37.
PIE-CHARTS
IMPORTANT
FACTS AND FORMULAE
The pie-chart or a pie-graph
is a method of representing a given numerical data in the form of sectors of a
circle.
The sectors of the circle are
constructed in such a way that the area of each sector is proportional to the
corresponding value of the component of the data.
From geometry, we know that the
area of a circle is proportional to the central angle.
So, the central angle of each
sector must be proportional to the corresponding value of the component.
Since the sum of all the central
angle is 360°, we have
Value of the component °
Central angle of the component
= ------------------------ *
360
Total value
SOLVED EXAMPLES
The procedure of solving problems
based on pie-charts will be clear from the following solved examples.
Example 1. The
following pie-chart shows the sources of funds to be collected by the National
Highways Authority of India (NHAI) for its Phase II projects. Study the
pie-chart and answer the questions that follow.
SOURCES OF FUNDS TO BE
ARRANGED BY NHAI
FOR PHASE II PROJECTS (IN
CRORES RS.)
Total funds to be arranged for
Projects (Phase II) =Rs.57,600 crores.
1. Near about 20% of the funds
are to be arranged through:
(a) SPVS (b) External Assistance
(c) Annuity (d) Market Borrowing
2. The central angle
corresponding to Market Borrowing is:
(a) 52° (b) 137.8°
(c) 187.2° (d) 192.4°
3. The approximate ratio of the
funds to be arranged through Toll and that through Market Borrowing is:
(a) 2:9 (b) 1:6
(c) 3:11 (d) 2:5
4. If NHAI could receive a total of Rs. 9695 crores
as External Assistance, by what percent (approximately) should it increase the
Market Borrowings to arrange for the shortage of funds ?
(a) 4.5% (b) 7.5%
(c) 6% (d) 8%
5.If the toll is to be collected
through an outsourced agency by allowing a maximum 10% commission, how much
amount should be permitted to be collected by the outsourced agency, so that
the project is supported with Rs. 4910 crores ?
(a) Rs.6213 crores (b) Rs. 5827 crores
(c) Rs. 5401 crores (d) Rs. 5216 crores
SOLUTION
1. (b): 20% of the total funds to
be arranged = Rs.(20% of 57600) crores
= Rs.11520 crores Rs.11486 crores.
2. (c):Central angle
corresponding to Market Borrowing =
29952
-------- * 360°
57600
=
187.2°
3. (b):
4910 1 1
Required ratio = ------- = ---- = ----
29952 6.1
6
4. (c):Shortage of funds arranged
through External Assistance
=Rs.(11486-9695)
crores =Rs. 1791 crores.
therefore, Increase required in Market
Borrowings =Rs. 1791 crores.
1791
Percentage increase required = -------- * 100 % =
5.98 % = 6%
29952
5. (c):Amount permitted = (Funds required from Toll for projects of
Phase II ) +
(10
% of these funds)
=Rs.
4910 crores + Rs. (10% of 4910) crores
=Rs.
(4910 + 491) crores = Rs. 5401 crores.
Example 2. The
pie-chart provided below gives the distribution of land (in a village) under
various food crops. Study the pie-chart carefully and answer the questions that
follow.
DISTRIBUTION OF AREAS (IN ACRES) UNDER
VARIOUS FOOD CROPS
1.Which combination of three
crops contribute to 50% of the total area under the food crops ?
(a) Wheat, Barley and Jowar
(b)Rice, Wheat and Jowar
(c) Rice, Wheat and Barley (d)Bajra, Maize and
Rice
2.If the total area under jowar
was 1.5 million acres, then what was the area (in million acres) under rice?
(a)6 (b)7.5 (c)9 (d)4.5
3.If the production of wheat is 6
times that of barley, then what is the ratio between the yield per acre of
wheat and barley ?
(a) 3:2 (b) 3:1 (c) 12:1 (d) 2:3
4.If the yield per acre of rice was 50% more than
that of barley, then the production of barley is what percent of that of rice
?
(a)30% (b)33 % (c)35% (d)36%
5.If the total area goes up by
5%, and the area under wheat production goes up by 12% , then what will be the angle for wheat in
the new pie-chart ?
(a) 62.4° (b) 76.8° (c) 80.6° (d) 84.2°
SOLUTIONS
1.(c):The
total of the central angles corresponding to the three crops which cover 50% of
the total area ,should be 180°.Now, the
total of the central angles for the given combinations are:
(i) Wheat,Barley and jowar =
(72°+36°+18°)=126°
(ii) Rice,Wheat and Jowar =
(72°+72°+18°)=162°
(iii) Rice,Wheat and Barley = (72°+72°+36°)=180°
(iv)Bajra,Maize and Rice = (18°+45°+72°) = 135°
Clearly:(iii) is the required combination.
2.(a): The area under any of
the food crops is proportional to the angle corresponding to that crop.
Let the area under the rice production be x million acres.
Then, 18:72 = 1.5:x Þ
x=(72*15/18)=6
Thus, the area under rice production be = 6 million acres.
3.(b): Let the total production of barley be T tones and let Z acres
of land be put under barley
production.
Then, the total production of wheat =(6T) tones.
Also,area under wheat production = (2Z) acres.
\Area Under Wheat Production
72°
------------------------------------- = ---- =2
Area Under Barley Production 36°
And therefore,Area under wheat =
2*Area under Barley = (2Z)acres
Now, yield per acre for wheat = (6T/2Z) tones/acre = (3T/Z)
tones/acre
And yield per acre for barley = (T/Z) tones/acre.
3T/Z
\Required ratio = -------- =
3:1.
T/Z
4. (b):Let Z acres of land be put under
barley production.
Area Under Rice Production
72 °
Then, ----------------------------------- =
---- = 2.
Area Under Barley Production
36°
\Area under rice production = 2 * area under barley production = (2Z)
acres.
Now,if
p tones be the yield per acre of barley then ,yield per acre of rice
=(p+50% of p) tones
=(3/2 p) tones.
\Total production of rice = (yield per acre) * (area under
production)
= (3/2 p)*2Z=(3pZ) tones.
And,Total
production of barley = (pz) tones.
\Percentage production of barley to that rice = (pZ/3pZ *100)%= 33
1/3% .
5.(b):
Initially,let t be the total area under considerations.
The
area under wheat production initially was =(72/360 * t)acres =
(t/5)acres.
Now,if
the total area under consideration be increased by 5%,
then
the new value of the total area= (105/100 t) acres.
Also,if the area
under wheat production be increased by 12%,
t t
then
the new value of area under wheat =
-- +(12% of --)
acres = (112t/500)acres.
5 5
\Central
angle corresponding to wheat in the pie-chart
Area Under Wheat (new) ° (112t/500) °
= ------------------------------- *360 =
------------*360 =76.8°
Total area
(new) (105t/100)
Example 3.The following
pie-charts show the distribution of students of graduate and post graduate
levels in seven different institute-M,N,P,Q,R,S and T in a town.
DISTRIBUTION OF STUDENTS AT
GRADUATE AND POST-GRADUATE
LEVELS IN SEVEN
INSTITUTES-M,N,P,Q,R,S AND T.
Total Number of students of Total Number of students of
graduate level post
graduate level
1.How many students of institutes M and S are studying at graduate
level?
(a) 7516 (b) 8463 (c) 9127 (d) 9404
2.Total number of students studying at post -graduate level from
institutes N and P is:
(a) 5601 (b) 5944 (c) 6669 (d) 7004
3.What is the total number of graduate and post-graduate level
students in institute R?
(a) 8320 (b) 7916 (c) 9116 (d) 8372
4 .What is the ratio between the number of students studying at
post graduate and graduate levels
respectively from institute S?
(a)
14:19 (b) 19:21 (c) 17:21 (d) 19:14
5.What is the ratio between the number of students studying post
graduate level from institute S and the
number of students studying at graduate level from institute Q?
(a) 13:19 (b) 21:13 (c) 13:8 (d)19:13
SOLUTION
1.(b):Students of institute M at graduate level = 17% of 27300 =
4641.
Students of institute
S at graduate level = 14% of 27300 = 3822
\Total number students at graduate level in institutes M and S =
4641+3822=8463
2.(c):Required number = (15% of 24700) + (12% of 24700) = 3705 +
2964 = 6669.
3.(d):Required number = (18% of 27300) + (14% of 24700) = 4914 +
3458 = 8372.
4.(d):Required ratio
= (21% of 24700) = 21 * 24700 = 19
(14% of 27300) 14 * 27300 14
5.(d):Required ratio = (21% of 24700) = 21 * 24700 = 19
( 13% of
27300) 13 * 27300 13
Example 4.Study the following pie-chart and the table and the answer
the questions based on them.
PROPORTION OF POPULATION OF
SEVEN VILLAGES IN 1997
  village |
% population below poverty
|
X
|
38
|
Y
|
52
|
   Z |
42
|
R
|
51
|
S
|
49
|
T
|
46
|
V
|
58
|
1.Find the population of villages S if the population of village X
below poverty line in 1997 is 12160.
(a). 18500 (b) 20500 (c) 22000 (d) 26000
2.The ratio of the population of the village T below poverty line to
that of village Z below poverty line in 1997 is:
(a) 11:23 (b) 13:11 (c) 23:11 (d) 11:13
3.If the population of village R in 1997 is 32000,then what will be
the population of village Y below poverty line in that year?
(a) 14100 (b)15600 (c) 16500 (d) 17000
4.If in 1998, the population of villages Y and V increases by 10%
each and the percentage of population below poverty line remains unchanged for
all the villages, then find the population of village V below poverty line in
1998,given that the population of village Y in 1997 was 30000.
(a) 11250 (b) 12760 (c) 13140 (d) 13780
5.If in 1998,the population of village R increases by 10% while that
of village Z reduces by 5% compared that in 1997 and the percentage of
population below poverty line remains unchanged for all the village,then find
the approximate ratio of population of village R below poverty line for the
year 1999.
(a) 2:1 (b) 3:2 (c) 4:3 (d) 5:4
SOLUTION
1.(c):Let the population of village X be x
Then,38% of
x=12160 Þ x = 12160 * 100 =3200
38
Now ,if s be the
population village S,then
11 *
32000
16:11 = 32000 : s Þ s= = 22000.
16
2.(c): Let N be the total population of all the seven villages.
Then ,population of village T below poverty line = 46% of (21% of N)
and population of village Z below
poverty line = 42% of (11% of N)
\Required ratio = 46% of (21% of N) =
46 * 21 = 23
42% of (11% of N) 42 * 11 11
3.(b): Population of village R = 32000(given)
Let the population
of village Y be y.
Then, 16:15 =
32000 : y Þ y = 15 * 32000
=30000
16
4.(b): Population of village Y in 1997 =
30000(given) .
Let the population
village V in 1997 be v.
Then, 15:10 =
30000:v Þ v = 30000 * 10 = 20000.
15
Now population of
village V in 1998 = 20000 + (10% 0f 20000) = 20000.
\Population of village V below poverty line in 1998 = 58%
of 22000 = 12760.
5.(a) : Let the total population of all the seven villages in 1997
be N.
Then,population of
village R in 1997 = 16% of N = 16/ 100 N
And population of
village Z in 1997 = 11% of N =11/100 N
\Population of village R in 1999 = {16/100N+(10% of
16/100 N)}=1760/10000 N
and population of
village Z in 1999 = {11/100 N-(5% of 11/100 N)} = 1045/10000 N.
Now,population of
village R below poverty line for 1999 = 51% of(1760/10000 N)
And population of
village Z below poverty line 1999 = 42% of (1045/10000 n)
51% of (1760/10000 N) 51 *
1760 2
Required
ratio =
-------------------------- =
------------- = ------
42%
of (1045/10000 N) 42 * 1045
