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Direct Direct ProportionProportion
Inverse Proportion
Direct Proportion (Variation) Graph
Direct Variation
Direct Proportion
Inverse Proportion (Variation) Graph
Inverse Variation
Joint Variation
Understanding Formulae
The Circumference of circle is given by the formula :
C = πDWhat happens to the Circumference if we double the diameter
C = π(2D)New D = 2D
The Circumference doubles
In real-life we often want to see what effect changing the value of one of the variables has on the subject.
= 2πD
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain the term Direct Proportion.
1. Understand the idea of Direct Proportion.
Direct Direct ProportionProportion
2. Solve simple Direct Proportional problems.
Direct Proportion
21 Apr 202321 Apr 2023
Direct Direct ProportionProportion
“ .. When you double the number of cakes you double the cost.”
Cakes Cost
Two quantities, (for example, number of cakes and totalcost) are said to be in DIRECT Proportion, if :
Direct Proportion
Example : The cost of 6 cakes is £4.20. find the cost of 5 cakes.
6 4.20 1 4.20 ÷ 6 = 0.70 5 0.70 x 5 = £3.50
Write down two quantities that
are in direct proportion.
Easier methodCakes Pence
6 420 5
Are we expecting more or less
5
6420
(less)
350 £3.50
Direct Direct ProportionProportion
Direct Proportion
Example : Which of these pairs are in proportion.
(a) 3 driving lessons for £60 : 5 for £90
(b) 5 cakes for £3 : 1 cake for 60p
(c) 7 golf balls for £4.20 : 10 for £6
Same ratio means in proportion
Direct Direct ProportionProportion
Direct Proportion
Which graph is a direct proportion graph ?
x
y
x
y
x
y
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain the term Inverse Proportion.
1. Understand the idea of Inverse Proportion.
Inverse Inverse ProportionProportion
2. Solve simple inverse Proportion problems.
Inverse Proportion
Inverse Inverse ProportionProportion
Inverse Proportion is when one quantity increasesand the other decreases. The two quantities are said
to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other.
Example : Fill in the following table given x and yare inversely proportional.
Inverse Proportion
XX 11 22 44 88
yy 8080 102040
Notice xxy = 80Hence inverse
proportion
Inverse Inverse ProportionProportion
Men Hours
Inverse Proportion is the when one quantity increasesand the other decreases. The two quantities are said
to be INVERSELY Proportional or (INDIRECTLY Proportional) to each other.
Example : If it takes 3 men 8 hours to build a wall.How long will it take 4 men. (Less time !!)
3 8 1 3 x 8 = 24 hours
4 24 ÷ 4 = 6 hours
Inverse Proportion
y
x
Easier methodWorkers Hours
3 8 4
Are we expecting more or less
3
48
(less)
6 hours
21 Apr 202321 Apr 2023
Inverse Inverse ProportionProportion
Men Months
Example : It takes 10 men 12 months to build a house.How long should it take 8 men.
10 12 1 12 x 10 = 120
8 120 ÷ 8 = 15 months
Inverse Proportion
y
x
Easier methodWorkers months
10 12 8
Are we expecting more or less
10
812
(more)
15 months
12 288 ÷ 12 = 24 mins
1 32 x 9 = 288 mins 9 32 mins
21 Apr 202321 Apr 2023
Inverse Inverse ProportionProportion
Speed Time
Example : At 9 m/s a journey takes 32 minutes.How long should it take at 12 m/s.
Inverse Proportion
y
x
Easier methodSpeed minutes
9 32 12
Are we expecting more or less
9
1232
(less)
24 minutes
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain how Direct Direct Proportion Graph is always a straight line.
1. Understand that Direct Proportion Graph is a straight line.
Direct Direct ProportionProportion
2. Construct Direct Proportion Graphs.
Direct Proportion Graphs
Direct Direct ProportionProportion
The table below shows the cost of packets of “Biscuits”.
Direct Proportion Graphs
No. of Pkts 1 2 3 4 5 6
Cost (p) 20 40 60 80 100 120
We can construct a graph to represent this data.
What type of graph do we expect ?
Notice C ÷ P = 20Hence direct proportion
21 Apr 202321 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Direct Proportion
0
20
40
60
80
100
120
140
0 1 2 3 4 5 6
No. of Packets
Cos
t (p
)
Direct Proportion GraphsNotice that the points lie on a straight line passing through the
originSo direct proportion
C α PC = k P
k = 40 ÷ 2 = 20C = 20 P
Direct Direct ProportionProportion
Direct Proportion Graphs
KeyPoint
Two quantities which are in Direct Proportion
always lie on a straight linepassing through the origin.
Direct Direct ProportionProportion
Ex: Plot the points in the table below. Show that they are in Direct Proportion.Find the formula connecting D and W ?
Direct Proportion Graphs
We plot the points (1,3) , (2,6) , (3,9) , (4,12)
WW 11 22 33 44
DD 33 66 99 1212
1
Direct Direct ProportionProportion
Plotting the points
(1,3) , (2,6) , (3,9) , (4,12)
Direct Proportion Graphs
0 1 2 3 4
101112
23456789
Since we have a straight linepassing through the origin
D and W are in Direct Proportion.
W
D
1
Direct Direct ProportionProportion
Finding the formula connecting D and W we have.
Direct Proportion Graphs
0 1 2 3 4
101112
23456789
D α W
W
D
Constant k = 6 ÷ 2 = 3
Formula is : D= 3W
D = kWD = 6W = 2
Direct Direct ProportionProportion
Direct Proportion Graphs
1. Fill in table and construct graph
2. Find the constant of proportion (the k value)
3. Write down formula
Direct Direct ProportionProportion
Q The distance it takes a car to brake depends on how fast it is going.
The table shows the braking distance for various speeds.
Direct Proportion Graphs
SS 1010 2020 3030 4040DD 55 2020 4545 8080
Does the distance D vary directly as speed S ?
Explain your answer
The table shows S2 and D
Fill in the missing S2 values.
SS22
SS 1010 2020 3030 4040DD 55 2020 4545 8080
Direct Direct ProportionProportion
Direct Proportion Graphs
Does D vary directly as speed S2 ?
Explain your answer
100 400 900 1600
D
S2
Direct Direct ProportionProportion
Find a formula connecting D and S2.
Direct Proportion Graphs
D α S2
Constant k = 5 ÷ 100 = 0.05
Formula is : D= 0.05S2
D = kS2D = 5S2 = 100
D
S2
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain how the shape and construction of a Inverse Proportion Graph.
1. Understand the shape of a Inverse Proportion Graph .
Inverse Inverse ProportionProportion
2. Construct Inverse Proportion Graph and find its formula.
Inverse Proportion Graphs
Inverse Inverse ProportionProportion
The table below shows how the total prize money of £1800 is to be shared depending on how many winners.
Inverse Proportion Graphs
We can construct a graph to represent this data.
What type of graph do we expect ?
Notice W x P = £1800Hence inverse proportion
Winners WWinners W 11 22 33 44 55
Prize PPrize P £1800£1800 £900£900 £600£600 £450£450 £360£360
Direct Proportion GraphsNotice that the points
lie on a decreasing curve
so inverse proportion
Inverse Proportion
1PN
kP
N
k PN
1800 1 1800k
1800P
N
Inverse Inverse ProportionProportion
Inverse Proportion Graphs
KeyPoint
Two quantities which are in Inverse Proportion
always lie on a decrease curve
Inverse Inverse ProportionProportion
Ex: Plot the points in the table below. Show that they are in Inverse
Proportion.Find the formula connecting V and N ?
Inverse Proportion Graphs
We plot the points (1,1200) , (2,600) etc...
NN 11 22 33 44 55
VV 12001200 600600 400400 300300 240240
Inverse Inverse ProportionProportion
Plotting the points
(1,1200) , (2,600) , (3,400) (4,300) , (5, 240)
Inverse Proportion Graphs
0 1 2 3 4
1000
1200
200
400
600
800
Since the points lie on adecreasing curveV and N are in
Inverse Proportion.
N
V
5
1
N
1
N
Note that if we plotted V against
then we would get a straight line.
because v directly proportional to1
N
1
NThese graphstell us the same thing
V
N
V
1
N
Inverse Inverse ProportionProportion
Finding the formula connecting V and N we have.
Inverse Proportion Graphs
k = VN = 1200 x 1 = 1200
V = 1200N = 1
0 1 2 3 4
1000
1200
200
400
600
800
V
5 N
1V
N
1200V
N
kV
N
Direct Direct ProportionProportion
Direct Proportion Graphs
1. Fill in table and construct graph
2. Find the constant of proportion (the k value)
3. Write down formula
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain how to work out direct variation formula.
1. Understand the process for calculating direct variation formula.
Direct VariationDirect Variation
2. Calculate the constant k from information given and write down formula.
Direct VariationDirect Variation
Given that y is directly proportional to x,and when y = 20, x = 4.
Find a formula connecting y and x.
Since y is directly proportional to x the formula is of the form
y = kx k is a constan
t20 = k(4)
k = 20 ÷ 4 = 5
y = 5x
y = 20x =4
y
x
Direct VariationDirect Variation
The number of dollars (d) varies directly as the
number of £’s (P). You get 3 dollars for £2. Find a formula connecting d and P.
Since d is directly proportional to P the formula is of the form
d = kP k is a constan
t3 = k(2)
k = 3 ÷ 2 = 1.5
d = 1.5P
d = 3P = 2
d
P
d = 1.5 x 20 = 30 dollars
Direct VariationDirect Variation
Q. How much will I get for £20
d = 1.5Pd
P
Direct VariationDirect Variation
Given that y is directly proportional to the square of x, and when y = 40, x = 2.
Find a formula connecting y and x .
Since y is directly proportional to x squaredthe formula is of the form
y = kx2
40 = k(2)2
k = 40 ÷ 4 = 10
y = 10x2
y
x2
y = 40x = 2
Harder Direct Variation
Direct VariationDirect Variation
Q. Calculate y when x = 5
y = 10x2
y = 10(5)2 = 10 x 25 = 250
y
x2
Harder Direct Variation
Direct VariationDirect Variation
Q. The cost (C) of producing a football magazine varies as the square root of the number of pages (P). Given 36 pages cost 48p to produce. Find a formula connecting C and P.
Since C is directly proportional to “square root of” P the formula is of the form
k = 48 ÷ 6 = 8
C =k P
48 =k 36
C =8 P
C
√P C = 48P = 36
Harder Direct Variation
Direct VariationDirect Variation
Q. How much will 100 pages cost.
C =8 100
C =8 100 =8 10 =80p
C
√P
Harder Direct Variation
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain how to work out inverse variation formula.
1. Understand the process for calculating inverse variation formula.
Inverse Inverse VariationVariation
2. Calculate the constant k from information given and write down formula.
Inverse Inverse VariationVariation
Given that y is inverse proportional to x,and when y = 40, x = 4.
Find a formula connecting y and x.
Since y is inverse proportional to x the formula is of the form
k is a constan
tk = 40 x 4 = 160y = 40x =4
ky
x
160y
x
1 α yx
y
x1
y
x
Inverse Inverse VariationVariation
Speed (S) varies inversely as the Time (T)When the speed is 6 kmph the Time is 2 hours Find a formula connecting S and
T.Since S is inversely proportional to T the formula is of the form
k is a constan
t
S
T
S
T1
k = 6 x 2 = 12S = 6T = 2
kS
T
12S
T
1 α ST
Inverse Inverse VariationVariation
Find the time when the speed is 24mph.
S
T1 S = 24
T = ?
12S
T
1224
T
120.5
24T hours
Inverse Inverse VariationVariation
Given that y is inversely proportional to the square of x, and when y = 100, x = 2.
Find a formula connecting y and x .
Since y is inversely proportional to x squaredthe formula is of the form
k is a constan
tk = 100 x 22 = 400y = 100x = 2
2
ky
x
2
400y
x
2
1 α yx
y
x21
y
x2
Harder Inverse variation
Inverse Inverse VariationVariation
Q. Calculate y when x = 5
y = ?x = 5
2
400y
x
2
400
5y
y
x21
16y
Harder Inverse variation
Inverse Inverse VariationVariation
The number (n) of ball bearings that can be made from afixed amount of molten metal varies inversely as the cube of the radius (r). When r = 2mm ; n = 168Find a formula connecting n and r.
Since n is inversely proportional to the cube of r the formula is of the form
k is a constan
tk = 168 x 23 = 1344n = 100
r = 2
3
kn
r3
1 α nr
n
r3
1
3
1344n
r
y
r3
Harder Inverse variation
Inverse Inverse VariationVariation
How many ball bearings radius 4mmcan be made from the this amount of metal.
n
r3
1r = 4
3
1344
4n
21n
Harder Inverse variation
Inverse Inverse VariationVariation
T varies directly as N and inversely as SFind a formula connecting T, N and S given T = 144 when N = 24 S = 50
Since T is directly proportional to N and inversely to S the formula is of the form
k is a constan
tk = 144 x 50 ÷ 24= 300T = 144N = 24S = 50
kNT
S
300NT
S
α N
TS
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