Year 8 math
2h3
Name:
date:
Indirectly proportional situations
1. Painters are hired to complete a job. Assuming that each painter is equally
productive (i.e. they paint at the same rate) fill in the following table:
Number of painters
Hours to complete
the job
1
2
3
6
12
18
18
a. Is this a directly proportional situation? Why or why not?
b. What do you notice about the relationship between the variables? (i.e. the
number of painters vs. the hours needed).
1
Year 8 math
2h3
Name:
date:
Indirectly proportional situations
2. A group of friends want to rent a bus to go skiing. Assuming that each person
pays the same amount, fill in the following table
Number people going
10
Cost per person (€)
12
20
20
15
30
40
12
a. What is the cost of renting the bus?
b. Graph the relationship:
2
Year 8 math
2h3
Indirectly proportional situations
Name:
date:
Two variables are said to be inversely proportional when their products always
results in the same constant, k. (In German: Indirekt proportionalität).
When one variable is multiplied by any number, n, the other variable is divided
by n.
In other words, when one variable is doubled, tripled, or halved etc. the other
variable is halved, divided by three, or doubled, etc.
The graph of an inversely propotional relationship is called a hyperbole (in
German: hyperbel)
Ex.
We write that y ∝
1
k
and the equation for the relationship is y =
.
x
x
Check out the song (video 3) at:
http://www.onlinemathlearning.com/direct-inverse-proportion.html
3
Year 8 math
2h3
Indirectly proportional situations
Name:
date:
Find the missing term in an inversely proportional situation:
Ex. The following is an inversely proportional situation. Fill in the table:
x
y
4
25
8
20
10
5
Method 1:
Find the constant product and use it to solve for the unknown.
Method 2:
set up the pair of equal products and solve. (ie. x1y1 = x2y2 )
Could you find an equation to represent how to find y if you are given x?
Ex.
Example:
It takes 14 hours for a faucet with a flow of 18 liters per minute to fill a reservoir with
water. How long will it take if its flow is reduced to 7 liters per minute?
4
Year 8 math
2h3
Name:
date:
Indirectly proportional situations
Practice Exercises:
1. Determine if each of the following tables represents an inversely proportional
situation, a directly proportional situation, or neither.
Extra: If you can, try to write the equation to represent the relationship.
a.
1
100
c.
0.4
1.6
e.
2.1
24
2
50
4
25
5
20
0.6 0.8 1.2
2.4 3.2 4.8
10
10
40
2.5
1.4
5.6
1.6
6.4
b.
4
15
d.
0
0
f.
0
0
3.6 0.7 0.3 4.8
6.3
14 72 168 10.5 8
5
12
6
8
8
6
10
6
30
2
1
1
2
4
4
16
5
25
6
36
0.2 0.4 0.6 0.8 1
0.5 1
1.5 2
2.5
2. Graph each of the relationships in a.
3. Each of the following are inversely proportional relationships. Fill in the tables.
a.
Quantity per
bottle (ml)
# of bottles
c.
Wassermenge pro
s (l)
Zeitdauer (min)
400
300
6
200
3
10
0,25 0,2
10
0,1
2,5
20
b.
# workers
Time (h)
d.
# of days
worked
Money per
day
25
20
40
4
16
10
8
10
15
6
20
40
4. A hostel has enough food for 125 students for 16 days. How long will the food
last if 75 more students join them?
5. The price, P, of a diamond is directly proportional to the square of the weight,
W. If a 1 carat diamond costs $2000, find the price of a 0.7 carat diamond.
6. Boyle’s law states that for a constant temperature, the pressure of a gas varies
inversely with its volume. A sample of hydrogen gas has a volume of 8.56 liters
at a pressure of 1.5 atmospheres. Find the pressure at a volume of 10.5 liters.
5
Year 8 math
2h3
Indirectly proportional situations
Name:
date:
7. Grefor surft pro Monat durchschnittlich 15 Studen im Internet. Er bezahlt für
den Internetanschluss keine Grundgebühr und pro Minute “online” 1,8 ct.
a. Finde auf zwei verschienden Arten heraus, wie lange Gregor bei
gleichbleibenden Gesamtkosten im Internet surfen könnte, wen er nur 1,5
ct pro Minute bezahlen müsste.
b. Lucas hat die gleichen monatlichen Gesamtkosten wie Gregor, obwohl er
pro Monat im Schnitt 3 Studen weniger im Internet surft. Auch er bezahlt
keine Grundgebühr, sondern einen festen Preis pro Minute. FInde mithilfe
einer Gleichung heraus, wie viel Lucas pro Minute bezahlen muss.
8. It is known that current (A) in an electric circuit is inversely proportional to the
resistance (R) in the circuit. When the resistance is 3 ohms, the current is 2
amperes. Find the resistance if the current is 5 amperes.
9. The following table represents the number of boxes required to package
chocolate bars, compared to the capacity (how much fits into) of each box.
x
Capacity of each
box
y
# of boxes (in
hundreds)
0
20
40
50
100
n/a
100
50
40
20
a. How many chocolate bars are being packed?
b. Without doing any calculations, explain in words why the situation is inversely
proportional.
b. Why does the column with x = 0 not make any sense for the situation?
6
Year 8 math
2h3
Indirectly proportional situations
Answers:
1.
a. inversely proportional (y = 100/x)
b. neither (no constant product or quotient)
c. directly proportional (k = 4, y = 4x)
d. neither (no constant product or quotient)
e. inversely proportional (y = 50.4/x)
f. directly proportional (k = 2.5, y = 2.5x)
Name:
date:
2.
a.
b.
c.
120"
7"
6"
5"
4"
3"
2"
1"
0"
100"
80"
60"
40"
20"
0"
0"
0"
10"
20"
30"
40"
0.5"
1"
1.5"
2"
50"
d.
e.
f.
40"
180"
35"
160"
30"
3"
2.5"
140"
25"
20"
15"
120"
2"
100"
1.5"
80"
10"
1"
60"
5"
40"
0"
0"
1"
2"
3"
4"
5"
6"
7"
0.5"
20"
0"
0"
0"
1"
2"
3"
4"
5"
6"
7"
0"
0.2"
0.4"
0.6"
0.8"
1"
1.2"
3. tables:
a.
Quantity per
bottle (ml)
# of bottles
c.
Wassermenge
pro s (l)
Zeitdauer (min)
400
300
800 24
200
6
8
3
10
12
1
0,125 0,1
0,25 0,2
10
12,5 2,5
20
25
b.
# workers
Time (h)
d.
# of days
worked
Money per
day
25
20
100 125 16
40
50
4
6
12
8
3
30
20
10
15
40
10
8
62.5
4. 10 days
7
Year 8 math
2h3
5. $980
price
Carat squared
Indirectly proportional situations
2000
(12 = )
1
980
(0.72
=)
0.49
Name:
date:
2000(1) = 0.49x
2000
=x
0.49
980 = x
6. c. 1.22 atmospheres
Volume (l)
Pressure of
hydrogen (at)
7.
a. 18 h
8.56
1.5
10.5
8.56(1.5) = 10.5x
12.84
=x
10.5
1.222857... = x
b. 2.25 ct
8. 1.2 ohms
9.
a. 200 000 chocolate bars
b. As the capacity of the boxes increase you naturally will need fewer boxes. If you
can fit twice as many chocolate bars inside each box, you will need half the number
of boxes, etc. Therefore they are inversely proportional.
c. The point at x = 0 makes no sense because that would mean the box can hold 0
chocolate bars. No number of boxes that hold 0 chocolate bars could be used to pack
up 2000 chocolate bars.
8