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In this Math tutorial, you will learn:
You are travelling by car at low speed but you notice that you are late. What do you do? How does the increase of speed affect the distance travelled? What about the time elapsed? How does the increase in speed affect the travelling time?
Suppose you have to complete a job alone in five days but you are not able to do so. What do you do in this case? Can you complete the task within this deadline if friends come and help you? How does the number of people affect the time to complete the task? How does the number of people affects the amount of work done?
In this tutorial we discuss proportion and its types. This is because proportion is a very important concept that shows how two quantities are related to each other. Proportions are very common in daily real life situations.
If two quantities involved in a situation (we call them variables) are related in such a way that if one quantity changes, the other quantity changes at the same (or opposite) degree, we say that these quantities are in proportion with (or proportional to) each other. In simpler words, a proportion indicates to us how one thing changes in relation to another.
For example, if a car travels 240 km with 8L of gasoline, it can travel other for other 720 km with 24 L of gasoline if moving at the same rate (speed). The unit rate u.r. of gasoline consumption therefore is
Since there two unit rates are equal, we say they are in proportion. Hence, it is clear that a proportion includes two equal fractions, i.e. if
then a and c are proportional (as well as b and d).
In other words, if a/b and c/d are two equal ratios or rates, they are in proportion.
We can also write the above proportion in the form
where a is the first term, b is the second term, c is the third term and d is the fourth term of the proportion. The terms a and d are called outer terms or extremes, while b and c are called inner terms or means of the proportion.
Which of the following pairs of fractions are in proportion?
First, we express each fraction in the simplest terms. In a certain sense, a fraction expressed in the simplest terms is similar to the unit rate (in fact, one of the two associated unit rates) we discussed earlier. Thus,
and3/15 = 3 ÷ 3/15 ÷ 3 = 1/5
and12/4 = 3/1 = 3
and9/15 = 9 ÷ 3/15 ÷ 3 = 3/5
There are two main types of proportion: direct and inverse proportion. In direct proportion, the quantities a and c change in the same way as well as b and d.
In the example given earlier, the number of kilometres travelled is proportional to the amount of gasoline consumption. From the numbers, it is clear that if the number of kilometres triples, the gasoline consumption triples too. This type of proportion is known as direct proportion.
By definition, direct proportion is the relation between quantities whose ratio (or rate) is constant.
(We will see later that two quantities that are in inverse proportion behave in the opposite way, i.e. when one increases, the other decreases).
In the example with fuel consumption, this constant was the unit rate of gas consumption expressed in km/L (in both cases this number was 30 km/L). Let's see another example involving quantities that are in direct proportion with each other.
Five workers can carry a 200 kg load from the point A to the point B. How many workers are needed to carry 760 kg load in the same route?
This is a direct proportion example where any increase by a certain factor of the load brings the necessity to increase the number of workers by the same factor. Hence, we can use two approaches to find the missing quantity:
Thus, using the first method, we have:
Load carried by a single worker = Reference load/Reference number of workers
Thus, we have
Since the actual load to be carried is 760 kg, we obtain for the number of workers needed for this work:
The second method is shorter. We write
As you see, the result obtained is the same with either method used.
In ratio tutorial 4.1 we proved that ratios produce a linear graph. Since the direct proportion is obtained by two equal ratios, it is easy to conclude that the graph representing a direct proportion is a straight (sloped) line that starts from the origin (otherwise it is not a direct proportion), where the slope is determined by the simplest form of the ratio or of the unit rate.
Let's consider one such a graph to clarify this point.
The graph below is not an example of direct proportion because the ratios of position at the two given instants are not the same as the ratios of time. In other words, 320/110 is different from 20/10, as the first ratio gives 1.5 and the second gives 2.
Therefore, the quantities involved in the above situation are not directly proportional. In simpler words, the graph must start from the origin in order to have a direct proportion, as shown below.
Here, both ratios are the same as 220/110= 20/10 = 2. Therefore, the quantities involved are directly proportional (or simply, proportional).
Remark! It is better to solve the situations with ratios, not with rates when dealing with proportions. In other words, it is better to divide two like quantities instead of the unlike ones.
Which of the following situations involves a direct proportion?
As we explained earlier, a proportion is written in the form
where a and b are similar quantities as well as c and d. This is the ratio form of expressing a proportion, where all units are simplified.
However, another form of writing a proportion is by expressing it as a rate, i.e. as two fractions where the numerator and denominator of each, represent two different quantities (the units cannot be simplified).
For example, if 16 tailors can make 40 shirts in one day, then obviously if the number of tailors doubles (16 × 2 = 32), the number of shirts they can produce doubles too (40 × 2 = 80). Therefore, if we use the above notation, we have
Hence, the two methods of expressing this proportion is
From the two above methods of expressing a proportion, we can see that a and b can switch their position as well as c and d and still the proportion is valid. From chapter 1, we know that the only two operations that possess the commutative property are addition and multiplication. Since ratios are expressed as divisions, which are multiplications by the inverse, we can write the ratio as two products on either side of the expression. Hence, we can write a direct proportion as:
This method of expressing a proportion as equality of products is known as cross product.
Considering the numbers written in the above example, we can write
The result in either side is 1280, which confirms the proportion.
The cross product method of writing a proportion allows us to check the veracity of proportion in an easier way without involving rational numbers, simplifications or GCF calculations.
Check the veracity of the following proportions by using cross products
First, we write the possible proportion in fractional form. We have:
Now, we can write the proportion as cross product:
Since the last equality is not true, it is clear that the original proportion is not true either.
Again, we write the possible proportion in the fractional form. We have:
Writing the expression as a cross product yields
Therefore, the original proportion is true.
This type of proportion occurs when the quantities involved change inversely, i.e. when one of them increases by a certain factor, the other quantity decreases by the same factor. For example, if 20 people can do a job in 5 days, it takes 50 days to 2 workers to complete the same job. This is because decreasing the number of workers increases by the same factor the time necessary to complete the work.
By definition, inverse (or indirect) proportion occurs when a decrease in one quantity or variable causes an increase by the same factor in another quantity or variable.
If we continue operating with the symbols of direct proportion, we can express an inverse proportion as
A car travelling at 60 km/h reaches the destination in 3 hours. How long does it take to this same car to reach the same destination if it travels at 20 km/h?
From Physics, it is known that distance = speed × time. Since the distance is the same (there is the same destination), this is a typical situation involving two quantities related to each other in an inversely proportional way, because any decrease in speed brings an increase in travelling time by the same factor. Thus, we have for the two situations given in the clues:
We can write the last equation in proportion form using fractions, i.e.
where v1 = 60 km/h, t1 = 3 h, v2 = 20 km/h and t2 is to be calculated. Thus, we have:
Using the cross product method described earlier, we obtain
From the above example, we have demonstrated that two quantities involved in an inverse proportion have a relation of type a × b = constant, which we can write as
The graph of this relation is called hyperbola, it is a curved line that approaches the axes without touching them with the increase in the values of a and b. To increase the accuracy of an inverse proportion graph, we must use as many points as possible. Thus, if we reconsider the above example, we can form the following table:
We can write the time axis in the horizontal direction as we do in Physics, while the speed values are placed in the vertical direction. The graph looks like this:
The example below provides a closer look at the graph with the part corresponding to the small values shown. This is provided to make the values of the table more evident.
The graph of an inverse proportion is shown in the figure below.
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