related quantities a) the cost of a lemonade is €1.10. complete the following table. (l is the...

DIRECTLY ANDINDIRECTLY

PROPORTIONAL

Related Quantities a) The cost of a Lemonade is €1.10. Complete the following table. (L is the number of Lemonade cans and C is the cost)

What is the gradient of the graph?

What is the equation connecting C and L?

Use the equation to find C when L = 150.

L 1 2 3 5C €1.10 €5.50 €0.00

0 1 2 3 4 5 60

1

2

3

4

5

6

Number Of Lemonade Cans

Costin €

€2.20 €3.30

0

Draw the graph of C against L

0 2 4 6 8 10 120

10

20

30

40

50

60

Hours

Wage

Related Quantities b) Emily works part time at a local store. Her wage is €5.50 per hour. (H is the number of hours and W is her weekly wage)

What is the gradient of the graph?

What is the equation connecting W

and H?

Use the equation to find H when W= €46.75

H 0 1 4 8W €5.50 €44.00 €55.00€0.00 €22.00

10

Draw the graph of H against W

The wage (W) is DIRECTLY PROPORTIONAL to the number of hours (H) worked.

W directly proportional to H

W is connected to H by the formula W = kH k is called the constant of proportion.

The graph of W against H is a straight line through the origin. The gradient represents the value per unit (e.g. wage per hour)

is used to mean ‘W is directly proportional to H’Thus is another way of writing

The ratio is constant (the same) for every pair of values of W and H

CHECKING FOR DIRECT PROPORTION

Which of the following graphs involve direct proportion? a) Revenue of a Quantity of Items Produced

Revenue (is, is not) directly proportional to quantity produced. REASON: Graph is a straight line through the origin.

CHECKING FOR DIRECT PROPORTIONb) Cost of a journey by Taxi.

Cost (is, is not) directly proportional to distance travelled. REASON: Graph is a straight line but NOT through the origin.

CHECKING FOR DIRECT PROPORTIONc) Distance travelled by a car at constant speed

Distance (is, is not) directly proportional to time. REASON: Graph is a straight line through the origin.

CHECKING FOR DIRECT PROPORTIONd) Temperature of a cup of tea

Temperature (is, is not) directly proportional to time. REASON: Graph is NOT a straight line


a) Is B directly proportional to A in the following table? If so give the equation connecting B and A.

The ratio is constant B is directly proportional to A Equation:

A 1 3 4 10

B 4 12 16 40Check the

ratio for each pair

of values


b) Is B directly proportional to A in the following table? If so give the equation connecting B and A.

?

The ratio is constant B is directly proportional to A Equation:

A 0 1 2 6

B 0 2.5 5 15Check the ratio

for each pair of values


c) Is B directly proportional to A in the following table? If so give the equation connecting B and A.

The ratio is not constant B is NOT directly proportional to A

A 1 2 3 4

B 2 3 6 10Check the

ratio for each pair

of values

CALCULATING WITH DIRECT PROPORTIONa) In the following table B is directly proportional to A. Find the equation connecting B and A. Hence complete the table.

A 3 5

B 10.5 28

B is directly proportional to A

17.5

When A = 5

𝐵=3.5 𝐴𝐵=3.5×5=17.5

When B = 28

𝐵=3.5 𝐴28=3.5 𝐴𝐴=

283.5

=8

Equation connecting B and A

8

Substitute values of A and B

Rearrange

Use the equation to find the missing

values

c) The mass, M kg, of oil is directly proportional to its volume, V litres. 6.5 litres of oil have a mass of 23.4kg. What is the equation connecting M and V? Hence calculate: i) the mass of 10 litres of oil ii) the volume of 171kg of oil

M is directly proportional to V

When V = 10

𝑀=3.6𝑉𝑀=3.6×10=36 kg

When M = 171

𝑀=3.6𝑉171=3.6𝑉V=

1713.6

=47.5 𝑙𝑖𝑡𝑟𝑒𝑠

Equation connecting M and V isSubstitute values of M and V

Rearrange


values

CALCULATING WITH DIRECT PROPORTION

a) In the following table B is directly proportional to the square of A. Find the equation connecting B and A. Hence complete the table.

A 2 5

B 14 61.74

B is directly proportional to

Equation connecting B and A is


Rearrange

CALCULATING WITH DIRECT PROPORTION a) In the following table B is directly proportional to the square of A. Find the equation connecting B and A. Hence complete the table.

A 2 5

B 14 61.7487.5

When A = 5

𝐵=3.5 𝐴2

𝐵=3.5×52=87.5

When B = 61.74

𝐵=3.5 𝐴2

61.74=3.5 𝐴2

𝐴2=61.743.5

4.2


values

𝐴=√17.64=4.2𝐴2=17.64

CALCULATING WITH DIRECT PROPORTIONb) and when P is 8 Q is 4. Find the equation connecting P and Q. Hence find i) P when Q = 7 ii) Q when P = 84.5

Equation connecting P and Q is

Substitute values of P and Q

Rearrange

CALCULATING WITH DIRECT PROPORTIONb) and when P is 8 Q is 4. Find the equation connecting P and Q. Hence find i) P when Q = 7 ii) Q when P = 84.5

When Q = 7

𝑃=0.5𝑄2

𝑃=0.5×72=24.5

When P = 84.5

𝑃=0.5𝑄2

84.5=0.5𝑄2

𝑄2=84.50.5


values

𝑄2=169

𝑄=√169=13

CALCULATING WITH DIRECT PROPORTION c) A factory produces spheres used as garden ornaments. The weight W kg is directly proportional to the cube of its diameter D cm.

i) Write down a formula connecting W and D

ii) Find the value of k given that an ornament of diameter 30cm weighs 9kg. (Give k as a fraction in its lowest terms).

Substitute values of W and D

Rearrange

W=k D 3

∴𝑊=13000

𝐷3

CALCULATING WITH DIRECT PROPORTION c) A factory produces spheres used as garden ornaments. The weight W kg is directly proportional to the cube of its diameter D cm.

iii) For safety reasons an ornament cannot weigh more than 30kg. Find the largest diameter of an ornament correct to the nearest cm

For SAFETY the largest diameter can be .

𝑊=13000

𝐷3

44cm

INVERSE PROPORTIONA inversely proportional to B

A is connected to B by the formula thus AB = kk is called the constant of proportion.

The product is constant

is used to mean ‘A is inversely proportional to B’Thus is another way of writing

A inversely proportional to

CALCULATING WITH INVERSE PROPORTION

a) In the following table A is inversely proportional to B. Find the equation connecting A and B. Hence complete the table.

A 5 8

B 10 4

A is inversely proportional to B


Rearrange

Equation connecting A and B is

CALCULATING WITH INVERSE PROPORTION a) In the following table A is inversely proportional to B. Find the equation connecting B and A. Hence complete the table.

A 5 8

B 10 46.25

When A = 8

𝐴=50𝐵

8=50𝐵

When B = 4

𝐴=50𝐵

𝐴=504

=12.5

12.5

Use the equation

to find the

missing values

8𝐵=50𝐵=

508

=6.25


b) In an electrical circuit, the resistance, R ohms, is inversely proportional to the square of the current, I amps. When the resistance is 4 ohms, the current flowing is 6 amps.

i) Find the equation connecting R and I

R is inversely proportional to the square of I

Substitute values of R and I

Rearrange

Equation connecting R and I is


ii) Find the resistance when the current is 7 amps (correct to 3 s.f.)

R=2.94 h𝑜 𝑚𝑠

ii) Find the current when the resistance is 3 ohms (correct to 3 s.f.)

3 I2=144

I2=1443

I2=48

I=√48=6.93𝑎𝑚𝑝𝑠


c) If varies inversely with the square of and when , find i) the equation connecting ii) when

is inversely proportional to the square of

𝑦𝑥2=108

Equation connecting and is

75 𝑥2=108

𝑥2=10875

𝑥2=1.44

𝑥2=√1.44𝑥2=1.2

related quantities a) the cost of a lemonade is €1.10. complete the following table. (l is the...

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