© t madas. what do we mean when we say two quantities are in proportion ? it means that if: one of...

© T Madas

What do we mean when we say two quantities are in proportion?

It means that if:

one of them doubles,the other one also doubles.one of them trebles,the other one also trebles.one of them x4,the other one also x4.one of them halves,the other one also halves.one of them ÷4,the other one also ÷4.

Can you give examples of directly proportional quantities from every day life?

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Directly proportional quantities:

They increase or decrease at the same rate

More formally:

Two variables are directly proportional if the ratio between them remains constant.

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Two variables v and t are directly proportional.

When t = 8, v =18.

Write a formula which links v and t, in the form v = …

Proportional

v t

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ProportionalityConstant

v t

v = kt This will be the formula when we find the value of k


When t = 8, v =18.


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v t

v = kt

v = kt

18= k x 8

8k = 18

k = 188 = 9

4 = 2.25v = t9

4

v = 9t4

v = 2.25t

So:

or:

or:


When t = 8, v =18.


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In a chemistry experiment, the reaction time t is directly proportional to the mass m of the compound present.

When the mass is 3 grams the reaction time is 0.2 seconds.

1. Write a formula which links t and m, in the form t = …

2. What is the reaction time when the mass is 8 grams?

Proportional

t m

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t m


t = km This will be the formula when we find the value of k

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t m

t = km

t = km

0.2= k x 3

3k =0.2

k = 0.23 = 2

30 ≈0.067= 115

t = m115

t = m15

t ≈0.067m

So:

or:

or:

t = m15using:

t = 815 ≈0.53 s

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What do we mean when we say two quantities are inversely proportional ?

It means that if:

one of them doubles,the other one halves.one of them x3,the other one ÷3.one of them x4,the other one ÷4.one of them ÷2,the other one x2.one of them ÷10,the other one x10.

Can you give an example of inversely proportional quantities from every day life?

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The Civic Centre is to be painted, so they call a firm of decorators. If this firm provide:

1 decorator 2 decorators3 decorators4 decorators5 decorators6 decorators

10 decorators12 decorators15 decorators20 decorators30 decorators60 decorators

120 decorators

will take 60 days for the jobwill take 30 days for the jobwill take 20 days for the jobwill take 15 days for the jobwill take 12 days for the jobwill take 10 days for the jobwill take 6 days for the jobwill take 5 days for the jobwill take 4 days for the jobwill take 3 days for the jobwill take 2 days for the jobwill take 1 day for the jobwill take ½ day for the job

1 x 60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10

10 x 612 x 515 x 420 x 330 x 260 x 1

120 x ½

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INVERSELY PROPORTIONAL QUANTITIES

One increases at the same rate as the other one decreases.

More formally:

Two variables are inversely proportional if their product remains constant.

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A variable P is inversely proportional to a variable A.

When A = 2, P = 36.

1. Write a formula which links P and A, in the form P = …

2. Find the value of P when A is 2.5.

InverselyProportional

P 1A

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When A = 2, P = 36.



P 1A

P 1A= k x


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When A = 2, P = 36.



P 1A

P 1A= k x

This will be the formula when we find the value of k

PkA=

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When A = 2, P = 36.



P 1A

P 1A= k x

PkA=

PkA=

36k2=

k = 72

P72A=So:

P72A=using

P722.5=

1445=

28810= = 28.8

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A variable F is inversely proportional to a variable t.

When t = 3, F = 12.

Find the value of t when F is 48.

InverselyProportional

F 1t

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When t = 3, F = 12.


F 1t

F 1t= k x


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When t = 3, F = 12.


F 1t

F 1t= k x

This will be the formula when we find the value of k

Fkt=

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When t = 3, F = 12.


F 1t

F 1t= k x

Fkt=

Fkt=

12k3=

k = 36

F36t=So:

F36t=using

4836t=

48 =t 36

=t 3648=

34

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When t = 3, F = 12.


F x t =constant12 x 3 = 36

36 ÷ 48=0.75

Since we do not require a formula in this example we could also have worked as follows:

The product of inversely proportional quantities remains constant

48 x t = 36

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Sometimes we may be asked to set and solve problems involving direct or inverse proportion to the:

• square of a variable• cube of a variable• square root of a variable

or simply combine 3 variables with direct and inverse proportion in the same problem.

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A variable A is directly proportional to the square of another variable r .

When r = 3, A = 36.

Find the value of A, when r = 2.5

A r

A = kr 2 36 = kx 32

9k = 36

k = 4

A = 4r 2So:

using:

2

A = kr 2

A = 4r 2

A = 4 x2.52

A = 4 x52

2

A = 4 x254

A = 25

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A variable y is directly proportional to the SQUARE ROOT of another variable x .

When x = 25, y = 3.

Find the value of x, when y = 1.2

y x

y = k x 3 = k x 25

5k = 3

k =35

y =35So:

using: y = k x

= 0.6

x

y =35 x

1.2=35 x

65 =

35 x5 x x 5

6 = 3 x

= 2x= 4x

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A variable W is directly proportional to a variable m and inversely proportional to another variable t.

When m = 2 and t = 8, W = 15.

Find the value of W when m = 6 and t = 4.

W 1t

W mt

= k x

So:

m x

W mt

W kmt

=

W kmt

=

15 k x 28

=

15 2k8

=

2k =120

k = 60W 60m

t=

using:W 60mt

=

W 60 x 64

=

W 3604

=

W = 90

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A variable F is directly proportional to a variable m and inversely proportional to the square of another variable r.

When m = 10 and r = 2, F = 15.

Find the value of F when m = 24 and r = 3.

F 1r

2

F mr

2= k x

So:

m x

F mr

2

F kmr

2=

F kmr

2=

15 k x 1022

=

15 10k4

=

10k=60

k = 6F 6m

r 2

=

using:F 6mr

2=

F 6 x 2432

=

F 1449

=

F = 16

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What does the graph of two directly proportional quantities looks like?

Cost of packets of pens 3 pens cost £2

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Cost of packets of pens

12108642Cost (£)

181512963Number of pens

3 pens cost £2

Let us plot the information of this table in a graph

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Cost of packets of pens

12108642Cost (£)

181512963Number of pens

3 pens cost £2

4 8 12 16 20 24

12

10

8

6

4

2

0 pens

£

© T Madas

4 8 12 16 20 24

12

10

8

6

4

2

0 pens

£

when graphed the points of Directly Proportional Quantities:

1. always form a straight line through the origin

2. always form the corners of similar rectangles whose opposite corner is at the origin.

3. the line is a diagonal of every rectangle

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12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

The data above has been obtained from a chemistry experiment and concerns two quantities, u and v.

Are u and v directly proportional quantities?

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0 5 10 15 20 25

12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

20

15

10

5

v

the quantities u and v are directly proportional u

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uv

vu

0 5 10 15 20 25

12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

20

15

10

5

17

25

What is the gradient of the line?gradient=

diff in ydiff in x

=2517

≈ 1.47

the ratio between directly proportional quantities remains constant.Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round?

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u

v0 5 10 15 20 25

12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

20

15

10

5

25

What is the gradient of the line?gradient=

diff in ydiff in x

=2517

≈ 1.47

the ratio between directly proportional quantities remains constant.Work the ratio v : u from the table and compare it with the gradient of this line. What would have happened if we plotted the data with the axes the other way round?

17

1725

= 0.68

u : v

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12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

u v

u = kv

u = kv

3.4 = k x 5

5k =3.4

k = 3.45 =6.8

10 =0.68u = 0.68vSo:

We could obtain a formula linking u and v

The proportionality constant is the gradient of the line in the graph

v =1

0.68or u

v ≈1.47u

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12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

v

u0 5 10 15 20 25

20

15

10

5

v ≈1.47u

© T Madas

12

8.16

10

6.8

8

5.44

5

3.4

v

u

25211916

1714.2812.9210.88

u

v0 5 10 15 20 25

20

15

10

5

u=0.68v

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What does the graph of two inversely proportional quantities looks like?

1 decorator takes 24 days to finish a job

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1 decorator takes 24 days to finish a job

1234681224Days

2412864321No of decorators

0 5 10 15 20 25

20

15

10

5

decorators

days

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0 5 10 15 20 25

20

15

10

5

decorators

days

The graphed points of Inversely Proportional Quantities:

1. always lie on a curve like the one shown below.

2. always form the corners of rectangles of constant area whose opposite corner is at the origin.

Hyperbola

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The data above has been obtained from the physics department and concerns two quantities, P and A.

Are P and A inversely proportional quantities?

3.754.567.51011.251518A

242015129865P

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1 2 3 4 5 6 7

4.5

4

3.5

3

2.5

2

1.5

1

0.5

When plotted, Inversely Proportional quantities, always show as Hyperbolas.

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Suppose we have a formula which contains 2 or more variables.

The data which produced this formula is not available.

Is it possible to establish if variables are directly proportional or inversely proportional?

This is how this is done.

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Directly proportional to variables which appear in the numerator of the R.H.S

Inversely proportional to variables which appear in the denominator of the R.H.S

The variable for which the formula is solved for is:

svt

= v s v and s are directly proportionalv 1

t v and t are inversely proportional

s=vt s t s and t are directly proportional

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Vmgh

=

V m V and m are directly proportional

m 1g m and g are inversely

proportional

V=mgh

Vhmg

=

V g V and g are directly proportionalV h V and h are directly proportional

m 1h m and h are inversely

proportional

h 1g h and g are inversely

proportional

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GMm

Fr 2

=

F G F and G are directly proportionalF M F and M are directly proportionalF m F and m are directly proportional

F 1r 2

F is inversely proportional to the square of r

To get relationships between any other 2 variables we appropriately rearrange the formula.

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4V3

=

V r 3 V is directly proportional to the cube

of r

Rearranging the formula for r gives:

πr 3

V π Because π is not a v_______;π is a c_______ n______

ariableonstant umber

3 3V4

r π=

CHALLENGEr is directly proportional to the c___ r___ of __ ube oot V

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u + v

St

= S 1t S and t are inversely

proportionalS u Because u and v

are not in a productS v

S is directly proportional to the sum of u and v

S u + v

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60 workers, working a 9 hour day produce 720 toys a day.1. Find a formula which relates the number of workers w, the number of hours they work h and the number of toys T they produce.

2. How many hours a day, do 90 workers need to work if they are to produce 1020 toys?

The formula must contain the 3 variables w, h and T

Suppose that:the workers work a constant number of hours per dayThen:If we double the workers, ___________________________the toys produced will also double

Toys and workers are directly proportional quantitiesT w

Suppose that:we keep the number of workers constantThen:doubling the hours they work, ___________________________the toys produced will also double

Toys and hours are directly proportional quantitiesT h

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Three variables u, v and w are related by a formula.The following table gives some of the values that these three variables can take:

Obtain the formula linking these variables, solved for u.

54321111w

121520308642v

120120120120161284u

u v

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Three variables u, v and w are related by a formula.The following table gives some of the values that these three variables can take:

Obtain the formula linking these variables, solved for u.

54321111w

121520308642v

120120120120161284u

v 1w

v 1w

v uw

= k x

So:u x

v uw

v kuw

=

v kuw

=

2 k x 41

=

4k = 2

k =

v u

w=

u v

12

u v= w

u 2= vw

© t madas. what do we mean when we say two quantities are in proportion ? it means that if: one of...

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