Transcript Inverse Proportion. - gcse-maths

Inverse Proportion.
Cows 2
4
8
16
32
64
Days 64
32
16
8
4
2
Cattle
Days
64
2
32
d
64
32
=
Days
d
Cows
2
= ………….
What Is Inverse Proportion ?
Consider the problem below:
Problem 1.
A farmer has enough cattle feed to feed 64 cows for 2 days.
(a) How long would the same food last 32 cows ?
4 days
(b) Complete the table below for the same amount of cattle feed :
Cows 2
4
8
16
32
64
Days 64
32
16
8
4
2
We can now see what Inverse Proportion is all about:
Look at the cattle feed problem again:
Cows 2
4
8
16
32
64
Days 64
32
16
8
4
2
If you double the number of cows , what happens to the number of
days you can feed them ?
The number of days is halved.
If you treble the number of cows , what happens to the number of
days you can feed them ?
You can only feed them for a third of the number of days.
If you have ten times the number of cows , what happens to the
number of days you can feed them ?
You can only feed them for a tenth of the number of days.
For Direct Proportion when one quantity is doubled , trebled
quadrupled etc the other quantity becomes a half , third, quarter etc.
There is another important fact to pick up on in the table:
Cows 2
4
8
16
32
64
Days 64
32
16
8
4
2
What do you notice happens if you multiply cows x days ?
Cows x days = 128 for all pairs of values in the table.
This gives us one method for solving Inverse Proportion problems.
To see this method complete the problem below :
Problem 2 .
It takes 96 hours for 3 strawberry pickers to clear a field .
Complete the table below:
Pickers
3
6
12
24
88
Hours
96
48
24
12
6
The Idea of Picker-Hours.
Using the table we have just completed we can now find out how
many hours any number of pickers will take.
Pickers
3
6
12
24
88
Hours
96
48
24
12
6
How many hours will 17 pickers take to complete the job ?
Solution.
For all pairs of values : Pickers x hours = 288 picker hours.
That is to say there are 288 hours of work to complete.
Divide these hours amongst our 17 pickers:
No of hours = 288  17 = 16.94 hours per picker.
It will take 16 hours and 57 minutes to the nearest minute.
Problem 3.
If a school kitchen has enough food for 234 pupils for 37 days, how
long will the same food last 73 pupils ?
Solution.
If you double, treble etc the number of pupils what happens to the
number of days ?
The days half, third etc.
This is inverse proportion.
What is the total number of “ pupil days” ?
Pupil days = 234 x37 = 8658 pupil days.
Divide the pupil days by the 73 pupils :
8658  73 = 118.6 days.
The kitchen can feed 73 pupils for 118 days at most.
Problem 4.
You can exactly fit 7 volumes of an encyclopaedia each 5.7cm
thick on a shelf. How many volumes each 6.7cm thick fit on the
V
V
V
V
V
V
same shelf ?
O
O
O
O
O
O
Solution
V
O
L
L
L
L
L
L
L
1
2
3
4
5
6
7
If you double , treble etc the thickness of the encyclopaedia , what
happens to the number of volumes you can place on the shelf ?
Volumes half, third etc. This is inverse proportion.
What is the total number of “volume-centimetres” ?
Volume-centimetres = 7 x 5.7 = 39.9 volume centimetres
Divide the volume centimetres by the new volume thickness .
39.9  6.7 = 5.95 volumes.
You can place 5 volumes 6.7cm thick on the same shelf.
What Goes In The Box?.
(1) If 7 electricians can wire some new houses in 17 days , how
many electricians would be required to do the job in 9 days ?
14 electricians.
(2) Jane can type at 6o words a minute and took 35minutes to
complete a letter. How long would John take who types at 43 words
a minute ?
48.8 minutes.
(3) A car travelling at 45km/hr takes 33 minutes for a journey. How
long does a car travelling at 55km/hr take for the same journey ?
27 minutes.
The Cross- Multiplication Method.
We are going to apply the same method that we used to solve direct
proportion questions to inverse proportion questions , with one additional
step to the process.
The method has the following strengths :
• It is consistent with the method used for Direct Proportion.
•It establishes a strong and consistent routine for problem solving.
Consider the first problem again :
Problem 1.
A farmer has enough cattle feed to feed 64 cows for 2 days. How long
would the same food last 32 cows ?
Solution
What two quantities are being talked about ?
Cattle
Days
What two numbers go together ?
64
2
Add in the additional information.
32
d
d
If you doubled, trebled etc the cattle, what
would happen to the number of days ?
2
Half, third etc.
64
32
=
32d = 2 x 64
d = 128  32 = 4 days.
Inverse proportion. Use
arrows.
Cross multiply to solve but flip over the
second column numbers to make it work.
Problem 2.
It takes 13 workers 27 hours to leaflet an area of houses. How long
would 22 workers take to do the same work ?
Solution
What two quantities are being talked about ?
Workers
Hours
What two numbers go together ?
13
27
Add in the additional information.
22
h
h
If you doubled, trebled etc the workers,
what would happen to the number of hours ?
27
Half, third etc.
13
32
=
32h = 13 x 27
h = 351  32 = 7.84
Inverse proportion. Use
arrows.
Cross multiply to solve but flip over the
second column numbers to make it work.
It would take at least 8 hours to the nearest hour to complete the task.
Problem 3.
A car travelling at an average speed of 67km/hr takes 14 hours to complete
a journey. How long would the same journey take at 42km/hr ?
Solution
What two quantities are being talked about ?
Speed
Time
67
14
Add in the additional information.
42
t
t
If you doubled, trebled etc the speed, what
would happen to the time ?
14
Half, third etc.
67
42
=
42t = 67 x 14
h = 938  42 = 22.33
What two numbers go together ?
Inverse proportion. Use
arrows.
Cross multiply to solve but flip over the
second column numbers to make it work.
It would take 22.4 hours to complete the journey at 42km/hr .
What Goes In The Box ? 2
(1) 24 workers take 14 days to deliver census forms to all
houses in a city . How many workers could do it in 20 days ?
17 workers.
(2) A farmer can feed 245 sheep for 50 days. How many days can
he feed 152 sheep for with the same amount of food ?
80 days
(3) I can afford 23 bars of chocolate at 27p a bar . How many bars at
19p can I buy with the same money ?
32 bars.
Graphs of Inverse Proportion.
Consider once again the first problem we looked at:
A farmer has enough cattle feed to feed 64 cows for 2 days.
Complete the table below for the same amount of cattle feed :
Cows 2
4
8
16
32
64
Days 64
32
16
8
4
2
We are going to draw a graph of the table.
Choose your scale carefully and allow each access to go at least up to 65.
Estimate the position of the points (2,64) (4,32) etc as accurately as
you can.
Days
70
(2,64)
60
50
40
30
20
(4,32)
(8,16)
10
0
(16,8)
10
20
(32,4)
30
40
(64,2)
50
60
70
Cows
The graph is a typical inverse proportion graph :
Days
Cows
It shows us that as the number of cows increases :
The number of days decreases :
Obviously the reverse is also true that if we decrease the number of
cows we will increase the number of days feed.
What Goes In The Box ? 3
160 markers take 3 hours to complete marking their examination
scripts.
(a) Complete the table below :
Markers
Time
5
10
20
40
80
160
96
48
24
12
6
3
(b) Draw a graph of the table.
Time
100
80
60
40
20
0
20
40
60
80
100
120
140
160
180 Markers