Direct Proportion

47
Direct Direct Proportion Proportion Inverse Proportion Direct Proportion (Variation) Graph Direct Variation Direct Proportion Inverse Proportion (Variation) Graph Inverse Variation Joint Variation

description

Direct Proportion. Direct Proportion. Inverse Proportion. Direct Proportion (Variation) Graph. Inverse Proportion (Variation) Graph. Direct Variation. Inverse Variation. Joint Variation. Understanding Formulae. - PowerPoint PPT Presentation

Transcript of Direct Proportion

Page 1: Direct Proportion

Direct Direct ProportionProportion

Inverse Proportion

Direct Proportion (Variation) Graph

Direct Variation

Direct Proportion

Inverse Proportion (Variation) Graph

Inverse Variation

Joint Variation

Page 2: Direct Proportion

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

Page 3: Direct Proportion

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

Page 4: Direct Proportion

20 Apr 202320 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

Page 5: Direct Proportion

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

Page 6: Direct Proportion

Direct Direct ProportionProportion

Direct Proportion

Which graph is a direct proportion graph ?

x

y

x

y

x

y

Page 7: Direct Proportion

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

Page 8: Direct 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

Page 9: Direct 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

Page 10: Direct Proportion

20 Apr 202320 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

Page 11: Direct Proportion

12 288 ÷ 12 = 24 mins

1 32 x 9 = 288 mins 9 32 mins

20 Apr 202320 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

Page 12: Direct Proportion

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

Page 13: Direct Proportion

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

Page 14: Direct Proportion

20 Apr 202320 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

Page 15: Direct Proportion

Direct Direct ProportionProportion

Direct Proportion Graphs

KeyPoint

Two quantities which are in Direct Proportion

always lie on a straight linepassing through the origin.

Page 16: Direct Proportion

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

Page 17: Direct Proportion

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

Page 18: Direct Proportion

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

Page 19: Direct Proportion

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

Page 20: Direct Proportion

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

Page 21: Direct Proportion

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

Page 22: Direct Proportion

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

Page 23: Direct Proportion

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

Page 24: Direct Proportion

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

Page 25: Direct Proportion

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

Page 26: Direct Proportion

Inverse Inverse ProportionProportion

Inverse Proportion Graphs

KeyPoint

Two quantities which are in Inverse Proportion

always lie on a decrease curve

Page 27: Direct Proportion

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

Page 28: Direct Proportion

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

Page 29: Direct Proportion

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

Page 30: Direct Proportion

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

Page 31: Direct Proportion

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.

Page 32: Direct Proportion

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

Page 33: Direct Proportion

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

Page 34: Direct Proportion

d = 1.5 x 20 = 30 dollars

Direct VariationDirect Variation

Q. How much will I get for £20

d = 1.5Pd

P

Page 35: Direct Proportion

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

Page 36: Direct Proportion

Direct VariationDirect Variation

Q. Calculate y when x = 5

y = 10x2

y = 10(5)2 = 10 x 25 = 250

y

x2

Harder Direct Variation

Page 37: Direct Proportion

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

Page 38: Direct Proportion

Direct VariationDirect Variation

Q. How much will 100 pages cost.

C =8 100

C =8 100 =8 10 =80p

C

√P

Harder Direct Variation

Page 39: Direct Proportion

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.

Page 40: Direct Proportion

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

Page 41: Direct Proportion

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

Page 42: Direct Proportion

Inverse Inverse VariationVariation

Find the time when the speed is 24mph.

S

T1 S = 24

T = ?

12S

T

1224

T

120.5

24T hours

Page 43: Direct Proportion

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

Page 44: Direct Proportion

Inverse Inverse VariationVariation

Q. Calculate y when x = 5

y = ?x = 5

2

400y

x

2

400

5y

y

x21

16y

Harder Inverse variation

Page 45: Direct Proportion

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

Page 46: Direct Proportion

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

Page 47: Direct Proportion

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