Equations and Graphs of Direct Proportion

Writing a direct proportion as the equation $y = kx$ , finding $k$ from one pair of values, drawing and reading the graph, and deciding from an equation or a graph whether two quantities are directly proportional.

Before you start

Lesson 6.1: $y$ is directly proportional to $x$ ( $y \propto x$ ) when $\frac{y}{x}$ is the same for every pair of values. That value is the constant of proportionality, $k$ .
Chapter 2: the gradient of a straight line is $\frac{\text{rise}}{\text{run}}$ , and the line $y = mx + c$ has gradient $m$ and meets the $y$ -axis at $(0, c)$ .

The Equation y = kx

If $\frac{y}{x} = k$ for every pair of values, then multiplying both sides by $x$ gives $y = kx$ . So a direct proportion can be written as an equation.

The equation of a direct proportion

If $y \propto x$ , then $y = kx$ , where $k$ is a constant and $k \neq 0$ .

$k$ is the constant of proportionality: $k = \frac{y}{x}$ for any pair of values. In this chapter $k$ is always positive.

To find $k$ , one pair of values is enough, because the question tells you that $y \propto x$ .

Example 1: Finding the equation, then values both ways

$y$ is directly proportional to $x$ , and $y = 12$ when $x = 4$ . Find
(a) an equation connecting $x$ and $y$ ,
(b) the value of $y$ when $x = 8$ ,
(c) the value of $x$ when $y = 21$ .

Solution

(a) Since $y \propto x$ , $y = kx$ . Put in $x = 4$ and $y = 12$ : $12 = k \times 4$ , so $k = \frac{12}{4} = 3$ . The equation is $y = 3x$ .

(b) When $x = 8$ : $y = 3 \times 8 = 24$ .

Check (c): $\frac{21}{7} = 3 = k$ .

(a) $y = 3x$ (b) $y = 24$ (c) $x = 7$

Example 2: k as a decimal or a fraction

(a) $y \propto x$ , and $y = 15$ when $x = 6$ . Find $y$ when $x = 9$ .
(b) $y \propto x$ , and $y = 4$ when $x = 10$ . Find $x$ when $y = 6$ .

Solution

(a) $15 = 6k$ , so $k = \frac{15}{6} = 2.5$ and $y = 2.5x$ . When $x = 9$ : $y = 2.5 \times 9 = 22.5$ .

(b) $4 = 10k$ , so $k = \frac{4}{10} = \frac{2}{5}$ and $y = \frac{2}{5}x$ (or $y = 0.4x$ ).

When $y = 6$ : $6 = \frac{2}{5}x$ , so $x = 6 \times \frac{5}{2} = 15$ .

Check (b): $\frac{6}{15} = \frac{2}{5}$ .

(a) $y = 22.5$ (b) $x = 15$

Example 3: Other letters, and what k means

The cost, $ $C$ , of $n$ kg of chicken wings is directly proportional to $n$ . 4 kg cost $22.
(a) Find an equation connecting $C$ and $n$ . What does $k$ stand for here?
(b) Find the cost of 11 kg.
(c) How many kilograms can be bought for $66?

Solution

(a) $C = kn$ . Put in $n = 4$ , $C = 22$ : $22 = 4k$ , so $k = 5.5$ and $C = 5.5n$ .

$k = \frac{C}{n}$ is the cost in dollars of 1 kg: the wings cost $5.50 per kilogram.

(b) When $n = 11$ : $C = 5.5 \times 11 = 60.5$ , so 11 kg cost $60.50.

(a) $C = 5.5n$ ; $k$ is the price per kilogram, $5.50. (b) $60.50 (c) 12 kg

Example 4: Completing a table

Given that $y \propto x$ , copy and complete the table.

$x$	$2$	$5$	?	$12$
$y$	?	$17.5$	$28$	?

Solution

The column $x = 5$ , $y = 17.5$ is complete: $k = \frac{17.5}{5} = 3.5$ , so $y = 3.5x$ .

When $x = 2$ : $y = 3.5 \times 2 = 7$ . When $y = 28$ : $x = \frac{28}{3.5} = 8$ . When $x = 12$ : $y = 3.5 \times 12 = 42$ .

$x$	$2$	$5$	$8$	$12$
$y$	$7$	$17.5$	$28$	$42$
$\dfrac{y}{x}$	$3.5$	$3.5$	$3.5$	$3.5$

The last row checks the table: $\frac{y}{x} = 3.5$ in every column.

The missing values are $y = 7$ , $x = 8$ and $y = 42$ .

The Graph of y = kx

The equation $y = kx$ is $y = mx + c$ with $m = k$ and $c = 0$ . So its graph is a straight line with gradient $k$ that meets the $y$ -axis at $(0, 0)$ , the origin.

The graph of a direct proportion

The graph of $y = kx$ is a straight line that passes through the origin.

Its gradient is $k$ .

To draw it, work out two or three points from the equation; the origin is always one of them. Plot them to the scale given and join them with a ruler.

Example 5: Drawing the graph

A printer prints $y$ pages in $x$ minutes, where $y = 12x$ . Draw the graph of $y = 12x$ for $0 \le x \le 6$ , using 1 square for 1 minute across and 1 square for 12 pages up. Use it to show that the gradient is 12.

Solution

Make a table: when $x = 0$ , $y = 0$ ; when $x = 2$ , $y = 12 \times 2 = 24$ ; and so on.

$x$	$0$	$2$	$4$	$6$
$y$	$0$	$24$	$48$	$72$

Plot the points and join them with a straight line.

From $(2, 24)$ to $(4, 48)$ : gradient $= \frac{\text{rise}}{\text{run}} = \frac{24}{2} = 12\text{.}$ Read the rise and the run from the numbers on the axes, not by counting squares: here one square up is 12 pages.

The graph is the straight line above, through the origin, with gradient $12 = k$ : the printer prints 12 more pages for each minute.

Example 6: Finding k from a graph

The graph shows a direct proportion between $x$ and $y$ . Find $k$ , write down the equation of the line, and find $y$ when $x = 7$ .

Solution

Find a point where the line crosses grid lines exactly, so that its coordinates can be read: the line passes through $(4, 10)$ .

$k = \frac{y}{x} = \frac{10}{4} = 2.5\text{,}$ so the equation is $y = 2.5x$ . (This is also the gradient: from the origin, a run of 4 and a rise of 10.)

When $x = 7$ : $y = 2.5 \times 7 = 17.5$ .

$k = 2.5$ ; $y = 2.5x$ ; $y = 17.5$ when $x = 7$ .

The value of $k$ sets how steep the line is. The graph below shows $y = 2x$ , $y = 4x$ and $y = 8x$ . All three pass through the origin; for each 1 across, $y = 8x$ goes up 8, $y = 4x$ goes up 4 and $y = 2x$ goes up 2.

For $k > 0$ , the larger the value of $k$ , the steeper the line.

Is It Direct Proportion?

Deciding

From an equation: $y \propto x$ only if the equation can be written $y = kx$ , with nothing added or taken away and $x$ not squared or under a fraction line.

From a graph: $y \propto x$ only if the graph is a straight line and it passes through the origin.

Example 7: A fixed charge

A mobile phone plan charges a fixed $20 a month plus $0.20 for every minute of calls. The charge for a month, $ $C$ , for $n$ minutes is $C = 0.2n + 20$ . Is $C$ directly proportional to $n$ ?

Solution

When $n = 0$ : $C = 20$ . When $n = 60$ : $C = 0.2 \times 60 + 20 = 32\text{.}$ When $n = 120$ : $C = 0.2 \times 120 + 20 = 44\text{.}$

The graph is a straight line, but it meets the $C$ -axis at 20, not at the origin. The equation has $+ 20$ , so it is not of the form $C = kn$ .

Check with pairs: $\frac{32}{60} \approx 0.53$ and $\frac{44}{120} \approx 0.37$ , which are not the same.

The 0.2 is the charge per minute in dollars. Only a plan with no fixed charge, $C = 0.2n$ , would make $C$ directly proportional to $n$ with $k = 0.2$ .

No. The graph is a straight line that does not pass through the origin, because of the fixed $20.

Example 8: Deciding from a graph

Which of the graphs $A$ , $B$ and $C$ shows $y$ directly proportional to $x$ ? Give a reason for each.

Solution

$A$ is a straight line through the origin: $y \propto x$ . It passes through $(3, 6)$ , so $k = \frac{6}{3} = 2$ and its equation is $y = 2x$ .

$B$ is a straight line, but it meets the $y$ -axis at $(0, 3)$ , not at the origin: not direct proportion.

$C$ passes through the origin, but it is a curve, not a straight line: $y$ is not directly proportional to $x$ . (Its equation is $y = 0.5x^2$ ; lesson 6.3 shows that $y$ is directly proportional to $x^2$ .)

Only $A$ .

Example 9: Deciding from an equation

In which of these is $y$ directly proportional to $x$ ? Give $k$ when it is.
(a) $y = 5x$ (b) $y = 5x + 2$ (c) $y = \frac{5}{x}$ (d) $y = \frac{x}{3}$

Solution

(a) Yes: $y = 5x$ is $y = kx$ with $k = 5$ .

(b) No: the $+ 2$ means that $y = 2$ when $x = 0$ , so the line does not pass through the origin.

(c) No: here $x$ is under the fraction line. When $x$ is doubled, $y$ is halved. (This is inverse proportion, lesson 6.4.)

(d) Yes: $\frac{x}{3} = \frac{1}{3}x$ , so $k = \frac{1}{3}$ .

(a) yes, $k = 5$ ; (b) no; (c) no; (d) yes, $k = \frac{1}{3}$

Common Mistakes

" $C = 0.2n + 20$ is a straight line, so $C \propto n$ ." A straight line shows direct proportion only if it passes through the origin.
Multiplying instead of dividing to find $x$ . If $y = 3x$ and $y = 21$ , then $x = \frac{21}{3} = 7$ , not $3 \times 21 = 63$ . Check: $3 \times 7 = 21$ .
Finding $k$ upside down. $k = \frac{y}{x}$ : if $y = 12$ when $x = 4$ , $k = 3$ , not $\frac{4}{12}$ .
Counting squares for the gradient when the axes have different scales. Use the numbers on the axes.
Calling a curve through the origin a direct proportion. The graph of $y \propto x$ is straight.

Practice Questions

Find $k$ first, then use the equation, and check each answer with $k = \frac{y}{x}$ . The questions get harder as you go.

Question 1

$y$ is directly proportional to $x$ , and $y = 10$ when $x = 2$ . Find (a) an equation connecting $x$ and $y$ , (b) $y$ when $x = 10$ , (c) $x$ when $y = 60$ .

ShowHide solution

(a) $10 = 2k$ , so $k = 5$ and $y = 5x$ .

(b) $y = 5 \times 10 = 50$ .

(a) $y = 5x$ (b) $y = 50$ (c) $x = 12$

Question 2

$y \propto x$ , and $y = 9$ when $x = 12$ . Find (a) $k$ as a fraction, (b) $y$ when $x = 20$ , (c) $x$ when $y = 4.5$ .

ShowHide solution

(a) $k = \frac{9}{12} = \frac{3}{4}$ , so $y = \frac{3}{4}x$ .

(b) $y = \frac{3}{4} \times 20 = 15$ .

(a) $k = \frac{3}{4}$ (b) $y = 15$ (c) $x = 6$

Question 3

$P$ is directly proportional to $t$ , and $P = 42$ when $t = 3.5$ . Find $P$ when $t = 5$ , and $t$ when $P = 90$ .

ShowHide solution

$P = kt$ : $42 = 3.5k$ , so $k = \frac{42}{3.5} = 12$ and $P = 12t$ .

When $t = 5$ : $P = 60$ . When $P = 90$ : $t = \frac{90}{12} = 7.5$ .

$P = 60$ ; $t = 7.5$

Question 4

Given that $q$ is directly proportional to $p$ , copy and complete the table.

$p$	$4$	$5$	$7$	?	?
$q$	?	$30$	?	$48$	$57$

ShowHide solution

From $p = 5$ , $q = 30$ : $k = 6$ , so $q = 6p$ .

$q = 6 \times 4 = 24$ ; $q = 6 \times 7 = 42$ ; $p = \frac{48}{6} = 8$ ; $p = \frac{57}{6} = 9.5$ .

$p$	$4$	$5$	$7$	$8$	$9.5$
$q$	$24$	$30$	$42$	$48$	$57$
$\dfrac{q}{p}$	$6$	$6$	$6$	$6$	$6$

$q = 24$ , $q = 42$ , $p = 8$ , $p = 9.5$

Question 5

In which of these is $y$ directly proportional to $x$ ? Give $k$ when it is.
(a) $y = 7x$ (b) $y = 7x - 1$ (c) $y = \frac{x}{4}$ (d) $y = \frac{4}{x}$ (e) $y = 2(x + 3)$

ShowHide solution

(a) Yes, $k = 7$ .

(b) No: $y = -1$ when $x = 0$ ; the line misses the origin.

(d) No: $x$ is under the fraction line.

(e) No: $y = 2x + 6$ , and $y = 6$ when $x = 0$ .

(a) yes, $k = 7$ ; (b) no; (c) yes, $k = \frac{1}{4}$ ; (d) no; (e) no

Question 6

Petrol costs $2.80 a litre. The cost of $n$ litres is $ $C$ , where $C = 2.8n$ . Draw the graph of $C$ against $n$ for $0 \le n \le 50$ , and use it to find the cost of 35 litres.

ShowHide solution

$n$	$0$	$10$	$20$	$30$	$40$	$50$
$C$	$0$	$28$	$56$	$84$	$112$	$140$

Read up from $n = 35$ to the line and across: $C = 98$ . Check: $2.8 \times 35 = 98$ .

35 litres cost $98.

Question 7

The graph shows $y$ against $x$ for a direct proportion. Find the equation of the line, and find $x$ when $y = 5$ .

ShowHide solution

The line passes exactly through the grid point $(3, 2)$ (and $(6, 4)$ and $(9, 6)$ ).

$k = \frac{2}{3}$ , so $y = \frac{2}{3}x$ .

When $y = 5$ : $5 = \frac{2}{3}x$ , so $x = 5 \times \frac{3}{2} = 7.5$ .

$y = \frac{2}{3}x$ ; $x = 7.5$ when $y = 5$ .

Question 8

Plumber A charges a call-out fee of $40 plus $30 for each hour. Plumber B charges $45 for each hour and no fee. For each, write the charge $ $C$ for $h$ hours as an equation, and say whether $C$ is directly proportional to $h$ .

ShowHide solution

A: $C = 30h + 40$ . Not direct proportion: the $40 is charged even for 0 hours, so the graph does not pass through the origin.

B: $C = 45h$ . Direct proportion, with $k = 45$ : B charges $45 per hour.

A: $C = 30h + 40$ , not directly proportional. B: $C = 45h$ , directly proportional with $k = 45$ .

Question 9

The lines $y = 1.5x$ , $y = 3x$ and $y = 0.5x$ are drawn on the same axes. (a) Which is the steepest? (b) Which one passes through the point $(4, 6)$ ?

ShowHide solution

(a) All three pass through the origin, and $k = 3$ is the largest, so $y = 3x$ is the steepest.

(b) At $x = 4$ : $1.5 \times 4 = 6$ , $3 \times 4 = 12$ , $0.5 \times 4 = 2$ . Only $y = 1.5x$ gives $6$ .

(a) $y = 3x$ (b) $y = 1.5x$

Question 10

Ben is told that $y \propto x$ and $y = 6$ when $x = 15$ . He writes " $k = \frac{15}{6} = 2.5$ , so $y = 2.5x$ ". Find his mistake, write the correct equation, and find $y$ when $x = 40$ .

ShowHide solution

He divided $x$ by $y$ . The constant is $k = \frac{y}{x} = \frac{6}{15} = 0.4\text{.}$

His equation fails its own pair: $2.5 \times 15 = 37.5$ , not $6$ .

So $y = 0.4x$ , and when $x = 40$ , $y = 0.4 \times 40 = 16$ .

$k = \frac{y}{x} = 0.4$ , not $\frac{x}{y}$ ; $y = 0.4x$ ; $y = 16$ when $x = 40$ .

Question 11

The mass, $m$ g, of a length of copper wire is directly proportional to its length, $L$ cm. A 25 cm length has a mass of 4 g. (a) Find $k$ and say what it means. (b) Find the mass of 1.5 m of the wire. (c) What length has a mass of 10 g?

ShowHide solution

(a) $m = kL$ : $4 = 25k$ , so $k = \frac{4}{25} = 0.16$ . Each centimetre of wire has a mass of 0.16 g.

(b) 1.5 m = 150 cm. $m = 0.16 \times 150 = 24$ .

(a) $k = 0.16$ , the mass in grams of 1 cm of wire (b) 24 g (c) 62.5 cm

Question 12

Pei Shan plots the table below and gets a straight line. She says, "The graph is a straight line, so $y$ is directly proportional to $x$ ." Is she right?

$x$	$0$	$2$	$4$	$6$
$y$	$3$	$7$	$11$	$15$

ShowHide solution

When $x = 0$ , $y = 3$ : the line meets the $y$ -axis at $(0, 3)$ , not at the origin.

Also $\frac{7}{2} = 3.5$ but $\frac{15}{6} = 2.5$ : $\frac{y}{x}$ is not the same. (The equation is $y = 2x + 3$ .)

No. The line is straight but does not pass through the origin, so $y$ is not directly proportional to $x$ .

Key Points

If $y \propto x$ , then $y = kx$ , where $k = \frac{y}{x}$ is the constant of proportionality ( $k \neq 0$ ).
Find $k$ from one pair of values, write the equation, then find $y$ from $x$ (multiply) or $x$ from $y$ (divide).
The graph of $y = kx$ is a straight line through the origin with gradient $k$ ; for $k > 0$ , a larger $k$ gives a steeper line.
To read $k$ from a graph, find a point on the line where it crosses grid lines exactly, and work out $\frac{y}{x}$ from the numbers on the axes.
A straight line that misses the origin ( $y = mx + c$ , $c \neq 0$ ) and a curve through the origin are not direct proportion.
In a setting, $k$ is a rate: pages per minute, dollars per kilogram, grams per centimetre.