Quantitative Aptitude - SPLessons

Pie Chart Problems

Chapter 40

SPLessons 5 Steps, 3 Clicks
5 Steps - 3 Clicks

Pie Chart Problems

shape Introduction

Pie chart Problems or pie graph is a circle, divided into different segments representing the area of each segment.


  • Given numerical values are represented in the form of sectors of a circle.

  • The sectors of the circle are drawn in such a way that the area of each segment is proportional to the size of the figure to be represented.

  • Degree of each component will be based on the proportional value it has to the whole circle.

  • The values are represented in the pie diagram only with percentages and it makes comparison easy.


shape Methods


Example 1:
How do you find the central angle of a pie chart for a section of the pie chart that represents 20% of the pie?


Solution:

    In any sector, there are 3 parts to be considered:

    The arc length,

    The sector area

    The sector angle

    They all represent the SAME fraction of the whole circle.

    The arc length is a fraction of the circumference

    The sector area is a fraction of the whole area

    The sector angle is a fraction of 360°

    If the sector is 20 % of the pie chart, then each of these parts is 20 % of the whole.

    The sector angle is therefore:

    20 % × 360°

    \(\frac{20}{100}\) x 360° = 72°


Example 2:
In a pie chart the central angle for a component value is 320 when the total value is 1440, is______


Solution:

    The total angle that 320 value will cover in a pie chart is 80 degree.

    We know that total angle is 360 degree.

    Hence 360 degree corresponds to 1440 value

    => 320 value corresponds to = 360x 320/1440

    = 80 degree

shape Formulae

The central angle of each sector must be proportional to the corresponding value of the component.

Since sum of all central angles is \({360}^{\circ}\). Then,

Central angle of the component = \((\frac{value of the component}{total value} * 360)^{\circ}\)

shape Samples

1. Distribution of land (in a village) under various food crops is provided in the below pie chart. Answer the following question.


    Here, Others = 99°
    Rice = 72°
    Wheat = 72°
    Barley = 36°
    Jowar = 18°
    Bajra = 18°
    Maize = 45°


1. 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) 33\(\frac{1}{3}\)%
    (B) 30%
    (C) 35%
    (D) 36%


2. Which combination of three crops contribute to 50% of the total area under the food crops?

    (A) Bajra, Maize and Rice.
    (B) Rice, Wheat and Jowar.
    (C) Rice, Wheat and Barley.
    (D) Wheat, Barley and Jowar.


3. If the total area under Jowar was 1.5 million acres, then what was the area (in million acres) under rice?

    (A) 4.5
    (B) 9
    (C) 7.5
    (D) 6


Solution:


    1. Let Z acres of land be put under Barley production.

    Then,

    \(\frac{Area under rice production}{Area under barley production}\) = \(\frac{72°}{36°}\) = 2

    Therefore,

    Area under rice production = 2 x area under barley production = 2Z acres.

    Now, if ‘m’ tonnes be the yield per acre of barley then, yield per acre of rice = (m + 50% of m) tonnes = \(\frac{3}{2}m\) tonnes.

    So, total production of rice = (yield per acre) x (area under production) = \(\frac{3}{2}m\) x 2 Z = 3mZ tonnes.

    And total production of barley = pZ tonnes.

    Therefore, percentage production of barley to that of rice = \(\frac{m Z}{3 m Z} * 100\)% = 33\(\frac{1}{3}\)%

    Therefore, option (A) is correct answer.


    2. 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, Rice, Wheat and Barley = 180° is required combination.


    3. let the area under rice production be \(x\) million acres.

    Then, 18 : 72 = 1.5 : \(x\)

    \(x\) = \(\frac{72 * 1.5}{18}\)

    \(x\) = 6

    Thus, the area under rice production = 6 million acres.

    So, option (D) is the correct answer.


2. The distribution of students of graduate and post-graduate levels in seven different institutes like M, N, P, Q, R, S and T in a town are listed in the below pie chart. Study the charts and answer the given questions.


Total number of graduate level students = 27300


Total number of post-graduate level students = 24700


1. What is the total number of graduate and post-graduate level students in institute R?

    (A) 8320
    (B) 7916
    (C) 9116
    (D) 8372


2. What is the ratio between the number of students studying at post-graduate level from institute S and the number of students studying at graduate level from the institute Q?

    (A) 13 : 19
    (B) 21 : 13
    (C) 13 : 8
    (D) 19 : 13


3. How many students of institutes M and S are studying at graduate level?

    (A) 7516
    (B) 8463
    (C) 9127
    (D) 9404


Solution:


    1. Required number = (18% of 27300) + (14% of 24700) = 4914 + 3458 = 8372.


    2. Required ratio = \(\frac{21% of 24700}{13% of 27300}\) = \(\frac{21 * 24700}{13 * 27300}\) = \(\frac{19}{13}\)


    3. Students of institute M at graduate level = 17% of 27300 = 4641.

    Students of institute S at graduate level = 14% of 27300 = 3822.

    Therefore,

    Total number of students at graduate level in institutes M and S = 4641 + 3822 = 8463.


3. Study the following pie graph and the table. Then, answer the given questions.


Village % Population below poverty line
A 52
B 38
C 58
D 46
E 49
F 51
G 42


1. If the population of village F in 1997 is 32000, then what will be the population of village A below poverty line of that year?

    (A) 14100
    (B) 15600
    (C) 16500
    (D) 17000


2. Find the population of village E if the population of village B below poverty line in 1997 is 12160 ?

    (A) 18500
    (B) 20500
    (C) 22000
    (D) 26000


3. Find the ratio of population of village D below poverty line to that of village Z below poverty line in 1997?

    (A) 11 : 23
    (B) 13 : 11
    (C) 23 : 11
    (D) 11 : 13


Solution:


    1. Given that, Population of village F = 32000

    Let the population of village A be \(y\)

    Then,

    16 : 15 = 32000 : \(y\)

    ⇒ \(y\) = \(\frac{15 * 32000}{16}\) = 30000

    Therefore, population of village A below poverty line = 52% of 30000 = 15600.

    So, option (B) is the correct answer.


    2. Let the population of village B be \(x\)

    Then,

    38% of \(x\) = 12160

    ⇒ \(x\) = \(\frac{12160 * 100}{38}\) = 32000

    Now, if p be the population of village E, then

    16 : 11 = 32000 : p

    ⇒ p = \(\frac{11 * 32000}{16}\) = 22000


    3. Let N be the total population of all the seven villages.

    Then, population of village D below poverty line = 46% of (21% of N)

    Population of village G below poverty line = 42% of (11% of N)

    Therefore, required ratio = \(\frac{46% of (21% of N)}{42% of (11% of N)}\) = \(\frac{46 * 21}{42 * 11}\) = \(\frac{23}{11}\)


4. The following pie chart shows the expenditure incurred, in bringing out a book, by a publisher:


1. What should be the central angle of the sector for transportation charges?

    (A) 4 degrees
    (B) 8.4 degrees
    (C) 12.4 degrees
    (D) 14.4 degrees
    (E) None of these.


2. If the author’s royalties amount to Rs.30,000 what is the binder’s charges amount?

    (A) 6,000
    (B) 10,500
    (C) 12,000
    (D) 15,000
    (E) None of these.


Solution:


    1. Central angle representing transportation charges of 4% = \(\frac{4}{100} * {360}^{\circ}\) = \({14.4}^{\circ}\)

    Hence, correct option is (D).


    2.If the author’s royalties is Rs. 30,000 which is 30% of total expenditure.

    Therefore, total of publisher’s expenditure = Rs. \(\frac{100}{30} * 30,000\) = Rs. 1,00,000

    Hence, Binder’s charges = 12% of Rs. 1,00,000

    = Rs. \(\frac{12}{100} * 1,00,000\)

    = Rs. 12,000.

    Hence, option (C) is the correct answer.


5. The following pie chart shows the amount of subscriptions generated for India Bonds from different categories of investors.


1. If the total investment other than by FII and corporate houses is Rs 3,35,000 crore, then what will be the investment by NRI’s and Offshore funds (approximately)?

    (A) 274,100
    (B) 285,600
    (C) 293,000
    (D) Cannot be determined


2. If the investment by NRI’s are Rs 4,000 crore, then what is the investments by corporate houses and FII’s together?

    (A) 24,000 crore
    (B) 24,363 crore
    (C) 25,423 crore
    (D) 25,643 crore


3. If the total investment flows from FII’s were to be doubled in the next year and the investment flows from all other sources had remained constant at their existing levels for this year, then what would be the proportion of FII investment in the total investment into India Bonds next year (in US $ millions)?

    (A) 40 %
    (B) 50 %
    (C) 60 %
    (D) 70 %


4. In the corporate sector, approximately how many degrees should be there in the central angle?

    (A) 120
    (B) 121
    (C) 122
    (D) 123


Solution:

    1. Investment other than NRI and corporate houses is 33% = 335000.

    Also, investment by offshore funds and NRI’s is equal to 27%.

    Hence, 27 x \(\frac{335,000}{33}\) = 274 090.909

    So, option (A) is correct answer.


    2. The investments by corporate houses and FII’s together = \(\frac{67}{11}\) x 4000 = 24 363.6364
    Hence, option (B) is correct.


    3. FII’s currently account for 33 out of 100.

    If their value is doubled and all other investments are kept constant then their new value would be 66 out of 133 = approximately equal to 50%

    Hence, option (B) is correct answer.


    4. 34 x 3.6 = 122.4 (since 1% = 3.6 degrees)

    Therefore, 122.4 degrees should be there in the central angle.