Proportion

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4.3Proportion

In this Math tutorial, you will learn:

  • What is proportion?
  • What is the difference between proportion and ratio?
  • What is the difference between proportion and rate?
  • What are the names of the outer and inner terms of a proportion?
  • How many types of proportion are there? What is the difference between them?
  • How do you express a direct proportion graphically?
  • How do you express an inverse (indirect) proportion graphically?
  • What is the graph shape for direct/indirect proportion? Why?

Introduction

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.

Definition of Proportion

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

u.r. = 240 km/8 L = 30 km/L

Also

u.r. = 720 km/24 L = 30 km/L

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

a/b = c/d

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

a:b = c:d

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.

Example 1

Which of the following pairs of fractions are in proportion?

  1. 4/18 and 3/15
  2. 21/7 and 12/4
  3. 24/36 and 9/15

Solution 1

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,

  1. 4/18 = 4 ÷ 2/18 ÷ 2 = 2/9

    and

    3/15 = 3 ÷ 3/15 ÷ 3 = 1/5
    Since these two fractions are not equal, the quantities they represent are not in proportion.
  2. 21/7 = 3/1 = 3

    and

    12/4 = 3/1 = 3
    These fractions are equal, so the quantities they represent are in proportion.
  3. 24/36 = 24 ÷ 12/36 ÷ 12 = 2/3

    and

    9/15 = 9 ÷ 3/15 ÷ 3 = 3/5
    Since these two fractions are not equal, the quantities they represent are not in proportion.

Direct Proportion

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.

Example 2

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?

Solution 2

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:

  1. To find the unit rate first (which in this specific example corresponds to the amount of load a single worker can carry) and then divide the total load to this number in order to find the number of workers needed, and
  2. To write the proportion in the form a/b = c/d where d is 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

L1 = Ltot/N1
= 200 kg/5 workers
= 40 kg/worker

Since the actual load to be carried is 760 kg, we obtain for the number of workers needed for this work:

Nactual = Lactual/L1
= 760 kg/40 kg/worker
= 19 workers

The second method is shorter. We write

Reference load/Reference number of workers = Actual load/Actual number of workers

Thus,

200 kg/5 workers = 760 kg/Actual number of workers
Actual number of workers = 760 kg × 5 workers/200 kg
= 19 workers

As you see, the result obtained is the same with either method used.

Graph of Direct Proportion

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. Math Tutorials: Proportion Example

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.

Math Tutorials: Proportion Example

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.

Example 3

Which of the following situations involves a direct proportion?

  1. A car can travel 200 km with 12 L of fuel and other 300 km with 20 L of fuel.
  2. 5 kg of paint can paint 30 m2 wall while 12 kg paint can paint 72 m2 wall.

Solution 3

  1. We will check whether the ratios are equal by dividing the like quantities. We have
    200 km distance/300 km distance and 12 L fuel/20 L fuel
    The first fraction gives 2/3 while the second gives 3/5. Therefore, this situation does not involve a direct proportion.
  2. Again, we take the ratios. We have
    5 kg paint/12 kg paint and 30 m2 wall/72 m2 wall
    The first ratio is already in the simplest form, and the second ratio gives 5/12 too when written in the simplest terms. Therefore, the quantities involved are proportional.

Cross Product in Direct Proportion

As we explained earlier, a proportion is written in the form

a:b = c:d

or

a/b = c/d

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

a = 40 shirts
b = 80 shirts
c = 16 workers
d = 32 workers

Hence, the two methods of expressing this proportion is

a/b = c/d

i.e.

40 shirts/80 shirts = 16 workers/32 workers

or

a/c = b/d

i.e.

40 shirts/16 workers = 80 shirts/32 workers

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:

a × d = b × c

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

40 shirts × 32 workers = 80 shirts × 16 workers

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.

Example 4

Check the veracity of the following proportions by using cross products

  1. 3:12 = 7:21
  2. 20:15 = 36:27

Solution 4

  1. First, we write the possible proportion in fractional form. We have:

    3/12 = 7/21

    Now, we can write the proportion as cross product:

    3 × 21 = 12 × 7
    63 = 84

    Since the last equality is not true, it is clear that the original proportion is not true either.

  2. Again, we write the possible proportion in the fractional form. We have:

    20/15 = 36/27

    Writing the expression as a cross product yields

    20 × 27 = 15 × 36
    540 = 540

    Therefore, the original proportion is true.

Inverse (Indirect) Proportion

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:b = d:c

or

a/b = d/c

Example 5

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?

Solution 5

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:

Reference speed × Reference time = Actual speed × Actual time

or

v1 × t1 = v2 × t2

We can write the last equation in proportion form using fractions, i.e.

v1/v2 = t2/t1

where v1 = 60 km/h, t1 = 3 h, v2 = 20 km/h and t2 is to be calculated. Thus, we have:

60 km/h/20 km/h = t2/3 h

Using the cross product method described earlier, we obtain

60 × 3 = 20 × t2
t2 = 60 × 3/20
= 9 h

Graph of Inverse Proportion

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

b = constant/a

or

b = C/a

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:

Math Tutorials: Proportion Example

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:

Math Tutorials: Proportion Example

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.

Math Tutorials: Proportion Example

Example 6

The graph of an inverse proportion is shown in the figure below.

Math Tutorials: Proportion Example
  1. What kind of proportionality is this?
  2. Calculate the constant C of this proportionality.
  3. How many people are needed to dig the same hole in half an hour?
  4. How long does it take to a single worker to dig the hole?

Solution 6

  1. The graph shows an inverse proportion as the graph's shape is a hyperbola.
  2. Since we have an inverse proportionality, the relation involved has the general form
    b = C/a
    where b represents the number of people needed to dig the hole, a represent the number of hours needed for this work and C is the constant of proportionality.
    From the graph, we see that 12 people can dig the hole in 2 hours (b = 12, a = 2). Other values that are clearly spotted in the graph include: 8 people in 3 hours (b = 8, a = 3), 6 people in 4 hours (b = 6, a = 4), etc. if we substitute one of the sets of values (for example the first from the above)in the general form of the proportion, we obtain
    12 = C/2
    Hence, the constant of proportionality C is
    C = 12 × 2 = 24
    Therefore, the inverse proportionality shown in the graph is
    b = 24/a
  3. Knowing the formula of this inverse proportionality helps us find other values not indicated in the graph. Thus the number of people a needed to dig the hole in half an hour (a = 0.5) is
    b = 24/0.5
    = 48 people
  4. If there is a single worker available (b = 1), it takes
    1 = 24/a
    a = 24/1
    = 24 hours
    to dig the hole.

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  5. Continuing learning ratio and proportion - read our next math tutorial: Properties of Proportion. Geometric Mean

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