In the previous section we completed the discussion on Histograms. In this section we will discuss about Pie charts. Pie charts show the relation between a 'part' to the 'whole'. Let us see an example:
Consider the construction of a boundary wall of a plot. The total length of the wall to be constructed is 35 m. On the first day, 14 m was completed. The owner of the plot wants to know the progress of the work at the end of the day. For that, we can make a special type of graph.
Consider the bar AB shown in fig.1.31 below:
It is drawn on a piece of paper. It's length from A to B is 7 cm. It's width is about 0.5 cm. [But since it is a 'bar', it's width is not important. The only point to note is that, the width should be small, so that it will appear as a 'bar' having only one dimension, which is the 'length'. If the width is more, it will appear as a 'rectangle' having two dimensions, the 'length' and 'width'.] This 7 cm long bar AB represents the total boundary wall.
• 14 m out of 35 m is completed. So 14/35 of the whole is completed.
• 14/35 =2/5. So two fifth of the work is completed.
• Two fifth of our bar is 7 x 2/5 = 14/5 =2.8 cm.
We will now shade 2.8 cm of our bar from the left end A as shown below:
AC is 2.8 cm in length. This represents the completed work. For the final presentation, the name of the bars and their lengths are not required. 2/5 is 40%. So this has to be indicated in the final presentation as shown below:
From the above fig.1.33, the owner can get a quick idea about the progress of the work. This 40% may not be apparant from a quick view at the actual site of the work. This is especially so if the plot is irregular. From the presentation, it is clear that 60% of the work is remaining, and so more work has to be done on the second day, if the time allowed for the entire work is 2 days.
Now let us see another method by which the above information can be presented. This method is called the Circle graph or the Pie chart. In this method, a circle is used instead of a bar. For a 'bar', length is the parameter. So we marked appropriate length for completed work. The remaining length indicate the incomplete work.
But for a circle, 'area' is the parameter. So the completed work has to be given an appropriate area within the circle. The incomplete work will occupy the remaining area. Let us first see how the total area of a circle is divided into parts:
In the fig.1.34 above, the circle is divided into 4 parts. Each is given a distinct colour. Green, blue, yellow and red.
• Each area is defined by 3 separate entities: Two 'straight lines' and one 'curve'.
• The straight lines radiate out from the centre of the circle. They are called 'radial lines'
• The curve is a part of the circle itself.
With these three entities, we can completely define an area inside a circle. In the fig.1.34, there are 4 radial lines, which give us 4 areas. Each of such areas is called a sector of the circle. More radial lines can be drawn to obtain more sectors.
For our problem, we want to mark an appropriate sector in the circle. It should indicate the completed work. When such a sector is marked off, the 'remaining sector' will indicate the incomplete work. We know that two fifth of the work is completed. So the 'completed work' has the claim to occupy two fifth of the area of the circle. Thus, we need a method to mark off two fifth of the area of the circle.
We have seen that two radial lines and a portion of the circle defines the sector.
• If the angle between these radial lines increases, the area of the sector increases.
• If the angle decreases, the area also decreases.
• This angle between two radial lines is measured by keeping the protractor at the centre of the circle. • So it is called the central angle of the sector.
We need that 'central angle' which gives two fifth of the area of the whole circle. We know that the central angle of a whole circle is 360o. This is shown in the fig.1.35(a) below
• This 360o gives the whole area of the circle.
• So 2/5 of 360 will give 2/5 of the whole area.
• Thus the required central angle is 2/5 x 360 = 144o.
• Keep the protractor at O. Mark an angle OAB =144o. This is shown in the fig.1.35(b).
• The radial lines OA and OB, and the curved portion AB define a sector, whose area is equal to 2/5 of the whole area of the circle.
The final presentation do not need to show the names of the radial lines. It is shown in fig.1.35(c).
Next we will see the application of Pie charts in another situation: The market shares of different brands of cars were obtained in a survey. It is as follows: Brand A has a market share of 25%, [This means that if, a total of 1000 cars were sold in a year, 250 of them will be Brand A. It is a theoretical value. The actual sale may be slightly different from 250. In our present discussion, we are concerned about theoretical values only] Brand B has a market share of 45%, Brand C has 10 %, Others have 20%. We will show these shares in a pie chart. The calculations are as follows:
• A has 25%. So it's sector has a claim to occupy 25% of the area of the whole circle.
• 25% =25/100 =0.25. So the sector for A will occupy one fourth of the whole area.
• 0.25 of 360 = 0.25 x 360 = 90o
• B has 45%. So it's sector has a claim to occupy 45% of the area of the whole circle.
• 45% =45/100 = 0.45 . So the sector for B will occupy 0.45 of the whole area.
• 0.45 of 360 = 0.45 x 360 = 162o
In the same way, angle for C = 0.1 x 360 =36o
angle for others = 0.20 x 360 = 72o
We can check that all the angles add up to give 360.: 90 +162 +36 +72 = 360o. The following fig.1.35(a) shows the method of marking off the sectors. Fig.(b) shows the final presentation.
From the presentation in Fig.1.35(b), the customers are able to get a comparison between the various brands.
Now we will see some solved examples:
Solved example 1.4
A survey was made to find the type of music that a certain group of young people liked in a city. The pie chart in fig.1.36 below shows the findings of this survey.
From this pie chart answer the following:
(i) If 20 people liked classical music, how many young people were surveyed?
(ii) Which type of music is liked by the maximum number of people?
(iii) If a cassette company were to make 1000 CD’s, how many of each type would they make?
Solution:
This problem tests our knowledge about the basics of pie charts and percentages. We can see that the percentages of each categories add up to 100:
40 +20 +10 +30 =100%
(i) It is given that 20 people liked classical music. So, if the total no. of people surveyed is x, then 10 % of x is 20. That is., x x 10/100 = 20. Which is same as 0.1x =20. From this we get x = 200. So a total of 200 people were surveyed
(ii) The sector coloured in red occupies the maximum area. It has the highest percentage of 40. So, the type of music liked by the maximum no. of people is Light music.
(iii) According to the survey, each person has his/her own liking for a particular type of music. No person will buy more than one type of music. So the 1000 CD's will be bought by 1000 people.
• 40% of a group of 1000 people will be asking for light music. This is equal to [1000 x (40/100)] =1000 x 0.4 =400. So the company has to make 400 light music CD's.
• Similarly, number of semi classical CD's = 1000 x 0.2 =200
• Number of classical CD's = 1000 x 0.1 =100
• Number of folk CD's = 1000 x 0.3 =300
Check: The total number adds up to 1000: 400 +200 +100 +300 =1000
Solved example 1.5
The monthly expenditure for a family is shown in the pie chart in fig.1.37 below.
If the monthly expenditure for food is 12000, find the expenditure for each item.
Solution:
Like the previous problem, this problem also tests our knowledge about the basics of pie charts and percentages. We can see that the percentage for 'others' is not given. But we can easily calculate it by using the fact that all percentages must add up to 100. So, if others have x%,
10 +22 + x +15 +40 =100 ⇒ 87 + x =100
∴ x = 13%
It is given that expenditure for food is 12000/-. So, if y is the total income, 40% of y =12000
⇒ y x (40/100) = 12000 ⇒ 0.4y =12000
∴ Total income = y = 30000/-
Now we can calculate each item:
Medicine: 30000 x 0.1 = 3000/-
Education: 30000 x 0.22 = 6600/-
Others: 30000 x 0.13 = 3900/-
House rent: 30000 x 0.15 = 4500/-
Food: 12000/-
Check: All the individual expenses should add up to 30000/-: 3000 +6600 +3900 +4500 +12000 = 30000/-
In the next section, we will learn about Probability.
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Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved
Consider the construction of a boundary wall of a plot. The total length of the wall to be constructed is 35 m. On the first day, 14 m was completed. The owner of the plot wants to know the progress of the work at the end of the day. For that, we can make a special type of graph.
Consider the bar AB shown in fig.1.31 below:
 |
| Fig.1.31 |
• 14 m out of 35 m is completed. So 14/35 of the whole is completed.
• 14/35 =2/5. So two fifth of the work is completed.
• Two fifth of our bar is 7 x 2/5 = 14/5 =2.8 cm.
We will now shade 2.8 cm of our bar from the left end A as shown below:
 |
| Fig.1.32 |
 |
| Fig.1.33 |
Now let us see another method by which the above information can be presented. This method is called the Circle graph or the Pie chart. In this method, a circle is used instead of a bar. For a 'bar', length is the parameter. So we marked appropriate length for completed work. The remaining length indicate the incomplete work.
But for a circle, 'area' is the parameter. So the completed work has to be given an appropriate area within the circle. The incomplete work will occupy the remaining area. Let us first see how the total area of a circle is divided into parts:
 |
| Fig.1.34 Sectors of a circle |
• Each area is defined by 3 separate entities: Two 'straight lines' and one 'curve'.
• The straight lines radiate out from the centre of the circle. They are called 'radial lines'
• The curve is a part of the circle itself.
With these three entities, we can completely define an area inside a circle. In the fig.1.34, there are 4 radial lines, which give us 4 areas. Each of such areas is called a sector of the circle. More radial lines can be drawn to obtain more sectors.
For our problem, we want to mark an appropriate sector in the circle. It should indicate the completed work. When such a sector is marked off, the 'remaining sector' will indicate the incomplete work. We know that two fifth of the work is completed. So the 'completed work' has the claim to occupy two fifth of the area of the circle. Thus, we need a method to mark off two fifth of the area of the circle.
We have seen that two radial lines and a portion of the circle defines the sector.
• If the angle between these radial lines increases, the area of the sector increases.
• If the angle decreases, the area also decreases.
• This angle between two radial lines is measured by keeping the protractor at the centre of the circle. • So it is called the central angle of the sector.
We need that 'central angle' which gives two fifth of the area of the whole circle. We know that the central angle of a whole circle is 360o. This is shown in the fig.1.35(a) below
• This 360o gives the whole area of the circle.
• So 2/5 of 360 will give 2/5 of the whole area.
• Thus the required central angle is 2/5 x 360 = 144o.
• Keep the protractor at O. Mark an angle OAB =144o. This is shown in the fig.1.35(b).
• The radial lines OA and OB, and the curved portion AB define a sector, whose area is equal to 2/5 of the whole area of the circle.
 |
| Fig.1.34 Method for drawing a Pie chart |
The final presentation do not need to show the names of the radial lines. It is shown in fig.1.35(c).
Next we will see the application of Pie charts in another situation: The market shares of different brands of cars were obtained in a survey. It is as follows: Brand A has a market share of 25%, [This means that if, a total of 1000 cars were sold in a year, 250 of them will be Brand A. It is a theoretical value. The actual sale may be slightly different from 250. In our present discussion, we are concerned about theoretical values only] Brand B has a market share of 45%, Brand C has 10 %, Others have 20%. We will show these shares in a pie chart. The calculations are as follows:
• A has 25%. So it's sector has a claim to occupy 25% of the area of the whole circle.
• 25% =25/100 =0.25. So the sector for A will occupy one fourth of the whole area.
• 0.25 of 360 = 0.25 x 360 = 90o
• B has 45%. So it's sector has a claim to occupy 45% of the area of the whole circle.
• 45% =45/100 = 0.45 . So the sector for B will occupy 0.45 of the whole area.
• 0.45 of 360 = 0.45 x 360 = 162o
In the same way, angle for C = 0.1 x 360 =36o
angle for others = 0.20 x 360 = 72o
We can check that all the angles add up to give 360.: 90 +162 +36 +72 = 360o. The following fig.1.35(a) shows the method of marking off the sectors. Fig.(b) shows the final presentation.
 |
| Fig.1.35 |
Now we will see some solved examples:
Solved example 1.4
A survey was made to find the type of music that a certain group of young people liked in a city. The pie chart in fig.1.36 below shows the findings of this survey.
 |
| Fig.1.36 |
From this pie chart answer the following:
(i) If 20 people liked classical music, how many young people were surveyed?
(ii) Which type of music is liked by the maximum number of people?
(iii) If a cassette company were to make 1000 CD’s, how many of each type would they make?
Solution:
This problem tests our knowledge about the basics of pie charts and percentages. We can see that the percentages of each categories add up to 100:
40 +20 +10 +30 =100%
(i) It is given that 20 people liked classical music. So, if the total no. of people surveyed is x, then 10 % of x is 20. That is., x x 10/100 = 20. Which is same as 0.1x =20. From this we get x = 200. So a total of 200 people were surveyed
(ii) The sector coloured in red occupies the maximum area. It has the highest percentage of 40. So, the type of music liked by the maximum no. of people is Light music.
(iii) According to the survey, each person has his/her own liking for a particular type of music. No person will buy more than one type of music. So the 1000 CD's will be bought by 1000 people.
• 40% of a group of 1000 people will be asking for light music. This is equal to [1000 x (40/100)] =1000 x 0.4 =400. So the company has to make 400 light music CD's.
• Similarly, number of semi classical CD's = 1000 x 0.2 =200
• Number of classical CD's = 1000 x 0.1 =100
• Number of folk CD's = 1000 x 0.3 =300
Check: The total number adds up to 1000: 400 +200 +100 +300 =1000
Solved example 1.5
The monthly expenditure for a family is shown in the pie chart in fig.1.37 below.
 |
| Fig.1.37 |
Solution:
Like the previous problem, this problem also tests our knowledge about the basics of pie charts and percentages. We can see that the percentage for 'others' is not given. But we can easily calculate it by using the fact that all percentages must add up to 100. So, if others have x%,
10 +22 + x +15 +40 =100 ⇒ 87 + x =100
∴ x = 13%
It is given that expenditure for food is 12000/-. So, if y is the total income, 40% of y =12000
⇒ y x (40/100) = 12000 ⇒ 0.4y =12000
∴ Total income = y = 30000/-
Now we can calculate each item:
Medicine: 30000 x 0.1 = 3000/-
Education: 30000 x 0.22 = 6600/-
Others: 30000 x 0.13 = 3900/-
House rent: 30000 x 0.15 = 4500/-
Food: 12000/-
Check: All the individual expenses should add up to 30000/-: 3000 +6600 +3900 +4500 +12000 = 30000/-
In the next section, we will learn about Probability.
PREVIOUS CONTENTS NEXT
Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved
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