What is direct variation?

Two quantities have a direct variation relationship 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.

The symbol for proportionality is "∝". We write:

$A \propto B$

This means "A varies directly as B."

To form an equation, we introduce a constant of proportionality (k):

$A = kB$

Where:

• k is a constant that does not change
• The ratio A/B is always equal to k

Finding the constant of proportionality

Given that A = kB, we can find k by:

$k = \frac{A}{B}$

If we know one pair of values for A and B, we can find k and then use it to find other values.

Worked Example 1: Direct Variation

Given that x varies directly as y. If x = 20 when y = 45, find:

(a) The value of y when x = 100.

(b) The value of x when y = 70.

Solution

Since x ∝ y, we have:

$x = ky$

Find k:

$k = \frac{x}{y} = \frac{20}{45} = \frac{4}{9}$

(a) When x = 100:

$\frac{4}{9} = \frac{100}{y}$

$4y = 900$

$y = \frac{900}{4} = 225$

(b) When y = 70:

$\frac{4}{9} = \frac{x}{70}$

$9x = 280$

$x = \frac{280}{9}$

Direct variation with powers

Sometimes a quantity varies directly as a power of another:

• $y \propto x^2$ means $y = kx^2$
• $y \propto x^3$ means $y = kx^3$

The same method applies — find k first, then use it to find unknown values.

What is inverse variation?

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

We write:

$p \propto \frac{1}{q}$

This means "p varies inversely as q."

The equation is:

$p = k \times \frac{1}{q} = \frac{k}{q}$

From this, we can see that:

$k = pq$

The product of the two quantities is always constant.

Worked Example 2: Inverse Variation

If y varies inversely as x and y = 60 when x = 1/12, find:

(a) The constant of variation.

(b) The value of x when y = 1/2.

(c) The value of y when x = 100.

Solution

Since y ∝ 1/x, we have:

$y = \frac{k}{x}$

Find k:

$k = y \times x = 60 \times \frac{1}{12} = 5$

(a) Constant of variation k = 5

(b) When y = 1/2:

$5 = \frac{k}{x} = \frac{5}{x}$

$x = 10$

(c) When x = 100:

$y = \frac{5}{100} = \frac{1}{20}$

What is joint variation?

Joint variation involves three or more quantities. One quantity varies directly or inversely with two or more other quantities at the same time.

Types of joint variation

1. Direct joint variation: If A varies directly with B and C:

$A \propto BC$

$A = kBC$

2. Mixed variation: If p varies directly as q and inversely as r:

$p \propto \frac{q}{r}$

$p = k\frac{q}{r}$

3. Inverse joint variation: If m varies inversely as both n and t:

$m \propto \frac{1}{nt}$

$m = \frac{k}{nt}$

Worked Example 3: Joint Variation

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 the value of m when l = 20 and h = 30.

Solution

Since h ∝ lm:

$h = klm$

(a) Find k:

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

(b) When l = 20 and h = 30:

From h = klm:

$30 = \frac{1}{2} \times 20 \times m$

$30 = 10m$

$m = 3$

• Always find the constant of proportionality (k) first using the given values
• In direct variation, the ratio of the two quantities is constant
• In inverse variation, the product of the two quantities is constant
• In joint variation, you work with three or more quantities at once
• The graph of direct variation is a straight line through the origin
• The graph of inverse variation is a curve that gets closer to the axes but never touches them

In Tanzania, variations are used in everyday life. For example, when planning a journey by daladala or bus, the time taken varies inversely with speed — if you travel faster, you reach your destination in less time. If a bus takes 2 hours to travel from Dar es Salaam to Morogoro at an average speed of 60 km/h, you can use inverse variation to calculate how long it would take if the speed increases to 80 km/h. This helps in planning travel schedules and estimating fuel costs for small business owners transporting goods to market.