Lecture 3 Proportionality Mark A. Magumba Lecture 3 Proportionality

Ratios If the ratio of A:B is 1:3 then A = 1/3 of B For instance if you were to divide 20 oranges between A and B in the ratio 1:3 it means you have a total of 1 + 3 or 4 equal parts and each part is equal to 20/4 or 5 oranges, A gets 1 * 5 oranges and B gets 3 * 5 oranges If the ratio of A:B:C is 1:3:4 then A = 1/3 of B and B:C is 3:4, A:C is 1:4 Ratios like fractions may be simplified by dividing both sides by the greatest common factor for instance the ratio 4:2 is the same as 2:1

Ratios contd To get the relationship between individual parts you have to convert to the fractional representation For instance if A:B:C is 2:3:4, it means you have 2+3+4 = 9 equal parts divided as per the ratio meaning expressed as fractions this would be 2/9:3/9:4/9 Therefore to express A (2/9) in terms of B (3/9) is simply solving for x in the equation A = xB This gives x = A/B = 2/9 ÷ 3/9 = 2/3 Therefore A = 2/3B and B = 3/2B

Proportionality B is said to be proportional to A if an increase in A leads to an increase in B by a uniform factor If in addition the ratio of A to B is constant then B is said to be directly proportional to A For direct proportionality the following equation is true for all values of A and B A = kB where k is a constant also known as the proportionality constant and k = A/B Mathematically this is denoted as A α B The ratio of A to B is 1:K If values of A and B are plotted against a Cartesian graph you get a straight line through the origin (0,0) whose gradient is k

Graph of y against x for yαx

Direct proportionality If y is directly proportional to x with a proportionality constant K then x is also proportional to y but with a proportionality constant 1/k Examples If you travel at a constant speed, the distance travelled is directly proportional to the time taken According to P Diddy Mo money, mo problems (Joke!!)

Inverse/ indirect proportionality With inverse proportionality an increase in one variable results in a reduction of the other by a uniform factor and vice versa and specifically the product of the two variables remains constant In other words if A and B are inversely proportional then A * B = K, where k is a constant for all values of A and B Mathematically this is written as A α 1/B For instance the time taken to dig a hole is inversely proportional to the number of people digging, the more people the shorter and the less people the longer

Exponential proportionality A is said to be exponentially proportional to B when it is equal to an exponential function of B The relationship between A and B can be expressed as A α KxB where k and B are non zero constants

Direct proportionality (red line) Vs exponential proportionality (green line)