, $k = 9$
✔ b) Inverse

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#### 8) d = 4t
- $d = kt$, $k = 4$
✔ a) Direct

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#### 9) c = $\frac{e}{-4}$
- This is $c = -\frac{1}{4}e$, which is direct variation.
✔ a) Direct

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#### 10) n = $\frac{1}{2}f$
- $n = kf$, $k = \frac{1}{2}$
✔ a) Direct

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#### 11) z = $\frac{-2}{t}$
- $z = \frac{k}{t}$, $k = -2$
✔ b) Inverse

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#### 12) c = 3v
- $c = kv$, $k = 3$
✔ a) Direct

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#### 13) u = $\frac{i}{18}$
- $u = \frac{1}{18}i$, which is direct variation.
✔ a) Direct

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Part 3: What is the constant of variation?

The constant of variation $k$ is the coefficient in direct ($y = kx$) or inverse ($y = \frac{k}{x}$) variation.

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#### 14) d = 4t
- $k = 4$
✔ Answer: 4

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#### 15) z = $\frac{-2}{t}$
- $k = -2$
✔ Answer: -2

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#### 16) n = $\frac{1}{2}f$
- $k = \frac{1}{2}$
✔ Answer: $\frac{1}{2}$

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#### 17) r = $\frac{9}{t}$
- $k = 9$
✔ Answer: 9

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✔ Final Answers Summary:

#### Find the Missing Variable:
1) 12
2) -128
3) $-\frac{28}{5}$ or -5.6
4) $-\frac{4}{3}$ or -1.333...
5) $\frac{25}{3}$ or 8.333...

#### Classify:
6) a) Direct
7) b) Inverse
8) a) Direct
9) a) Direct
10) a) Direct
11) b) Inverse
12) a) Direct
13) a) Direct

#### Constant of Variation:
14) 4
15) -2
16) $\frac{1}{2}$
17) 9

Let me know if you'd like this formatted as a printable answer key!