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Welcome to our Math lesson on Properties of Proportion, this is the first lesson of our suite of math lessons covering the topic of Properties of Proportion. Geometric Mean, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
First, it is worth highlighting that in all properties of proportion explained in the following part, we will assume the main proportion in the form
or
The following properties are true for direct proportion:
In symbols,
For example, if
then
In the first case, both ratios have the value 3/5 when written in the simplest form, while in the second case the values of both ratios are 3/2 when written in the simplest form.
The advantage of this property is evident when the terms in each ratio of the original proportion are difficult to deal with. Thus, when we swap the means we obtain fractions that are easier to handle, so the operations become shorter.
Prove that the proportion 33:54 = 44:72 is true.
First, we write the proportion in fraction form. We have
We cannot simplify the fractions expressing the individual ratios in the actual form. Thus, we can simplify by 3 the fraction on the left and by 4 that on the right. Therefore, we swap the means to obtain fractions that can be simplified. Hence, we obtain
Now, the task is easier, as the fraction on the left can be simplified by 11 while that on the right by 18 to obtain
Since the two final fractions representing the possible proportion are equal, the original proportion is true.
In symbols we write
The purpose of using this property is again to make the operations simpler. Let's consider an example:
Check whether the fractions 7/24 and 21/72 are in proportion or not.
One method could be simplifying both fractions by 3 and 7 respectively to obtain 3/8 in both cases. However, not everybody is able to figure out what is the GCF in either fraction (moreover, in the specific case, the terms of the first fraction are relatively prime, i.e. they cannot be simplified). Therefore, we use the second property of proportion to write
Both fractions give 3 when simplified (they express whole number, not improper fractions), so the original proportion is true.
In symbols, we write:
This property is particularly useful when the inverted ratios give whole numbers. An example of the use of this method (inverting ratios) in practice is when dealing with parallel combination of resistors, when we have to invert the fractions in the final step to obtain the result, as in the example below.
Two resistors, R1 = 12 Ω and R2 = 18 Ω are connected in parallel in an electrical circuit. Calculate the equivalent resistance of the circuit. Recall that the equivalent resistance Req of a parallel combination is given by the formula
Using the above formula, we write
Using the method of adding two fractions with different denominators, we obtain
Now, it is the time to explore the advantage the third property that ratios offer. Inverting down the two ratios (fractions) we obtain
In symbols, we write
Let's try a numerical example to confirm this property.
Prove that the proportion 12:30 = 24:60 is true.
From the result obtained in (a) prove that 42:30 = 84:60
First, we write the two ratios forming the proportion in fraction form. We have
We use the first property of ratios (swapping the means) to make the operations easier. Thus,
Simplifying the first ratio by 12 and the second one by 30, yields
Therefore, the original proportion is true.
Adding the denominators to the corresponding numerator of each ratio in the original proportion as the fourth property of proportion suggests, yields
Applying the second property of ratios (swapping the extremes), yields
Thus,
The last expression is a proof that the new ratio given in (b) is also true.
Another example with difference. The condition in this case is that the fraction must be improper; otherwise the difference becomes negative. To illustrate this property, let's take a known ratio, for example 4:3 = 8:6. Hence, we obtain
The last proportion is also true as when 2/6 is simplified by 2 gives 1/3.
Now, let's see an application of the above property in practice.
Two brothers are 8 and 11 years old respectively. After how many years the ratio of their ages will be 5:6?
Let's denote the number of years to come until the condition above will be fulfilled by x. Thus, we can write
Applying the cross product to remove fractions, we obtain
Using the distributive property of addition, we obtain
In symbols,
The condition for this property to be true is that both m and n must not be zero.
In the previous examples, we dealt with this property silently but using division instead of multiplication because we wanted to make simplifications. However, since division is a multiplication by the inverse, it is obvious that this new property is also true. Let's consider an example to clarify this point.
Prove that the proportion 32:54 = 80:135 is true
Prove that 96:162 = 160:270 is also true
We use the above property to confirm the ratio. We have
The first fraction is simplified by 2 while the second by 5. Hence, we obtain
It is easy to observe that the proportion
is obtained by multiplying up and down the first ratio by 3 and the second ratio by 2, i.e.
If we have the following proportion
then, we can express it as
The sides a, b and c of a triangle are in a ratio a : b : c = 5 : 9: 13 to each other and the perimeter (the sum of all side lengths) of the triangle is 108 cm. Calculate the length of each side.
Applying the 6th property of proportions, we can write
Thus,
Moreover, we have
Thus,
Therefore, the length of the three sides of the triangle are
Remark! We have silently used the 6th property of proportions too when explaining the definition of proportion in the previous tutorial.
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