Direct proportion is when two quantities increase and decrease at the same rate and/or that the ratio between the two quantities is always the same. An example of this is that the speed of a train is directly proportional to the distance it covers.
Therefore if the train travels at double the speed, it will cover double the distance and vice versa.
Two questions to ask about direct proportionality is:
If one of the variables doubles, does the other one double?
If one of the variable is zero, is the other one zero?
Direct proportion problems
A direct proportion problem is as follows: If 3 bars of chocolate weigh 84 g. How many do 15 bars weigh?
If all the bars weight the same, then the ratio between the number of bars and weight is constant. Therefore if the number of bars is multiplied by 6, then the weight is multiplied by 6 also.
The following example below is called the unitary method this is where the value of one unit must be worked out first.
If 5 bars of chocolate weigh 96 g, how much does 1 packet weigh?
- The number of bars is divided by the total weight to find the weight of a single bar
- Therefore 96 g ÷ 5 = 19.2 g
- Once the weight of a bar is found, if it possible to work out any number of bars.
The unitary method can also be calculated using a single method though a number of ways:
3 bars of chocolate weighs 84 g and 7 bars weigh 196 g. How do you work out the weight of a single bar?
- Divide 7 by 3 or 196 by 84 to get
- Scale 3 to 84 by multiplying by 28 (3 bars x 28 = 84 g or 7 bars x 28 = 196 g)
- 28 is also known as the constant multiplier
Note that 7/3 is the reciprocal of 3/7.
Equations and direct proportion
If two quantities y and x are directly proportional to each other then they can be linked with the symbol .
This is expressed as:
These variables can also be linked with the equation:
y = kx
k = constant of proportionality (constant multiplier as shown above, or represents the ratio between the two quantities in direct proportion)
y & x = variables (note that studying the way two variables changes in relation to each other is called variation)
If the equation was rearranged, it will be expressed as: k =
If a and b are in direct proportion, and a = 6 and b = 15 what is k?
a and b are directly proportional therefore, a = kb
When a = 6, b = 15, so 6 = 15k
k = 6 ÷ 15 = 0.4
This can also be written as: a = 0.4b or b = 2.5a