Direct and Inverse Variation
Understanding how quantities change together
What is Variation?
Variation describes how one quantity changes in relation to another. When two quantities are related, a change in one causes a predictable change in the other.
Types of Variation
Direct Variation
\(y \propto x\)
\(y = kx\)
As \(x\) increases, \(y\) increases
As \(x\) decreases, \(y\) decreases
Graph: Straight line through origin
Inverse Variation
\(y \propto \frac{1}{x}\)
\(y = \frac{k}{x}\)
As \(x\) increases, \(y\) decreases
As \(x\) decreases, \(y\) increases
Graph: Rectangular hyperbola
The Constant of Variation (k)
In both types of variation, \(k\) is called the constant of variation or constant of proportionality.
For direct variation: \(k = \frac{y}{x}\)
For inverse variation: \(k = xy\)
Direct Variation
Example 1: Finding the Constant
If \(y\) varies directly as \(x\), and \(y = 15\) when \(x = 3\), find:
(a) the constant of variation, (b) the value of \(y\) when \(x = 7\)
\[y = kx\]
\[15 = k(3)\]
\[k = \frac{15}{3} = 5\]
\[y = 5x\]
\[y = 5(7) = 35\]
Example 2: Direct Variation with Powers
The area \(A\) of a circle varies directly as the square of its radius \(r\). If \(A = 78.5\) cm² when \(r = 5\) cm, find \(A\) when \(r = 8\) cm.
\[A \propto r^2 \quad \Rightarrow \quad A = kr^2\]
\[78.5 = k(5)^2 = 25k\]
\[k = \frac{78.5}{25} = 3.14\]
\[A = 3.14(8)^2 = 3.14(64) = 200.96 \text{ cm}^2\]
Inverse Variation
Example 3: Inverse Variation
If \(y\) varies inversely as \(x\), and \(y = 6\) when \(x = 4\), find \(y\) when \(x = 8\).
\[y = \frac{k}{x}\]
\[6 = \frac{k}{4}\]
\[k = 6 \times 4 = 24\]
\[y = \frac{24}{8} = 3\]
Notice: When x doubled (4 to 8), y halved (6 to 3). This is characteristic of inverse variation.
Example 4: Real-World Inverse Variation
The time \(t\) to complete a journey varies inversely as the speed \(s\). If it takes 4 hours at 60 km/h, how long would it take at 80 km/h?
\[t = \frac{k}{s}\]
\[4 = \frac{k}{60}\]
\[k = 240\]
(This represents the distance: 240 km)
\[t = \frac{240}{80} = 3 \text{ hours}\]
Interactive Variation Explorer
Visualize Variation
See how direct and inverse variation differ graphically
Equation & Properties
y = 10x
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 10 | 20 | 30 | 40 | 50 |
Joint and Combined Variation
Joint Variation
When a quantity varies directly as the product of two or more other quantities:
\[z \propto xy \quad \Rightarrow \quad z = kxy\]
Example 5: Combined Variation
The force \(F\) between two magnets varies directly as the product of their strengths \(m_1\) and \(m_2\), and inversely as the square of the distance \(d\) between them.
Write the equation for this relationship.
\[F \propto \frac{m_1 m_2}{d^2}\]
\[F = \frac{k m_1 m_2}{d^2}\]
Practice Problems
Question 1: If \(y\) varies directly as \(x\), and \(y = 12\) when \(x = 4\), find \(y\) when \(x = 10\).
Show Solution
\(y = kx\)
\(12 = k(4) \Rightarrow k = 3\)
\(y = 3x\)
When \(x = 10\): \(y = 3(10) = 30\)
Question 2: The number of workers \(w\) needed to complete a job varies inversely as the time \(t\) taken. If 6 workers can complete the job in 10 days, how many workers are needed to complete it in 4 days?
Show Solution
\(w = \frac{k}{t}\)
\(6 = \frac{k}{10} \Rightarrow k = 60\)
When \(t = 4\): \(w = \frac{60}{4} = 15\) workers
Question 3: The volume \(V\) of a cone varies jointly as its height \(h\) and the square of its radius \(r\). If \(V = 48\pi\) when \(h = 9\) and \(r = 4\), find \(V\) when \(h = 12\) and \(r = 5\).
Show Solution
\(V = khr^2\)
\(48\pi = k(9)(16) = 144k\)
\(k = \frac{48\pi}{144} = \frac{\pi}{3}\)
When \(h = 12, r = 5\):
\(V = \frac{\pi}{3}(12)(25) = 100\pi\)
Question 4: The gravitational force \(F\) between two objects varies directly as the product of their masses and inversely as the square of the distance between them. If \(F = 100\) N when \(m_1 = 10\) kg, \(m_2 = 20\) kg, and \(d = 2\) m, find \(F\) when \(d\) is doubled.
Show Solution
\(F = \frac{km_1m_2}{d^2}\)
\(100 = \frac{k(10)(20)}{4} = 50k\)
\(k = 2\)
When \(d = 4\): \(F = \frac{2(10)(20)}{16} = \frac{400}{16} = 25\) N
(Force becomes 1/4 when distance doubles)
CSEC Exam Tips
1. "varies directly" or "is proportional to" means \(y = kx\)
2. "varies inversely" means \(y = \frac{k}{x}\)
3. Always find \(k\) first using given values
4. Check your answer makes sense (inverse: bigger x = smaller y)
5. Watch for "varies as the square" which means \(x^2\)