Math 8

Inversely proportional relationships

Three workers take eight hours to lay a floor. Twice as many should take half as long, but what exactly stays the same, and how do you tell this pattern from an ordinary proportion?

You need

The constant of proportionality and proportional tablesMath 7

Where you are 4 in Proportional relationships

Fractions, ratio and percent3

Ratio6

Proportional relationships4

The constant of proportionality and proportional tables

Inversely proportional relationships

Recognizing a proportional relationship in a graph Recognizing a proportional relationship in an equation

Why this exists

Some pairs of amounts grow together: buy twice as many notebooks and you pay twice as much. That is a proportional relationship, which Direct variation writes as y = kx.

Other pairs pull in opposite directions. More workers, less time. A faster drive, a shorter trip. More winners sharing a prize, a smaller share each. No fixed number times the number of workers gives the hours, so these are not proportional in the old sense.

Something is still fixed, and it is a product, not a ratio. Three workers working 8 hours do 3 \times 8 = 24 worker-hours of work; six working 4 hours also do 24.

“One goes up, the other goes down” is not enough on its own. A gift card loses 5 dollars a week as it is spent, yet doubling the weeks does not halve what is left. This lesson is about the exact pattern, where doubling one amount halves the other.

Definition

Two quantities are inversely proportional when their product is always the same number. In letters, y is inversely proportional to x when

xy = k

for every matching pair of values, where k is a fixed number that is not 0. Dividing both sides by x gives the rule solved for y:

y = \frac{k}{x}, \qquad x \neq 0

The number k is still called the constant of proportionality (or constant of variation), but here it is a product rather than a ratio. “y varies inversely with x” means the same thing, and the relationship runs both ways: \displaystyle x = \frac{k}{y}.

	direct proportion	inverse proportion
what stays fixed	the quotient \displaystyle \frac{y}{x} = k	the product xy = k
equation	y = kx	\displaystyle y = \frac{k}{x}
when x doubles	y doubles	y halves
graph	a straight line through (0, 0)	a curve that never touches either axis

In this lesson both quantities are positive amounts (people, speeds, times, lengths, money), so k is positive too. The reciprocal of a number is 1 divided by that number: the reciprocal of 4 is \displaystyle \frac{1}{4}, and a number times its reciprocal is 1. Since \displaystyle \frac{k}{x} = k \times \frac{1}{x}, the amount y is a fixed number times the reciprocal of x. That is where the word inverse comes from, and it is what \displaystyle y \propto \frac{1}{x} says.

Intuition

Picture rectangles that all have area 48 square centimeters. One 3 cm wide is 16 cm long, one 6 cm wide is 8 cm long, one 12 cm wide is only 4 cm long. Width times length is the fixed area, so width and length are inversely proportional with k = 48.

Figure 1Every rectangle here has area 48 square centimeters. Their far corners all lie on one curve, the graph of xy = 48.

Stand them on one corner, as in Figure 1, and their far corners line up along a curve. That is the shape of every inverse proportion: falling, bending, and never reaching either axis.

The fixed area plays the part of any fixed total: the work in a job, the length of a trip. Whatever x is multiplied by, y is divided by the same number.

Results

The product rule. x and y are inversely proportional exactly when xy = k for every pair, with k \neq 0 fixed. Then \displaystyle y = \frac{k}{x}, and neither x nor y is ever 0.

Scaling. If x is multiplied by a nonzero number c, then y is divided by c.

The table test. Multiply each pair. If every product is the same, the table is inversely proportional and that product is k. If instead every quotient \displaystyle \frac{y}{x} is the same, it is directly proportional. If neither, it is neither kind.

A missing value. Any two pairs (x_1, y_1) and (x_2, y_2) have the same product,

x_1 y_1 = x_2 y_2

so three of the four numbers fix the fourth.

The graph. For positive x, the graph of \displaystyle y = \frac{k}{x} is a falling curve, not a straight line. It never meets either axis, though the further out it runs the closer it comes to each one.

Why it is true

Scaling. If xy = k and x is replaced by cx, the new y must still make the product k:

y_{\text{new}} = \frac{k}{cx} = \frac{1}{c} \times \frac{k}{x} = \frac{y}{c}

The missing value. x_1 y_1 = k and x_2 y_2 = k, and two numbers equal to the same number are equal to each other.

Never on an axis. A point on either axis has x = 0 or y = 0, so its product is 0, not k. Yet the curve comes as close as you like: on \displaystyle y = \frac{48}{x}, x = 4800 gives y = 0.01.

Not a line. A straight line changes by the same amount for every equal step across; that is its constant rate of change. On \displaystyle y = \frac{48}{x} take (4, 12), (8, 6), (12, 4) and (16, 3), equally spaced across. The drops are 6, then 2, then 1. Unequal drops for equal steps: not a line, and the shrinking drops are why the curve flattens out.

Worked examples

Example 1

Workers laying a floor

Odalys hires workers to lay a wooden floor. Three workers, all working at the same steady pace and never slowing one another down, finish it in 8 hours. (a) Write an equation for the time h, in hours, that p workers take. (b) How long do 4 workers take? 5 workers? (c) How many workers are needed to finish in 3 hours?

Solution

The floor is the same whoever lays it, and each worker does the same amount an hour, so the total work is fixed:

3 \times 8 = 24 \text{ worker-hours}

(a) Workers times hours is always 24, so ph = 24, and dividing by p gives \displaystyle h = \frac{24}{p}.

(b) \displaystyle h = \frac{24}{4} = 6 hours. For five, \displaystyle h = \frac{24}{5} = 4.8 hours, and since 0.8 \times 60 = 48 minutes, that is 4 hours 48 minutes.

Check. Every product is 24: 4 \times 6, 5 \times 4.8, 8 \times 3. Going from 4 workers to 8 doubles the crew and halves the time, from 6 hours to 3.

Example 2

Reading three tables

Which table is an inverse proportion, which a direct proportion, and which neither? Give the equation where there is one.

Table A:

x	1	2	4	5	10
y	40	20	10	8	4

Table B:

x	2	3	5	8
y	6	9	15	24

Table C:

x	1	2	3	4
y	12	10	8	6

Solution

Table A. The products are 1 \times 40, 2 \times 20, 4 \times 10, 5 \times 8 and 10 \times 4, all 40. So it is inversely proportional with k = 40, and \displaystyle y = \frac{40}{x}.

Table B. Here y rises with x, so try quotients: \displaystyle \frac{6}{2}, \displaystyle \frac{9}{3}, \displaystyle \frac{15}{5} and \displaystyle \frac{24}{8} are all 3. It is directly proportional, y = 3x.

Table C. Here y falls as x rises, as in Table A, but the products are 12, 20, 24 and 24, and the quotients 12, 5, \displaystyle \frac{8}{3} and 1.5. Neither is constant, so it is neither kind. What it keeps fixed is the drop, 2 at every step: its points lie on the straight line y = 14 - 2x.

Figure 2The three tables of Example 2. Only the inverse proportion, A, is a curve.

Figure 2 plots all three. Falling is not the test; a constant product is.

Example 3

A fixed trip, from table to graph

Vasco drives the same 180 km route at different steady speeds. (a) Make a table of the times at 30, 45, 60 and 90 km/h, and write the equation. (b) Sketch the graph. (c) What steady speed makes the trip take 2.5 hours?

Solution

(a) At a steady speed, distance is speed times time. The distance is fixed, so speed s times time t is always 180: they are inversely proportional, with \displaystyle t = \frac{180}{s}.

speed s (km/h)	30	45	60	90
time t (hours)	6	4	3	2

(b) Plot (30, 6), (45, 4), (60, 3) and (90, 2) and join them with a smooth curve, not straight pieces.

Figure 3Time against speed for Vasco’s 180 km route in Example 3, \displaystyle t = \frac{180}{s}.

The equation gives values between the points too. At 36 km/h the trip takes \displaystyle \frac{180}{36} = 5 hours, where a straight piece from (30, 6) to (45, 4) would say 5.2 hours. The true graph sags below the straight pieces.

Faster, the time shrinks but never reaches 0; slower, it grows without end (18 hours at 10 km/h). So the curve touches neither axis.

Example 4

From one pair of values

y is inversely proportional to x, and y = 15 when x = 4. Find y when x = 12 and when x = 10, and find x when y = 2.4.

Solution

12 is 3 times 4, so x has been multiplied by 3, and by scaling y is divided by 3: \displaystyle y = \frac{15}{3} = 5.

10 is not a whole-number multiple of 4, so use the product rule instead:

4 \times 15 = 10 \times y

so 10y = 60 and y = 6.

For the last, 4 \times 15 = x \times 2.4, so \displaystyle x = \frac{60}{2.4} = 25.

All three rest on k = 4 \times 15 = 60, so \displaystyle y = \frac{60}{x}, and (12, 5), (10, 6) and (25, 2.4) each have product 60.

Example 5

Balancing a see-saw

Treat each person as sitting at a single point, on a see-saw that balances by itself when empty. Then it balances when weight times distance from the pivot is the same on both sides. Saskia weighs 36 kg and sits 1.5 m from the pivot. (a) Yannick weighs 30 kg. How far out must he sit, on the other side, to balance her? (b) Show that for anyone balancing her, weight and distance are inversely proportional, and find the distances for 27 kg, 45 kg and 54 kg. (c) Each arm of the see-saw is 2.5 m long. Who is the lightest person who can balance her?

Solution

Saskia’s side gives 36 \times 1.5 = 54 kilogram-meters, and the other side must match it.

(a) 30d = 54, so \displaystyle d = \frac{54}{30} = 1.8 m. He is lighter, so he sits further out.

(b) Weight w times distance d must be 54, a fixed product, so \displaystyle d = \frac{54}{w}:

weight w (kg)	27	45	54
distance d (m)	2	1.2	1

w = \frac{54}{2.5} = 21.6 \text{ kg}

Anyone lighter would need more than 2.5 m.

Boundaries

Neither amount can be 0, and neither can k. \displaystyle y = \frac{k}{x} has no value at x = 0, and y is never 0 because k is not.

The product must be fixed, not just the direction. Table C in Example 2 falls without being inverse, and pages read and pages left in a book have a fixed sum, not a fixed product.

A story fits only under its assumptions. The workers must all keep the same pace without getting in each other’s way; the trip needs a steady speed (or the average speed); the see-saw treats each person as a point. Counts of people allow only whole numbers, so such a story’s graph is a row of separate dots along the curve.

Positive amounts only, here. With negative numbers allowed, the graph of \displaystyle y = \frac{k}{x} has a second piece in the opposite corner of the grid. That comes later.

Where this goes next

This is the first relationship here whose graph is not a straight line. A line has one rate of change, which slope measures from any two points; the curve \displaystyle y = \frac{k}{x} has none, which is the subject of linear and nonlinear functions.

The pattern recurs: filling time against flow rate for a fixed volume, and time against speed for a fixed distance in speed, distance and time. When a word problem’s quantities move in opposite directions, setting up a proportion is the wrong tool and a constant product is the right one. Looking ahead, the whole curve \displaystyle y = \frac{1}{x}, negative side included, is the reciprocal function, and inverse, joint and combined variation lets a quantity depend on several others at once.

Exercises

Reading tables

For Exercises 1 to 4, decide whether the table is inversely proportional, directly proportional or neither. Where it is one of the two, give k and the equation.

x: 1, 3, 6, 9; y: 54, 18, 9, 6
x: 2, 5, 6, 10; y: 9, 22.5, 27, 45
x: 2, 4, 6, 8; y: 20, 16, 12, 8
x: 0.5, 2, 4, 16; y: 32, 8, 4, 1
y is inversely proportional to x. In the table x: 3, 6, a, 24 with y: 16, b, 4, c, find a, b and c.

Equations and missing values

y is inversely proportional to x, and y = 9 when x = 4. Find k, write the equation, and find y when x = 6.
y is inversely proportional to x, and y = 2.5 when x = 8. Find y when x = 5, and x when y = 10.
Which are inverse proportions, which direct, and which neither? Give k where there is one. (a) \displaystyle y = \frac{7}{x} (b) \displaystyle y = \frac{x}{7} (c) xy = 15 (d) y = 7 - x (e) \displaystyle y = \frac{1}{2x}
y is inversely proportional to x. What happens to y when x is (a) multiplied by 4, (b) halved, (c) multiplied by \displaystyle \frac{2}{3}?

Stories

Five workers, all working at the same pace, take 6 hours to tile a floor. (a) How long would 3 workers take? (b) How many workers would finish in 2 hours?
Winifred walks a 12 km trail at a steady 4 km/h. (a) How long does the walk take? (b) How long would it take at 3 km/h? At 5 km/h, in hours and minutes?
A prize shared equally among 6 winners gives each 45 dollars. (a) How much would each receive if 9 shared it? If 10 did? (b) Could each receive exactly 40 dollars?
A rectangle has an area of 60 square centimeters. (a) Find its length when its width is 4 cm, and when it is 2.5 cm. (b) The width changes from 5 cm to 10 cm. What happens to the length?
On a see-saw, Vasco, who weighs 40 kg, sits 1.2 m from the pivot. Umeko weighs 32 kg. How far from the pivot must she sit on the other side to balance him?
Decide whether each pair is inversely proportional, directly proportional or neither, and say what stays fixed. (a) The number of people sharing a 2 kg bag of rice equally, and the amount each gets. (b) The number of notebooks bought at 3 dollars each, and the total cost. (c) The number of pages Saskia has read in a 300-page book, and the number she has left. (d) The number of equal pieces a 3 m rope is cut into, and the length of each piece.

Graphs

Make a table for \displaystyle y = \frac{18}{x} at x = 1, 2, 3, 6, 9 and 18. Plot the points, sketch the graph, and describe its shape.
The point (4, 5) lies on the graph of an inverse proportion. Find k, and name two more points on the graph with whole-number coordinates.
Yannick sketches \displaystyle y = \frac{10}{x} and draws his curve crossing the horizontal axis at x = 20. Explain what is wrong.

Reasoning

In \displaystyle y = \frac{12}{x} + 1, the value of y falls as x rises. Explain why it is still not an inverse proportion.
In Example 1 the floor takes \displaystyle h = \frac{24}{p} hours. Explain why the graph of this story never touches either axis, and why the formula stops describing real workers when p is very large.

Answers

Inversely proportional, k = 54: 1 \times 54 = 3 \times 18 = 6 \times 9 = 9 \times 6 = 54. The equation is \displaystyle y = \frac{54}{x}.
Directly proportional, k = 4.5: \displaystyle \frac{9}{2} = \frac{22.5}{5} = \frac{27}{6} = \frac{45}{10} = 4.5. The equation is y = 4.5x.
Neither. The products are 40, 64, 72 and 64, and the quotients 10, 4, 2 and 1. The values fall by 4 at each step, so they lie on the line y = 24 - 2x.
Inversely proportional, k = 16: 0.5 \times 32 = 2 \times 8 = 4 \times 4 = 16 \times 1 = 16. The equation is \displaystyle y = \frac{16}{x}.
k = 3 \times 16 = 48, so \displaystyle b = \frac{48}{6} = 8, \displaystyle a = \frac{48}{4} = 12 and \displaystyle c = \frac{48}{24} = 2.
k = 4 \times 9 = 36, so \displaystyle y = \frac{36}{x}, and at x = 6, \displaystyle y = \frac{36}{6} = 6.
k = 8 \times 2.5 = 20. At x = 5, \displaystyle y = \frac{20}{5} = 4. When y = 10, \displaystyle x = \frac{20}{10} = 2.
(a) Inverse, k = 7. (b) Direct: \displaystyle y = \frac{1}{7}x, so \displaystyle k = \frac{1}{7}. (c) Inverse, k = 15, since \displaystyle y = \frac{15}{x}. (d) Neither: x = 1 gives y = 6 and x = 2 gives y = 5, with products 6 and 10 and quotients 6 and 2.5. (e) Inverse, \displaystyle k = \frac{1}{2}, since \displaystyle xy = \frac{1}{2} for every x.
(a) y is divided by 4. (b) y is doubled. (c) y is divided by \displaystyle \frac{2}{3}, which multiplies it by \displaystyle \frac{3}{2}.
The job is 5 \times 6 = 30 worker-hours. (a) \displaystyle \frac{30}{3} = 10 hours. (b) \displaystyle \frac{30}{2} = 15 workers.
(a) \displaystyle \frac{12}{4} = 3 hours. (b) \displaystyle \frac{12}{3} = 4 hours. At 5 km/h, \displaystyle \frac{12}{5} = 2.4 hours, and 0.4 \times 60 = 24, so 2 hours 24 minutes.
The prize is 6 \times 45 = 270 dollars. (a) \displaystyle \frac{270}{9} = 30 dollars and \displaystyle \frac{270}{10} = 27 dollars. (b) No: \displaystyle \frac{270}{40} = 6.75, not a whole number of winners.
(a) \displaystyle \frac{60}{4} = 15 cm and \displaystyle \frac{60}{2.5} = 24 cm. (b) The length goes from \displaystyle \frac{60}{5} = 12 cm to \displaystyle \frac{60}{10} = 6 cm: the width doubles and the length halves.
Vasco’s side gives 40 \times 1.2 = 48 kilogram-meters, so Umeko sits \displaystyle \frac{48}{32} = 1.5 m from the pivot.
(a) Inverse: people times each share is always 2 kg. (b) Direct: cost divided by notebooks is always 3 dollars. (c) Neither: pages read plus pages left is always 300, a fixed sum, while the product changes, for instance 100 \times 200 = 20000 but 150 \times 150 = 22500. (d) Inverse: pieces times the length of each is always 3 m.
The values of y are 18, 9, 6, 3, 2 and 1; each product is 18. The curve falls steeply, then flattens, coming closer to each axis without touching either.
k = 4 \times 5 = 20. Any whole-number pair with product 20 works: (1, 20), (2, 10), (5, 4), (10, 2) or (20, 1).
A point on the horizontal axis has y = 0, and then xy = 0, not 10. At x = 20, \displaystyle y = \frac{10}{20} = 0.5: small, but above the axis.
The product is not fixed. At x = 1, y = 13 and xy = 13; at x = 2, y = 7 and xy = 14. Doubling x took y from 13 to 7, not to 6.5: the added 1 spoils the pattern.
The time axis needs p = 0, and \displaystyle \frac{24}{0} has no value: with no workers the floor is never laid. The workers axis needs h = 0, but p \times 0 = 0, not 24. For very large p the formula promises tiny times (\displaystyle \frac{24}{96} = 0.25 hours for 96 workers), but that many people would get in each other’s way, which the model assumes never happens.