Directly and Indirectly proportional

Related Quantities a) The cost of a Lemonade is €1. 10 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? Gradient= change in y change in x Gradient= 5.5 5 =1.1 What is the equation connecting C and L? 𝐶=1.1𝐿 Use the equation to find C when L = 150. C=150×€1.10=€165 L 1 2 3 5 C €1.10 €5.50 €0.00 €2.20 €3.30 Draw the graph of C against L

Related Quantities b) Emily works part time at a local store 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? Gradient= change in y change in x Gradient= 55 10 =5.5 What is the equation connecting W and H? W=5.5H Use the equation to find H when W= €46.75 €46.75=€5.5H 𝐻= 46.75 5.5 𝐻=8.5 ℎ𝑜𝑢𝑟𝑠 H 1 4 8 W €5.50 €44.00 €55.00 10 €0.00 €22.00 Draw the graph of H against W

W directly proportional to H → 𝑊=𝑘𝐻 The wage (W) is DIRECTLY PROPORTIONAL to the number of hours (H) worked. W directly proportional to H → 𝑊=𝑘𝐻 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) 𝑊∝𝐻 is used to mean ‘W is directly proportional to H’ Thus 𝑊∝𝐻 is another way of writing 𝑊=𝑘𝐻 The ratio 𝑊 𝐻 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 PROPORTION b) 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 PROPORTION c) Distance travelled by a car at constant speed Distance (is, is not) directly proportional to time. REASON: CHECKING FOR DIRECT PROPORTION Graph is a straight line through the origin.

CHECKING FOR DIRECT PROPORTION d) Temperature of a cup of tea Temperature (is, is not) directly proportional to time. REASON: Graph is NOT a straight line

CHECKING FOR DIRECT PROPORTION a) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 4 1 =4 12 3 =4 16 4 =4 40 10 =4 The ratio 𝐵 𝐴 is constant ∴ B is directly proportional to A Equation: 𝐵=4𝐴 A 1 3 4 10 B 12 16 40 Check the ratio 𝐵 𝐴 for each pair of values

CHECKING FOR DIRECT PROPORTION b) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 ? 2.5 1 =2.5 5 2 =2.5 15 6 =2.5 The ratio 𝐵 𝐴 is constant ∴ B is directly proportional to A Equation: 𝐵=2.5𝐴 A 1 2 6 B 2.5 5 15 Check the ratio 𝐵 𝐴 for each pair of values

CHECKING FOR DIRECT PROPORTION c) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 2 1 =2 3 2 =1.5 6 3 =2 10 4 =2.5 The ratio 𝐵 𝐴 is not constant ∴ B is NOT directly proportional to A A 1 2 3 4 B 6 10 Check the ratio 𝐵 𝐴 for each pair of values

CALCULATING WITH DIRECT PROPORTION a) In the following table B is directly proportional to A. Find the equation connecting B and A. Hence complete the table. ∴𝐵=𝑘𝐴 10.5=𝑘×3 k= 10.5 3 k=3.5 A 3 5 B 10.5 28 8 17.5 B is directly proportional to A Equation connecting B and A 𝐵=3.5𝐴 Substitute values of A and B Rearrange When A = 5 When B = 28 Use the equation 𝐵=3.5𝐴 to find the missing values 𝐵=3.5𝐴 𝐵=3.5𝐴 𝐵=3.5×5=17.5 28=3.5𝐴 𝐴= 28 3.5 =8

M is directly proportional to V Equation connecting M and V is 𝑀=3.6𝑉 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=𝑘𝑉 23.4=𝑘×6.5 k= 23.4 6.5 k=3.6 M is directly proportional to V Equation connecting M and V is 𝑀=3.6𝑉 Substitute values of M and V Rearrange When V = 10 When M = 171 Use the equation M=3.6V to find the missing values 𝑀=3.6𝑉 𝑀=3.6𝑉 𝑀=3.6×10=36kg 171=3.6𝑉 V= 171 3.6 =47.5 𝑙𝑖𝑡𝑟𝑒𝑠

CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 2 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. ∴𝐵=𝑘 𝐴 2 14=𝑘× 2 2 14=4𝑘 k= 14 4 k=3.5 A 2 5 B 14 61.74 B is directly proportional to 𝐴 2 Substitute values of A and B Rearrange Equation connecting B and A is 𝐵=3.5 𝐴 2

CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 2 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 4.2 87.5 When B = 61.74 Use the equation 𝐵=3.5 𝐴 2 to find the missing values 𝐵=3.5 𝐴 2 61.74=3.5 𝐴 2 𝐴 2 = 61.74 3.5 When A = 5 𝐴 2 =17.64 𝐵=3.5 𝐴 2 𝐴= 17.64 =4.2 𝐵=3.5× 5 2 =87.5

CALCULATING WITH DIRECT PROPORTION b) 𝑃∝ 𝑄 2 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 ∴𝑃=𝑘 𝑄 2 8=𝑘× 4 2 8=16𝑘 k= 8 16 k=0.5 𝑃∝ 𝑄 2 Substitute values of P and Q Rearrange Equation connecting P and Q is 𝑃=0.5 𝑄 2

CALCULATING WITH DIRECT PROPORTION b) 𝑃∝ 𝑄 2 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 Use the equation P=0.5 Q 2 to find the missing values When P = 84.5 𝑃=0.5 𝑄 2 When Q = 7 84.5=0.5 𝑄 2 𝑃=0.5 𝑄 2 𝑄 2 = 84.5 0.5 𝑃=0.5× 7 2 =24.5 𝑄 2 =169 𝑄= 169 =13

CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 3 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). W=𝑘 𝐷 3 9=𝑘× 30 3 9=27000𝑘 k= 9 27000 k= 1 3000 W=k D 3 Substitute values of W and D Rearrange ∴𝑊= 1 3000 𝐷 3

CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 3 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 𝑊= 1 3000 𝐷 3 30= 1 3000 𝐷 3 D 3 =30×3000 D 3 =90000 D= 3 90000 =44.81404 𝑊= 1 3000 𝐷 3 For SAFETY the largest diameter can be . 44cm

INVERSE PROPORTION A inversely proportional to B → 𝐴= 𝑘 𝐵 →𝐴𝐵=𝑘 A is connected to B by the formula 𝐴= 𝑘 𝐵 thus AB = k k is called the constant of proportion. The product 𝐴×𝐵 is constant 𝐴∝ 1 𝐵 is used to mean ‘A is inversely proportional to B’ Thus 𝐴∝ 1 𝐵 is another way of writing 𝐴= 𝑘 𝐵 A inversely proportional to 𝐵 2 → 𝐴= 𝑘 𝐵 2

CALCULATING WITH INVERSE PROPORTION A∝ 1 𝐵 a) In the following table A is inversely proportional to B. Find the equation connecting A and B. Hence complete the table. ∴𝐴= 𝑘 𝐵 5= 𝑘 10 5×10=𝑘 k=50 A 5 8 B 10 4 A is inversely proportional to B Substitute values of A and B Rearrange Equation connecting A and B is A= 50 𝐵

CALCULATING WITH INVERSE PROPORTION 𝐴∝ 1 𝐵 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 4 12.5 6.25 When B = 4 Use the equation 𝐴= 50 𝐵 to find the missing values When A = 8 𝐴= 50 𝐵 𝐴= 50 𝐵 𝐴= 50 4 =12.5 8= 50 𝐵 8𝐵=50 𝐵= 50 8 =6.25

CALCULATING WITH INVERSE PROPORTION 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= k I 2 4= 𝑘 6 2 4= 𝑘 36 k=4×36=144 R is inversely proportional to the square of I Substitute values of R and I Rearrange Equation connecting R and I is R= 144 I 2

CALCULATING WITH INVERSE PROPORTION ii) Find the resistance when the current is 7 amps (correct to 3 s.f.) ii) Find the current when the resistance is 3 ohms (correct to 3 s.f.) R= 144 I 2 R= 144 I 2 3= 144 I 2 R= 144 7 2 3 I 2 =144 R= 144 49 I 2 = 144 3 I 2 =48 R=2.94𝑜ℎ𝑚𝑠 I= 48 =6.93𝑎𝑚𝑝𝑠

CALCULATING WITH INVERSE PROPORTION c) If 𝑦 varies inversely with the square of 𝑥 and 𝑦=12 when 𝑥=3, find i) the equation connecting 𝑦 and 𝑥 ii) 𝑥 when 𝑦=75 ∴ 𝑦 𝑥 2 =𝑘 𝑦= 𝑘 𝑥 2 12× 3 2 =𝑘 108=𝑘 𝑦 is inversely proportional to the square of 𝑥 𝑦𝑥 2 =108 75𝑥 2 =108 𝑥 2 = 108 75 𝑥 2 =1.44 Equation connecting 𝑦 and 𝑥 is 𝑦𝑥 2 =108 𝑥 2 = 1.44 𝑥 2 =1.2