Direct variation

Relationships of the form y = kx, finding the constant of variation from one pair of values, conversion graphs such as currency, and telling direct variation apart from other linear models.

Double the hours, double the pay. Buy three times the fuel, pay three times as much. When one quantity is always the same multiple of another, we say it varies directly with it.

The rule y = kx

Direct variation

$$y = kx \qquad k = \frac{y}{x} \text{ is the constant of variation}$$

This is $y = mx + c$ with $c = 0$. The graph is a straight line through the origin, and the gradient is $k$. The ratio $\dfrac{y}{x}$ is the same for every pair of values, which is a quick test from a table.

Example 1●

Leila's pay varies directly with the hours she works. She earns $\$228$ for $8$ hours. How much does she earn for $13.5$ hours?

$k = \dfrac{228}{8} = 28.5$

k is pay divided by hours: her hourly rate.

$P = 28.5h$

Write the rule.

$P = 28.5 \times 13.5 = 384.75$, so she earns $\$384.75$.

Substitute h = 13.5.

Conversion graphs

Unit and currency conversions are direct variation. A conversion graph lets you read either way. Exchange rates change every day, so questions give you the rate to use.

Example 2●●

Suppose $1$ Australian dollar buys $0.65$ US dollars. (a) How many US dollars do you get for $\$800$ AUD? (b) A hotel in Hawaii costs $\$260$ USD a night. What is that in AUD?

(a) $U = 0.65A = 0.65 \times 800 = 520$ USD

AUD to USD: multiply by the rate.

(b) $A = \dfrac{260}{0.65} = 400$ AUD

USD to AUD: go backwards, so divide by the rate.

InteractiveConversion lines through the origin

Try this

Pick kilometres to miles and slide to $42.2$ km, the length of a marathon. Read the miles. Next, pick AUD to USD and switch on the $\$10$ exchange fee. Watch the ratio $y \div x$. Why is the line with a fee no longer direct variation?

Linear, but not direct

A taxi fare with a flagfall, a gym with a joining fee, a phone plan with a monthly charge: all are linear, but none is direct variation. They don't pass through the origin, so doubling the distance doesn't double the cost.

Example 3●●

A taxi charges $F = 2.3d + 4.2$ dollars for $d$ km. Show that a $20$ km trip doesn't cost twice as much as a $10$ km trip.

$10$ km: $2.3 \times 10 + 4.2 = \$27.20$. $20$ km: $2.3 \times 20 + 4.2 = \$50.20$.

Substitute both distances.

$2 \times 27.20 = 54.40 \ne 50.20$, so the fare doesn't vary directly with distance.

Twice 27.20 is 54.40, which is more than 50.20, because the flagfall is only paid once.

Your turn

On a map, $3$ cm represents $7.5$ km. The map distance varies directly with the real distance. How far is a trip that measures $11$ cm on the map?

Show the working

$k = \dfrac{7.5}{3} = 2.5$ km per cm. Real distance $= 2.5 \times 11 = 27.5$ km.

Flashcards

Direct variation

$y = kx$: a straight line through the origin

Finding $k$

$k = \dfrac{y}{x}$ from any one pair of values

Linear but not direct

A straight line with a fixed starting amount: it misses the origin

Practice

$y$ varies directly with $x$, and $y = 12$ when $x = 4$. Find $k$.
For the same relationship, find $y$ when $x = 10$.
The graph of a direct variation is:
Pay varies directly with hours. Someone earns $\$171$ for $6$ hours. How much, in dollars, for $15$ hours?
If $1$ AUD buys $0.65$ USD, how many USD do you get for $250$ AUD?
If $1$ AUD buys $0.65$ USD, how many AUD is $91$ USD?
Which of these is not direct variation?
Fuel cost varies directly with litres bought. $40$ L costs $\$78$. What do $55$ L cost, in dollars?
Exam style: On a map, $3$ cm represents $7.5$ km. How many km does $11$ cm represent?
Exam style: The mass of a stack of identical pavers varies directly with the number of pavers. $12$ pavers weigh $30$ kg. A trailer can carry $425$ kg. How many pavers can it carry?