Number & Algebra
Variations
Model direct, inverse, joint, and partial variation and solve combined variation problems.
Variation describes how one quantity changes with another. It turns word statements into useful algebraic models.
Direct variation
\(y\) varies directly as \(x\): \[ y\propto x \quad\Rightarrow\quad y=kx \] where \(k\) is the constant of variation.
Example: If \(y=18\) when \(x=6\), then \(k=3\), so \(y=3x\).
Inverse variation
\(y\) varies inversely as \(x\): \[ y\propto\frac1x \quad\Rightarrow\quad y=\frac{k}{x}. \]
Example: If \(y=5\) when \(x=4\), then \(k=20\), so \(y=\frac{20}{x}\).
Variation with powers
You may see statements like:
- \(y\propto x^2 \Rightarrow y=kx^2\)
- \(y\propto \sqrt{x} \Rightarrow y=k\sqrt{x}\)
- \(y\propto \frac{1}{x^2} \Rightarrow y=\frac{k}{x^2}\)
Joint variation
If \(y\) varies jointly as \(x\) and \(z\): \[ y\propto xz \Rightarrow y=kxz. \]
Example: \(y=24\) when \(x=2,z=3\Rightarrow k=4\), so \(y=4xz\).
Partial variation
If \(y\) varies partly as \(x\), model has constant and variable part: \[ y=mx+c,\quad c\ne0. \]
This appears in pricing: fixed fee + per-unit charge.
Solving workflow
- Translate words into proportional equation with \(k\).
- Use given data to find \(k\).
- Substitute required values to answer question.
- Check units and reasonableness.
Common word-problem forms
- “\(A\) is directly proportional to \(B\)”
- “\(A\) varies inversely as \(B\)”
- “\(A\) varies jointly as \(B\) and \(C\)”
- “\(A\) varies directly as square/cube of \(B\)”
Exam strategy
- Always introduce \(k\) first before substituting numbers.
- Keep symbolic form until \(k\) is found.
- For inverse models, domain often excludes zero.
- State final answer with units when applicable.
History
Variation models grew from early physics and astronomy, where scientists needed formulas for how one quantity depends on another. The language of proportionality became a bridge between algebra and real measurements.
Derivation and reasoning
Why we use a constant \(k\)
“Directly proportional” means ratio \(\frac{y}{x}\) stays constant, so \(\frac{y}{x}=k\) and \(y=kx\). For inverse variation, product \(xy\) stays constant, so \(xy=k\).
Checkpoints
If \(y\propto x\) and \(y=15\) when \(x=3\), find \(y\) when \(x=10\).
Answer: \(k=5\Rightarrow y=5x\Rightarrow y=50\).
If \(y\propto \frac1x\) and \(y=8\) when \(x=5\), find \(y\) when \(x=20\).
Answer: \(k=40\Rightarrow y=\frac{40}{x}\Rightarrow y=2\).
\(y\) varies as \(x^2\). If \(y=27\) when \(x=3\), find \(y\) when \(x=5\).
Answer: \(y=kx^2,\ k=3\Rightarrow y=3(25)=75\).
\(y\) varies jointly as \(x\) and \(z\). If \(y=30\) when \(x=3,z=2\), find \(y\) when \(x=5,z=4\).
Answer: \(y=kxz,\ k=5\Rightarrow y=5(5)(4)=100\).
A taxi fare \(F\) varies partly as distance \(d\): \(F=md+c\). If \(F=26\) at \(d=8\), and \(F=38\) at \(d=14\), find \(m,c\).
Answer: Solve two equations: \(8m+c=26,\ 14m+c=38\).
\(6m=12\Rightarrow m=2,\ c=10\). So \(F=2d+10\).
Applications
- Physics: pressure, speed, and force models often use direct/inverse variation.
- Business: cost models combine fixed and variable components.
- Engineering: scaling laws use power variation relations.
References
- Standard high-school algebra units on direct and inverse variation.
- Elementary modeling texts for joint and partial variation.
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