Variation (Direct, Inverse, Joint, Partial & Percentage Changes)

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Variation — Edwin Ogie Library

Direct, inverse, joint, partial variations & percentage changes — worked examples and CBT for students.

Keywords: variation, proportionality Edwin Ogie Library

Introduction to Variation

Variation describes how one quantity changes with another. We study direct, inverse, joint, partial variations and percentage increases/decreases to solve many real-world and exam problems.

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Direct Variation

If y varies directly as x then y ∝ x or y = kx, where k is constant of proportionality.

Example

Q: y = 20 when x = 4. Find y when x = 10.

k = 20/4 = 5 ⇒ y = 5×10 = 50.

Inverse Variation

If y varies inversely as x: y ∝ 1/x or y = k / x.

Example

Q: y = 3 when x = 8. Find y when x = 16.

k = 3×8 = 24 ⇒ y = 24 / 16 = 1.5.

Joint & Partial Variation

Joint:

If y varies jointly as x and z: y = k x z.

Partial:

Partial/combined: y = k x + c (kx varies, c is constant).

Example (Joint)

Q: z = 30 when x=2 and y=5; find z when x=3 and y=4 (assuming z = kxy).

k = 30/(2×5)=3 ⇒ z = 3×3×4 = 36.

Percentage Increase & Decrease

Increase = [(New − Old)/Old] × 100%

Decrease = [(Old − New)/Old] × 100%

Example

Q: 80 → 100: percentage increase?

(100−80)/80 ×100 = 25%.

10 Worked Examples (Solutions Hidden)

Example 1 — Direct Variation

Q: If y varies directly as x and y=20 when x=4, find y when x=10.

k=5 ⇒ y=50.

Example 2 — Inverse Variation

Q: y=3 when x=8; find y when x=16.

k=24 ⇒ y=1.5.

Example 3 — Joint Variation

Q: z=30 when x=2, y=5; find z when x=3, y=4.

k=3 ⇒ z=36.

Example 4 — Partial Variation

Q: If y = 2x + 5, find y when x = 7.

y = 2×7 + 5 = 19.

Example 5 — Percentage Increase

Q: 80 → 100: % increase?

25%.

Example 6 — Percentage Decrease

Q: 120 → 90: % decrease?

(120−90)/120 ×100 = 25%.

Example 7 — Direct Variation % Change

Q: If x increases 20% in y = kx, what % does y increase?

y increases by 20% (same percent).

Example 8 — Inverse Variation % Change

Q: y = k/x; x increases 50% → % change in y?

New y = (2/3) original ⇒ y decreases by 33.33% (approx).

Example 9 — Joint Variation with change

Q: z=48 when x=4,y=3. If x+25% and y−20%, what's z?

k=48/(4×3)=4; x→5, y→2.4 ⇒ z = 4×5×2.4 = 48 (unchanged).

Example 10 — Partial Variation & %

Q: y = 3x + 10; x increases 10% — approximate % change in y?

Variable part increases 10% → 3x increases 10%. Overall % change depends on magnitude of 3x vs constant 10; compute with values for exact percent.

Summary

• Direct: y = kx
• Inverse: y = k / x
• Joint: y = k x z (or more variables)
• Partial: y = kx + c
• Percentage change: Use original value as base.

30-Question CBT Quiz (15 minutes)

Designed for JSS to early senior students. Click Start Quiz to begin.

Time Remaining: 15:00

Attempts: 0

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