Solved DIRECT, INVERSE AND JOINT VARIATION WORKSHEET Direct ... - Free Printable

Let's go through each problem step by step, using the definitions and methods provided in the worksheet.

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Part 1: Identify the type of variation and name the constant of variation

1) $ y = 2x $
- This is in the form $ y = kx $, so it’s a direct variation.
- Constant of variation $ k = 2 $

✔ Answer: Direct variation; $ k = 2 $

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2) $ \frac{x}{5} = y $
- Rewrite: $ y = \frac{1}{5}x $ → same as $ y = kx $
- So it’s a direct variation
- $ k = \frac{1}{5} $

✔ Answer: Direct variation; $ k = \frac{1}{5} $

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3) $ xy = 12 $
- Rearranged: $ y = \frac{12}{x} $ → this is $ y = \frac{k}{x} $
- So it’s an inverse variation
- $ k = 12 $

✔ Answer: Inverse variation; $ k = 12 $

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4) $ D = \frac{3}{4}gh $
- This is in the form $ D = kgh $, where $ g $ and $ h $ are variables (assuming).
- So $ D $ varies jointly with $ g $ and $ h $ → joint variation
- Constant $ k = \frac{3}{4} $

✔ Answer: Joint variation; $ k = \frac{3}{4} $

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Part 2: Translate statements into formulas

5) E varies jointly as M and the square of V.
- "Jointly" means product of both variables
- Square of V → $ V^2 $
- So: $ E = kM V^2 $

✔ Answer: $ E = kM V^2 $

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6) The volume, V, of a gas varies directly as the temperature, T, and inversely as the pressure P.
- Directly as T → $ V \propto T $
- Inversely as P → $ V \propto \frac{1}{P} $
- Combined: $ V = k \cdot \frac{T}{P} $

✔ Answer: $ V = k \frac{T}{P} $

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7) The mass, M, of a cement block varies jointly as the length, L, width, W, and thickness, T.
- "Jointly" means all multiplied together
- So: $ M = kLWT $

✔ Answer: $ M = kLWT $

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8) P varies directly as the square of V and inversely as R.
- Directly as $ V^2 $ → $ P \propto V^2 $
- Inversely as R → $ P \propto \frac{1}{R} $
- So: $ P = k \cdot \frac{V^2}{R} $

✔ Answer: $ P = k \frac{V^2}{R} $

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Part 3: Write an equation and solve

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9) If y varies inversely as x and $ y = 2 $ when $ x = 8 $, find x when $ y = 14 $.

- Step 1: General formula for inverse variation: $ y = \frac{k}{x} $
- Step 2: Use given values to find $ k $:
$$
2 = \frac{k}{8} \Rightarrow k = 2 \times 8 = 16
$$
- Step 3: Rewrite formula: $ y = \frac{16}{x} $
- Step 4: Find $ x $ when $ y = 14 $:
$$
14 = \frac{16}{x} \Rightarrow x = \frac{16}{14} = \frac{8}{7}
$$

✔ Answer: $ x = \frac{8}{7} $

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10) Suppose y varies jointly with x and z. If $ y = 20 $ when $ x = 2 $ and $ z = 5 $, find y when $ x = 14 $ and $ z = 8 $.

- Step 1: Joint variation: $ y = kxz $
- Step 2: Plug in known values:
$$
20 = k \cdot 2 \cdot 5 = 10k \Rightarrow k = \frac{20}{10} = 2
$$
- Step 3: Formula: $ y = 2xz $
- Step 4: Plug in $ x = 14 $, $ z = 8 $:
$$
y = 2 \cdot 14 \cdot 8 = 2 \cdot 112 = 224
$$

✔ Answer: $ y = 224 $

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11) If y varies inversely as x and $ x = 7 $ when $ y = 21 $, find y when $ x = 42 $.

- Step 1: Inverse variation: $ y = \frac{k}{x} $
- Step 2: Use given:
$$
21 = \frac{k}{7} \Rightarrow k = 21 \cdot 7 = 147
$$
- Step 3: Formula: $ y = \frac{147}{x} $
- Step 4: Find $ y $ when $ x = 42 $:
$$
y = \frac{147}{42} = \frac{7}{2} = 3.5
$$

✔ Answer: $ y = 3.5 $ or $ \frac{7}{2} $

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12) Find y when $ x = 1.5 $, if y varies directly as x and $ y = -16 $ when $ x = 6 $.

- Step 1: Direct variation: $ y = kx $
- Step 2: Use known values:
$$
-16 = k \cdot 6 \Rightarrow k = \frac{-16}{6} = -\frac{8}{3}
$$
- Step 3: Formula: $ y = -\frac{8}{3}x $
- Step 4: Plug in $ x = 1.5 = \frac{3}{2} $:
$$
y = -\frac{8}{3} \cdot \frac{3}{2} = -\frac{24}{6} = -4
$$

✔ Answer: $ y = -4 $

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✔ Final Answers Summary:

1. Direct variation; $ k = 2 $
2. Direct variation; $ k = \frac{1}{5} $
3. Inverse variation; $ k = 12 $
4. Joint variation; $ k = \frac{3}{4} $
5. $ E = kM V^2 $
6. $ V = k \frac{T}{P} $
7. $ M = kLWT $
8. $ P = k \frac{V^2}{R} $
9. $ x = \frac{8}{7} $
10. $ y = 224 $
11. $ y = \frac{7}{2} $ or $ 3.5 $
12. $ y = -4 $

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