Proportion is simply saying we have a relationship between two things. For example, if one variable increases and this causes an equal increase in another variable, this means that these two variables are in proportion. If two things are in proportion, there exists a proportionality constant between the two variables. We will see more on this later.
What symbol do we use for proportion?
To represent that two variables are proportional to one another, we use the symbol \(\propto\). For example, Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Letting voltage be represented by V, and current by I, then we can write \(V \propto I\).
Whenever we see a proportionality symbol, we can replace this with an equals sign and a proportionality constant. This means we could write Ohm's law as \(V = kI\), where k is our constant.
What are direct proportions?
If two variables are in direct proportion, as one variable increases, then so does the other. Conversely, it means as one variable decreases, then so does the other. Any direct relationship, for variables A and B, can be written as \(A = kB\). This means on a graph, this relationship will be represented as a straight line, passing through the origin. This is shown below.
Graph showing a direct relationship of the form \(A = kB\)
The weight of a piece of string is directly proportional to its length. When the piece of string is 30cm long, it weighs 0.2N. Find the weight of the piece of string when the string is 50 cm long.
We know that this is directly proportional, so we know the relationship W\(\propto\) L, when W represents weight, and L represents length. Let a be our constant of proportionality, so that \(W = aL\). From the first part of the question, we know \(0.2 = a \cdot 30\), so \(a = \frac{1}{150}\). We can now use this to find the weight when the string is 50cm long. The same relation holds, so \(W = \frac{1}{150} L\). Substituting in our length of 50 cm, we get \(W = \frac{50}{150} = \frac{1}{3}\), so to two decimal places, W = 0.33N.
What are inverse proportions?
Inverse proportion occurs when one variable increasing causes another variable to decrease. If this relationship were to occur between the variables c and d, we would write \(c \propto \frac{1}{d}\). An example of an inverse proportion would be as speed increases, then the time to travel a distance will decrease. Graphically, this means that the shape of the relationship will be represented by \(y = \frac{x}{k}\), with k constant, and x, y variables. This means that the graph will never touch the axis, but it will get very very close as we put a very big x value in, or an x value extremely close to 0. This is shown below.
Graph showing an inversely proportional relationship
Two variables, b and n are inversely proportional to one another. When b = 6, n = 2. Find the value of n when b is 15.
We know \(b \propto \frac{1}{n}\), so \(b = \frac{k}{n}\), when k is our constant of proportionality. Filling in the values of b and n, we get \(6 = \frac{k}{2}\), so k = 12. This means for all values of b and n, \(b = \frac{12}{n}\). We need to find n when b = 15, so we can fill this in, to get \(15 =\frac{12} {n}\). Rearranging this for n, we get \(n = {12}{15} = 0.8\).
Proportions and shapes
If two shapes are in proportion, this means that both shapes are the same, with the exception that one of these shapes will have been scaled either up or down. For two shapes to be similar, it is necessary for all the angles in the shape to be the same, and all the sides to be in proportion. Again, here we will have a proportionality constant, which relates the two shapes. In one dimension, this is called a length scale factor, in two dimensions we will call this an area scale factor, and in three dimensions this is called a volume scale factor. We are able to translate between length scale factor and volume or area scale factor. To get the area scale factor, we must multiply the length scale factor in two dimensions, so\((\text{length scale factor})^2 = \text{area scale factor}\) To get the volume scale factor, we must multiply the length scale factor in three dimensions, so
((\text{length scale factor})^3 = \text{volume scale factor}\)
Two cubes are mathematically similar. The first cube has a face area of 16m². The sides on the second cube are half the length of the sides on the first cube. Find the volume of the second cube. The length scale factor between the shapes is \(\frac{1}{2}\), which implies that the volume scale factor is \(\big( \frac{1}{2} \big)^3 = \frac{1}{8}\). If the first cube has a face area of 16m², this means that it must have side lengths of 4m, which implies it has a volume of 64m³. As the volume scale factor is \(\frac{1}{8}\), this gives the volume of the second cube as \(\frac{64}{8} = 8m^3\).
The triangles ABE and ACD are similar. Find the length of CD. (All lengths are in cm)
The constant of proportionality, k, between AB and AC is given by (AC) = k (AB), which gives 12 = k 8, so k = 1.5. This means that because the triangles are similar, CD = k (BE), so \(CD = 1.5 \cdot 10 = 15 cm\)
Proportion - Key takeaways
The symbol for proportion is ∝
If two things are in proportion, this means that there is a relationship between them.
Direct proportions are of the form \(y \propto x\)
Inverse proportion is of the form \(y \propto \frac{1}{x}\)
If two variables/shapes are in proportion, a proportionality constant exists.
(length scale factor) ² = area scale factor.
- (length scale factor) ³ = volume scale factor .
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