Equations and Graphs of Inverse Proportion

Writing an inverse proportion as the equation $y = \frac{k}{x}$ , finding $k$ from one pair of values, drawing the curve for positive values of $x$ , and telling its graph apart from the graphs of direct proportion.

Before you start

Lesson 6.4: $y$ is inversely proportional to $x$ ( $y \propto \frac{1}{x}$ ) when $xy$ is the same for every pair; that value is $k$ .
Lesson 6.2: the graph of the direct proportion $y = kx$ is a straight line through the origin.

The Equation y = k/x

If $xy = k$ for every pair of values, then dividing both sides by $x$ gives $y = \frac{k}{x}$ . So an inverse proportion can be written as an equation, in either form.

The equation of an inverse proportion

If $y \propto \frac{1}{x}$ , then $y = \frac{k}{x}$ , or $xy = k$ , where $k$ is a constant and $k \neq 0$ . In this chapter $k$ is always positive.

$k = xy$ for any pair of values, so one pair is enough to find it.

Compare: in direct proportion, $y = kx$ , $x$ multiplies $k$ ; in inverse proportion, $y = \frac{k}{x}$ , $x$ divides $k$ .

Example 1: Finding the equation, then values both ways

$y$ is inversely proportional to $x$ , and $y = 6$ 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 $y = 12$ .

Solution

(a) $y = \frac{k}{x}$ , so $k = xy = 4 \times 6 = 24$ . The equation is $y = \frac{24}{x}$ .

(b) When $x = 8$ : $y = \frac{24}{8} = 3$ .

Check: $8 \times 3 = 24$ and $2 \times 12 = 24$ .

(a) $y = \frac{24}{x}$ (b) $y = 3$ (c) $x = 2$

Example 2: Other letters and decimals

$T$ is inversely proportional to $n$ , and $T = 1.5$ when $n = 8$ . Find $T$ when $n = 5$ , and $n$ when $T = 0.4$ .

Solution

$T = \frac{k}{n}$ , so $k = nT = 8 \times 1.5 = 12$ and $T = \frac{12}{n}$ .

When $n = 5$ : $T = \frac{12}{5} = 2.4$ .

When $T = 0.4$ : $0.4n = 12$ , so $n = \frac{12}{0.4} = 30$ .

$T = 2.4$ when $n = 5$ ; $n = 30$ when $T = 0.4$ .

Example 3: Finding the equation from a table

Show that $y$ is inversely proportional to $x$ , and write down the equation.

$x$	$2$	$3$	$5$	$6$
$y$	$15$	$10$	$6$	$5$

Solution

$x$	$2$	$3$	$5$	$6$
$y$	$15$	$10$	$6$	$5$
$xy$	$30$	$30$	$30$	$30$

$xy = 30$ for every pair, so $y \propto \frac{1}{x}$ with $k = 30$ .

$y = \frac{30}{x}$ (or $xy = 30$ )

Example 4: Does a point lie on the curve?

Which of the points $(10, 12)$ , $(15, 9)$ and $(0.5, 240)$ lie on the graph of $y = \frac{120}{x}$ ?

Solution

A point lies on the graph when $xy = 120$ .

$(10, 12)$ : $10 \times 12 = 120$ . It lies on the graph.

$(15, 9)$ : $15 \times 9 = 135$ , not 120. It does not. (At $x = 15$ the graph has $y = 8$ .)

$(0.5, 240)$ : $0.5 \times 240 = 120$ . It lies on the graph.

$(10, 12)$ and $(0.5, 240)$ lie on the graph; $(15, 9)$ does not.

The Graph of y = k/x

Drawing the graph of y = k/x

1. Make a table for the values of $x$ asked for. In this chapter $x$ is always positive.

2. Draw and number the axes from 0, to the scale given, and plot every point.

3. Join the points with one smooth curve, not with straight lines. Do not take the curve to either axis.

Example 5: Drawing the graph

Draw the graph of $y = \frac{24}{x}$ for $1 \le x \le 8$ .

Solution

At $x = 1$ , $y = 24$ ; at $x = 2$ , $y = 12$ ; at $x = 3$ , $y = 8$ ; and so on. The $xy$ row checks each value.

$x$	$1$	$2$	$3$	$4$	$6$	$8$
$y$	$24$	$12$	$8$	$6$	$4$	$3$
$xy$	$24$	$24$	$24$	$24$	$24$	$24$

The graph is the smooth curve above, falling from $(1, 24)$ to $(8, 3)$ .

The curve y = k/x for positive x

It does not pass through the origin: when $x = 0$ , $\frac{k}{x}$ has no value.

As $x$ increases, $y$ decreases, steeply at first and then more and more slowly. But $y$ never reaches 0, because $\frac{k}{x}$ is never 0.

So the curve never meets either axis. It is called a hyperbola.

Example 6: Finding k from a graph

The graph shows $y$ against $x$ , where $y$ is inversely proportional to $x$ . Find its equation, and find $y$ when $x = 4$ .

Solution

Find a point where the curve crosses grid lines exactly: it passes through $(3, 6)$ .

$k = xy = 3 \times 6 = 18$ , so $y = \frac{18}{x}$ . Check with another point on a grid crossing: $(9, 2)$ gives $9 \times 2 = 18$ .

When $x = 4$ : $y = \frac{18}{4} = 4.5$ .

$y = \frac{18}{x}$ ; $y = 4.5$ when $x = 4$ .

The value of $k$ sets how far the curve is from the axes. The graph below shows $y = \frac{12}{x}$ , $y = \frac{24}{x}$ and $y = \frac{48}{x}$ . At every $x$ , the second is twice the first and the third twice the second: at $x = 4$ , they give $3$ , $6$ and $12$ .

For $k > 0$ , the larger the value of $k$ , the farther the curve lies from the axes. None of the curves meets an axis.

Which Graph Is Which?

Equation	Graph	Through the origin?
$y = kx$ (direct proportion)	straight line	yes
$y = mx + c$ , $c \neq 0$ (not proportion)	straight line	no
$y = kx^2$ (direct proportion to $x^2$ )	curve, rising more and more steeply	yes
$y = \frac{k}{x}$ (inverse proportion)	curve, falling more and more slowly	no

The test from a table

If $\frac{y}{x}$ is the same for every pair, $y \propto x$ . If $xy$ is the same for every pair, $y \propto \frac{1}{x}$ .

If neither is the same, $y$ is neither directly nor inversely proportional to $x$ . (It may still be proportional to $x^2$ , or to $\frac{1}{x^2}$ : lessons 6.3 and 6.6.)

A falling curve is not a test: always work out $xy$ .

Example 7: A falling curve that is not inverse proportion

Hui Min plots the points in the table, joins them with a smooth falling curve, and says, "The curve falls, so $y$ is inversely proportional to $x$ ." Is she right?

$x$	$1$	$2$	$4$	$5$
$y$	$20$	$14$	$8$	$6$

Solution

$x$	$1$	$2$	$4$	$5$
$y$	$20$	$14$	$8$	$6$
$xy$	$20$	$28$	$32$	$30$

$xy$ is not the same for every pair. A curve that falls is not enough.

No: $xy$ takes the values 20, 28, 32 and 30, so $y$ is not inversely proportional to $x$ .

Example 8: Matching graphs

The graphs $A$ , $B$ , $C$ and $D$ are those of $y = 1.5x$ , $y = x + 5$ , $y = x^2$ and $y = \frac{8}{x}$ , in some order. Match each graph to its equation, and say which shows direct proportion and which shows inverse proportion.

Solution

$A$ is a straight line through the origin, passing through $(4, 6)$ , so $k = \frac{6}{4} = 1.5$ : $y = 1.5x$ , direct proportion.

$B$ is a straight line meeting the $y$ -axis at $(0, 5)$ , and rising 1 for each 1 across: $y = x + 5$ , not proportion.

$C$ is a curve through the origin, passing through $(3, 9)$ : $y = x^2$ . It shows $y$ directly proportional to $x^2$ , not to $x$ .

$D$ is a curve that meets neither axis, passing through $(1, 8)$ and $(4, 2)$ , where $xy = 8$ : $y = \frac{8}{x}$ , inverse proportion.

$A$ : $y = 1.5x$ (direct); $B$ : $y = x + 5$ (neither); $C$ : $y = x^2$ (direct, to $x^2$ ); $D$ : $y = \frac{8}{x}$ (inverse).

Common Mistakes

"The curve falls, so $y \propto \frac{1}{x}$ ." Work out $xy$ for every pair first.
Finding $k$ as $\frac{y}{x}$ . For inverse proportion, $k = xy$ : if $y = 4$ when $x = 2$ , then $k = 8$ , not 2.
Drawing the curve through the origin, or on to an axis. The curve $y = \frac{k}{x}$ never meets either axis.
Joining the points with straight lines. Draw one smooth curve.
Writing the equation as $y = kx$ or $y = \frac{x}{k}$ . Inverse proportion is $y = \frac{k}{x}$ .

Practice Questions

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

Question 1

$y$ is inversely proportional to $x$ , and $y = 5$ when $x = 6$ . Find (a) the equation, (b) $y$ when $x = 10$ , (c) $x$ when $y = 7.5$ .

ShowHide solution

(a) $k = 6 \times 5 = 30$ , so $y = \frac{30}{x}$ .

(b) $y = \frac{30}{10} = 3$ .

(a) $y = \frac{30}{x}$ (b) $y = 3$ (c) $x = 4$

Question 2

The number of tiles needed to cover a floor is inversely proportional to the area of each tile. 400 tiles, each of area 900 cm², are needed. How many tiles of area 1600 cm² are needed?

ShowHide solution

Let $n$ tiles of area $A$ cm² be needed: $nA = k$ , the area of the floor. $k = 400 \times 900 = 360\,000\text{.}$

When $A = 1600$ : $n = \frac{360\,000}{1600} = 225\text{.}$

225 tiles

Question 3

Is $y$ directly proportional to $x$ , inversely proportional to $x$ , or neither? Find the equation if there is one.

$x$	$3$	$4$	$6$	$8$
$y$	$16$	$12$	$8$	$6$

ShowHide solution

$x$	$3$	$4$	$6$	$8$
$y$	$16$	$12$	$8$	$6$
$\dfrac{y}{x}$	$5.33$	$3$	$1.33$	$0.75$
$xy$	$48$	$48$	$48$	$48$

$\frac{y}{x}$ is not the same, but $xy = 48$ for every pair.

Inversely proportional: $y = \frac{48}{x}$ .

Question 4

Is $y$ directly proportional to $x$ , inversely proportional to $x$ , or neither?

$x$	$1$	$2$	$3$	$4$
$y$	$12$	$6$	$3$	$1.5$

ShowHide solution

$x$	$1$	$2$	$3$	$4$
$y$	$12$	$6$	$3$	$1.5$
$\dfrac{y}{x}$	$12$	$3$	$1$	$0.375$
$xy$	$12$	$12$	$9$	$6$

The first two pairs give $xy = 12$ , but the others do not; $\frac{y}{x}$ is not the same either.

Neither.

Question 5

Which of the points $(4, 9)$ , $(6, 5)$ , $(12, 3)$ and $(0.5, 72)$ lie on the graph of $y = \frac{36}{x}$ ?

ShowHide solution

$4 \times 9 = 36$ (yes); $6 \times 5 = 30$ (no); $12 \times 3 = 36$ (yes); $0.5 \times 72 = 36$ (yes).

$(4, 9)$ , $(12, 3)$ and $(0.5, 72)$ ; not $(6, 5)$ .

Question 6

Draw the graph of $y = \frac{12}{x}$ for $1 \le x \le 6$ , using $x = 1, 1.5, 2, 3, 4, 6$ . Use your graph to find $y$ when $x = 5$ , and check by working it out.

ShowHide solution

$x$	$1$	$1.5$	$2$	$3$	$4$	$6$
$y$	$12$	$8$	$6$	$4$	$3$	$2$
$xy$	$12$	$12$	$12$	$12$	$12$	$12$

From $x = 5$ up to the curve and across: $y$ is a little below 2.5, about 2.4. Check: $\frac{12}{5} = 2.4$ .

$y = 2.4$ when $x = 5$ .

Question 7

The graph shows $y$ against $x$ , where $y \propto \frac{1}{x}$ . Find the equation, and find $y$ when $x = 8$ .

ShowHide solution

The curve passes exactly through the grid points $(2, 10)$ , $(4, 5)$ , $(5, 4)$ and $(10, 2)$ .

$k = 4 \times 5 = 20$ (and $2 \times 10 = 20$ ), so $y = \frac{20}{x}$ .

When $x = 8$ : $y = \frac{20}{8} = 2.5$ .

$y = \frac{20}{x}$ ; $y = 2.5$

Question 8

The curves $y = \frac{6}{x}$ , $y = \frac{12}{x}$ and $y = \frac{24}{x}$ are drawn for positive $x$ on the same axes. (a) Which passes through $(3, 4)$ ? (b) Which lies farthest from the axes?

ShowHide solution

(a) $3 \times 4 = 12$ , so the point lies on $y = \frac{12}{x}$ .

(b) The largest $k$ , $y = \frac{24}{x}$ : at every $x$ its $y$ is the largest.

(a) $y = \frac{12}{x}$ (b) $y = \frac{24}{x}$

Question 9

Match each equation to a description.
Equations: (a) $y = 5x$ (b) $y = \frac{5}{x}$ (c) $y = 5x + 1$ (d) $y = 5x^2$ .
Descriptions: (1) a curve that does not pass through the origin; (2) a straight line through the origin; (3) a curve through the origin; (4) a straight line that does not pass through the origin.

ShowHide solution

(a) $y = kx$ : a straight line through the origin, (2).

(b) $y = \frac{k}{x}$ : a curve that never meets an axis, (1).

(d) $y = kx^2$ : a curve through the origin, (3).

(a) (2); (b) (1); (c) (4); (d) (3)

Question 10

Ali is told that $y \propto \frac{1}{x}$ and $y = 4$ when $x = 2$ . He writes " $k = \frac{y}{x} = 2$ , so $y = \frac{2}{x}$ ". Find his mistake and the correct equation.

ShowHide solution

For inverse proportion, $k = xy$ , not $\frac{y}{x}$ : $k = 2 \times 4 = 8$ .

His equation fails its own pair: $\frac{2}{2} = 1$ , not 4.

$y = \dfrac{8}{x}$

Question 11

The time, $t$ minutes, to fill a tank is inversely proportional to the rate, $r$ litres per minute, at which water flows in. At 15 litres per minute the tank fills in 40 minutes. (a) Find the equation connecting $t$ and $r$ . (b) How long does it take at 24 litres per minute? (c) What rate fills it in 30 minutes?

ShowHide solution

(a) $k = rt = 15 \times 40 = 600$ (the tank holds 600 litres), so $t = \frac{600}{r}$ .

(b) $t = \frac{600}{24} = 25$ .

(a) $t = \frac{600}{r}$ (b) 25 minutes (c) 20 litres per minute

Key Points

If $y \propto \frac{1}{x}$ , then $y = \frac{k}{x}$ , or $xy = k$ , where $k = xy$ for any pair of values.
Find $k$ from one pair, write the equation, then find $y$ from $x$ or $x$ from $y$ ; check by multiplying.
A point lies on the graph of $y = \frac{k}{x}$ when the product of its coordinates is $k$ .
For positive $x$ , the graph of $y = \frac{k}{x}$ is a smooth curve that falls more and more slowly, does not pass through the origin, and never meets either axis.
The larger the value of $k$ , the farther the curve lies from the axes.
$\frac{y}{x}$ the same: direct proportion. $xy$ the same: inverse proportion. A falling curve alone proves nothing.