Direct variation

The cost of rice varies directly as the weight bought. The distance travelled at constant speed varies directly as the time. When one quantity grows in proportion to another — both double together, both triple together — we have direct variation. WAEC tests this in every paper.

This lesson formalises direct variation, introduces the constant of proportionality, and works through the standard problem types: finding the constant from given values, then using it to find unknowns.

Learning objectives

By the end of this lesson you should be able to:

• explain direct variation and the symbol $\propto$
• write a direct variation as an equation $y = kx$
• find the constant of variation and use it
• recognise direct variation from a table or graph

Reading the notation

The symbol $\propto$ means "varies as" or "is proportional to."

• $y \propto x$: $y$ varies directly as $x$. $y = kx$.
• $y \propto x^2$: $y$ varies directly as the square of $x$. $y = kx^2$.
• $C \propto t$: cost varies directly as time. $C = kt$.
• $A \propto r^2$: area varies directly as the square of the radius. $A = kr^2$ (and $k = \pi$ for circle area).

Step-by-step technique

For any direct variation problem:

1. Write the relationship with the constant: $y = kx$ (or $y = kx^n$).
2. Find $k$ using the given pair of values.
3. Use the equation $y = kx$ to find unknowns.

Basic direct variation

Example 1

$y$ varies directly as $x$. When $x = 4$, $y = 12$. Find $y$ when $x = 7$.

Step 1: $y = kx$. Step 2: $12 = k(4)$, so $k = 3$. Step 3: When $x = 7$, $y = 3 \times 7 = 21$.

Example 2

$P \propto Q$. When $Q = 5$, $P = 20$. Find $P$ when $Q = 12$.

$P = kQ$. $20 = 5k$, $k = 4$. At $Q = 12$: $P = 48$.

Example 3

The cost $C$ varies directly as the weight $W$. If $C = 60$ NLe when $W = 4$ kg, find $C$ when $W = 11$ kg.

$C = kW$. $60 = 4k$, $k = 15$ NLe/kg. At $W = 11$: $C = 165$ NLe.

Example 4

$d \propto t$. When $t = 3$, $d = 75$. Find $d$ when $t = 8$.

$d = kt$. $75 = 3k$, $k = 25$. At $t = 8$: $d = 200$.

Example 5 — Finding $x$ given $y$

$y \propto x$, $y = 5$ when $x = 2$. Find $x$ when $y = 30$.

$k = 2.5$. $y = 2.5 x$. $30 = 2.5 x$. $x = 12$.

Direct variation with higher powers

Example 6 — $y \propto x^2$

The area of a circle varies directly as the square of its radius. A circle of radius $4$ has area $50.27$. Find the area when radius is $7$.

$A = kr^2$. $50.27 = 16 k$. $k \approx 3.14$ (this is $\pi$, of course). At $r = 7$: $A = \pi \times 49 \approx 153.94$.

Example 7

$y \propto x^2$. $y = 50$ when $x = 5$. Find $y$ when $x = 8$.

$k = 50/25 = 2$. $y = 2x^2$. At $x = 8$: $y = 128$.

Example 8 — $y \propto x^3$

The volume of a sphere varies directly as the cube of its radius. A sphere of radius $3$ has volume $113.1$. Find volume when radius is $6$.

$V = kr^3$. $113.1 = 27 k$. $k \approx 4.19$ (i.e., $\frac{4}{3}\pi$). At $r = 6$: $V = 4.19 \times 216 = 904.78$.

Note: doubling the radius multiplies the volume by $2^3 = 8$. ($113.1 \times 8 = 904.8$. ✓)

Example 9

$y \propto x^2$. $y = 100$ when $x = 5$. Find $x$ when $y = 144$.

$k = 100/25 = 4$. $y = 4x^2$. $144 = 4x^2$, $x^2 = 36$, $x = 6$.

Example 10 — $y \propto \sqrt{x}$

The period of a pendulum varies directly as the square root of its length. A $1$ m pendulum has period $2$ s. Find the period of a $4$ m pendulum.

$T = k\sqrt{L}$. $2 = k \sqrt{1}$, $k = 2$. At $L = 4$: $T = 2\sqrt{4} = 4$ s.

Four times the length, double the period.

Example 11

$y \propto \sqrt{x}$. $y = 6$ when $x = 4$. Find $y$ when $x = 25$.

$k = 6/\sqrt{4} = 3$. $y = 3\sqrt{x}$. At $x = 25$: $y = 15$.

Multiple-value problems

Sometimes you get two pairs of values and use ratios.

Example 12

$y \propto x$. If $y_1 = 6$ when $x_1 = 4$, find $y_2$ when $x_2 = 10$.

For direct variation: $\dfrac{y_2}{y_1} = \dfrac{x_2}{x_1}$. $\dfrac{y_2}{6} = \dfrac{10}{4}$, so $y_2 = 15$.

Example 13 — $y \propto x^2$

$y_1 = 12$ when $x_1 = 3$. Find $y_2$ when $x_2 = 5$.

$\dfrac{y_2}{y_1} = \dfrac{x_2^2}{x_1^2}$. $\dfrac{y_2}{12} = \dfrac{25}{9}$. $y_2 = \dfrac{300}{9} \approx 33.33$.

Or find $k$ first: $k = 12/9 = 4/3$. $y_2 = 4/3 \times 25 \approx 33.33$.

Word problems

Example 14

The wages of a worker varies directly as the hours worked. If a worker earns $400$ NLe in $5$ hours, how much in $12$ hours?

$W = kh$. $400 = 5k$, $k = 80$. At $h = 12$: $W = 960$ NLe.

Example 15

The cost of fuel varies directly as the kilometres travelled. A car uses $30$ NLe of fuel for $50$ km. Cost for $200$ km?

$C = kd$. $30 = 50k$, $k = 0.6$. $C = 120$ NLe.

Example 16

The mass of metal varies directly as the volume. $20$ litres of metal weighs $80$ kg. Mass of $35$ litres?

$M = kV$. $80 = 20k$, $k = 4$. $M = 140$ kg.

Example 17

The area of a square varies directly as the square of its side. A square of side $3$ has area $9$. Find the side when area is $64$.

$A = ks^2$. $9 = 9k$, $k = 1$. $s^2 = 64$, $s = 8$.

Example 18

The kinetic energy of an object varies directly as the square of its velocity. At $v = 10$ m/s, $KE = 50$ J. Find $KE$ at $v = 25$ m/s.

$KE = kv^2$. $50 = 100k$, $k = 0.5$. At $v = 25$: $KE = 0.5 \times 625 = 312.5$ J.

Example 19 — Sierra Leonean context

A taxi fare varies directly as the distance. Aminata paid $150$ NLe for $10$ km. Mohamed paid $375$ NLe. How far did Mohamed travel?

$F = kd$. $150 = 10k$, $k = 15$ NLe/km. Mohamed: $375 = 15 d$, $d = 25$ km.

Distinguishing direct variation from other relationships

Example 20

Is the following a case of direct variation? A taxi charges a NLe 50 flag-down fee plus NLe 15 per kilometre.

No — this is NOT direct variation. The relationship is $F = 50 + 15d$, which has a non-zero intercept. (At $d = 0$, fare is $50$, not $0$.) This is partial variation, covered in the next lesson.

Example 21

Is the following direct variation? The area of a square scales with the square of its side.

Yes — direct variation with the square. $A = ks^2$. Pure power-law relationship through the origin.

Ratio interpretation of direct variation

For direct variation $y \propto x$, the ratio $y/x$ is constant.

Example 22

Test if these data fit a direct variation: $(x, y) = (2, 8), (5, 20), (3, 12), (10, 40)$.

Ratios $y/x$: $8/2 = 4$, $20/5 = 4$, $12/3 = 4$, $40/10 = 4$.

All equal. Direct variation with $k = 4$.

Example 23

Test: $(1, 4), (2, 16), (3, 36), (4, 64)$.

Ratios $y/x$: $4, 8, 12, 16$. Not constant.

Ratios $y/x^2$: $4/1 = 4$, $16/4 = 4$, $36/9 = 4$, $64/16 = 4$. All equal.

So $y \propto x^2$ with $k = 4$.

WASSCE examination focus

Practice problems

1. $y \propto x$. $y = 24$ when $x = 6$. Find $y$ when $x = 11$.

2. $P \propto Q$. $P = 150$ when $Q = 5$. Find $Q$ when $P = 300$.

3. $C \propto t$. $C = 100$ at $t = 4$. Find $C$ when $t = 7$.

4. A car travels $250$ km in $5$ hours at constant speed. How far in $8$ hours?

5. A worker earns NLe 3,000 in $8$ hours. How much in $12$ hours?

6. $y \propto x^2$. $y = 18$ when $x = 3$. Find $y$ when $x = 5$.

7. $y \propto x^3$. $y = 16$ when $x = 2$. Find $y$ when $x = 5$.

8. $A \propto r^2$. Circle of radius $3$ has area $28.27$. Find area when radius is $5$.

9. $V \propto r^3$. Sphere of radius $2$ has volume $33.5$. Find volume when radius is $5$.

10. $y \propto \sqrt{x}$. $y = 4$ when $x = 16$. Find $y$ when $x = 36$.

11. The cost of fabric varies directly as length. $3$ metres costs NLe 45. Find the cost of $7$ metres.

12. A water tank's volume varies as the cube of its radius. A radius $2$ m tank holds $33.5$ m³. Find tank volume when radius is $4$ m.

13. Are these in direct variation? $(2, 6), (5, 15), (8, 24), (10, 30)$.

14. Are these in direct variation as $x^2$? $(2, 12), (3, 27), (4, 48)$.

Worked solutions

1. $k = 4$. $y = 44$.

2. $k = 30$. $Q = 10$.

3. $k = 25$. $C = 175$.

4. $k = 50$. $d = 400$ km.

5. $k = 375$. $4500$ NLe.

6. $k = 2$. $y = 50$.

7. $k = 2$. $y = 250$.

8. $k \approx 3.14$ ($= \pi$). $A = 78.54$.

9. $V = (4/3)\pi r^3$. $k = 33.5/8 = 4.19$. $V = 4.19 \times 125 = 523.6$ m³.

10. $k = 1$. $y = 6$.

11. $k = 15$. $C = 105$ NLe.

12. $V = kr^3$. $k = 33.5/8 \approx 4.19$. $V \approx 4.19 \times 64 = 268$ m³.

13. $y/x$ ratios: $3, 3, 3, 3$. Yes, direct variation with $k = 3$.

14. $y/x^2$ ratios: $12/4 = 3$, $27/9 = 3$, $48/16 = 3$. Yes, $y = 3x^2$.

What's next

In the next lesson we cover inverse variation, joint variation and partial variation — when two quantities have the opposite relationship: one grows while the other shrinks. Examples include speed and time, workers and time, pressure and volume.