What is direct variation?

Two quantities are in direct variation when an increase in one causes an increase in the other, and a decrease in one causes a decrease in the other. They are said to be directly proportional.

If $x$ varies directly as $y$, we write: $x \propto y$

This can be written as an equation by introducing a constant $k$: $x = ky$

Where $k$ is the constant of proportionality.

Solving direct variation problems

Step 1: Write the proportionality statement.

Step 2: Replace $\propto$ with $=$ and introduce $k$.

Step 3: Find the value of $k$ using the given values.

Step 4: Use the equation to find the unknown quantity.

Worked Example 1

Given that $y$ varies directly as $x$, and $y = 10$ when $x = 15$, find the value of $y$ when $x = 7$.

Solution:

Step 1: $y \propto x$

Step 2: $y = kx$

Step 3: Find $k$: $10 = k \times 15$ $k = \frac{10}{15} = \frac{2}{3}$

Step 4: When $x = 7$: $y = \frac{2}{3} \times 7 = \frac{14}{3} = 4\frac{2}{3}$

Therefore, $y = \frac{14}{3}$ or approximately $4.67$.

Direct Variation with Powers

Sometimes one quantity varies directly as a power of another (square, cube, square root, etc.).

Example: If $y$ varies directly as the square of $x$, and $y = 8$ when $x = 2$, find $y$ when $x = 1$.

Solution:

$y \propto x^2$ means $y = kx^2$

Find $k$: $8 = k \times 2^2$ $8 = 4k$ $k = 2$

When $x = 1$: $y = 2 \times 1^2 = 2$

What is inverse variation?

Two quantities are in inverse variation when an increase in one causes a decrease in the other, and vice versa. They are said to be inversely proportional.

If $x$ varies inversely as $y$, we write: $x \propto \frac{1}{y}$

This becomes: $x = \frac{k}{y}$

Where $k$ is the constant of proportionality. Notice that $k = xy$.

Solving inverse variation problems

Step 1: Write the proportionality statement.

Step 2: Replace $\propto$ with $=$ and introduce $k$.

Step 3: Find $k$ using the relationship $k = xy$ (multiply the given paired values).

Step 4: Use the equation to find the unknown.

Worked Example 2

If $y$ varies inversely as $x$ and $y = 60$ when $x = \frac{1}{12}$, find:

(a) The constant of variation.

(b) The value of $x$ when $y = \frac{1}{2}$.

Solution:

(a) Since $y \propto \frac{1}{x}$, we have $y = \frac{k}{x}$

Find $k$: $k = xy = 60 \times \frac{1}{12} = 5$

(b) When $y = \frac{1}{2}$: $\frac{1}{2} = \frac{5}{x}$ $x = 10$

What is joint variation?

Joint variation occurs when one quantity varies directly or inversely with two or more other quantities.

• If $z$ varies directly as $x$ and $y$: $z \propto xy$, so $z = kxy$
• If $z$ varies directly as $x$ and inversely as $y$: $z \propto \frac{x}{y}$, so $z = k\frac{x}{y}$

Worked Example 3

If $h$ varies jointly as $l$ and $m$, such that $h = 10$ when $l = 4$ and $m = 5$:

(a) Find the constant of proportionality.

(b) Find $m$ when $l = 20$ and $h = 30$.

Solution:

(a) $h \propto lm$ means $h = klm$

$10 = k \times 4 \times 5$ $10 = 20k$ $k = \frac{1}{2}$

(b) When $l = 20$ and $h = 30$: $30 = \frac{1}{2} \times 20 \times m$ $30 = 10m$ $m = 3$

1. Identify the type of variation from the problem statement (direct, inverse, or joint).

2. Write the proportionality using $\propto$ and the correct powers (square, cube, square root, etc.).

3. Convert to an equation by replacing $\propto$ with $=$ and adding the constant $k$.

4. Find $k$ by substituting the given values.

5. Solve for the unknown using the equation with the new values.

6. Check your answer by verifying it makes logical sense (for direct: increase gives increase; for inverse: increase gives decrease).

In Tanzania, variation is used in many practical situations. For example, when building a house, the number of workers and the time needed to complete work are in inverse variation — if you employ more workers, the time to finish decreases. If 10 workers can complete a construction task in 30 days, using 15 workers (with the same ability) would finish in $\frac{10 \times 30}{15} = 20$ days. This helps small business owners and project managers plan budgets and timelines accurately.