1. Lecture 3 Proportionality
Mark A. Magumba
Lecture 3
Proportionality

2. 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 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

3. 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 = 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

4. 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

5. Graph of y against x for yαx

6. 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!!)

7. 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

8. 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

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