Maharashtra Board Class 10 Maths Chapter 6 Statistics Set 6.6 Solutions

Question 1. The age group and number of persons, who donated blood in a blood donation camp is given below. Draw a pie diagram from it.
Answer: Solution: Total number of persons = 80 + 60 + 35 + 25 = 200
Measure of central angle (\(\theta\)) = \(\frac{\text{Number of scores in the components}}{\text{Total number of scores}} \times 360^\circ\)

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो रक्तदान शिविर में विभिन्न आयु समूहों द्वारा किए गए रक्त दान को दर्शाता है। यह दर्शाता है कि 20-25 वर्ष के आयु वर्ग ने कुल रक्त का 144° हिस्सा दान किया, 25-30 वर्ष वालों ने 108°, 30-35 वर्ष वालों ने 63° और 35-40 वर्ष वालों ने 45° दान किया। आरेख प्रत्येक आयु वर्ग के अनुपात को एक वृत्त के खंडों के रूप में दृश्य रूप से प्रस्तुत करता है। In simple words: To draw a pie diagram, first calculate the central angle for each category by finding its proportion of the total and multiplying by 360 degrees. Then, use a protractor to draw sectors with these angles in a circle.

🎯 Exam Tip: Always show the calculation for central angles clearly in your solution for full marks. Ensure the sum of central angles equals 360 degrees as a verification step.

Question 2. The marks obtained by a student in different subjects are shown. Draw a pie diagram showing the information.
Answer: Solution: Total marks obtained = 50 + 70 + 80 + 90 + 60 + 50 = 400
Measure of central angle (\(\theta\)) = \(\frac{\text{Number of scores in the components}}{\text{Total number of scores}} \times 360^\circ\)

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो एक छात्र द्वारा विभिन्न विषयों में प्राप्त अंकों के अनुपात को दर्शाता है। प्रत्येक विषय के अंक कुल अंकों का एक खंड बनाते हैं, जिसमें गणित 81° के साथ सबसे बड़ा भाग है, उसके बाद विज्ञान (72°), मराठी (63°), सामाजिक विज्ञान (54°), और अंग्रेजी व हिंदी प्रत्येक 45° के साथ। यह आरेख छात्र के शैक्षणिक प्रदर्शन का एक त्वरित दृश्य सारांश प्रदान करता है। In simple words: To create a pie diagram for marks, calculate the central angle for each subject by dividing the subject's marks by the total marks and multiplying by 360 degrees. Then, draw these angles as sectors in a circle to represent the proportions visually.

🎯 Exam Tip: When dealing with marks or similar data, double-check that your total sum of individual values matches the denominator used in the central angle calculation. This prevents errors in proportionality.

Question 3. In a tree plantation programme, the number of trees planted by students of different classes is given in the following table. Draw a pie diagram showing the information.
Answer: Solution: Total number of trees planted = 40 + 50 + 75 + 50 + 70 + 75 = 360
Measure of central angle (\(\theta\)) = \(\frac{\text{Number of scores in the components}}{\text{Total number of scores}} \times 360^\circ\)

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो एक वृक्षारोपण कार्यक्रम में विभिन्न कक्षाओं के छात्रों द्वारा लगाए गए पेड़ों की संख्या को दर्शाता है। प्रत्येक कक्षा द्वारा लगाए गए पेड़ों की संख्या एक केंद्रीय कोण के रूप में दर्शायी गई है। 7वीं और 10वीं कक्षा के छात्रों ने सबसे अधिक पेड़ (प्रत्येक 75°), उसके बाद 9वीं (70°), 6वीं और 8वीं (प्रत्येक 50°), और 5वीं कक्षा (40°) के छात्रों ने लगाए। यह आरेख विभिन्न कक्षाओं के योगदान का एक स्पष्ट दृश्य तुलनात्मक रूप प्रस्तुत करता है। In simple words: To draw a pie diagram for tree plantation data, first sum up all the trees planted to get the total. Then, for each class, calculate the central angle by dividing the number of trees they planted by the total and multiplying by 360 degrees. Finally, represent these angles as sectors in a circle.

🎯 Exam Tip: When the total number of items (like trees here) equals 360, the number of items directly corresponds to the central angle, simplifying calculations. However, always show the formula and substitution steps.

Question 4. The following table shows the percentages of demands for different fruits registered with a fruit vendor. Show the information by a pie diagram.
Answer: Solution: Total percentage = 30 + 15 + 25 + 20 + 10 = 100%
Measure of central angle (\(\theta\)) = \(\frac{\text{Percentage of components}}{100} \times 360^\circ\)

ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो एक फल विक्रेता के पास विभिन्न फलों की मांग के प्रतिशत को दर्शाता है। प्रत्येक फल की मांग केंद्रीय कोणों के रूप में प्रदर्शित की गई है। आम की मांग 108° के साथ सबसे अधिक है, उसके बाद सेब 90° के साथ, चीकू 72° के साथ, स्वीट लाइम 54° के साथ, और संतरे 36° के साथ सबसे कम मांग वाले हैं। यह आरेख स्पष्ट रूप से दिखाता है कि कौन से फल सबसे अधिक लोकप्रिय हैं। In simple words: When data is given in percentages, the central angle for each category is found by dividing its percentage by 100 and then multiplying by 360 degrees. These calculated angles are then used to draw sectors in a circle to represent the pie diagram.

🎯 Exam Tip: For percentage-based pie diagrams, always ensure the sum of percentages is 100% and the sum of central angles is 360 degrees. This provides an important check for accuracy.

Question 5. The pie diagram in the given figure shows the proportions of different workers in a town. Answer the following questions with its help.
(i) If the total workers is 10,000, how many of them are in the field of construction?
(ii) How many workers are working in the administration?
(iii) What is the percentage of workers in production?
Answer: Solution:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो एक शहर में विभिन्न क्षेत्रों में कार्यरत श्रमिकों के अनुपात को दर्शाता है। वृत्त को कई खंडों में विभाजित किया गया है: कृषि (24°), खाद्य (48°), निर्माण (72°), होटल (72°), उत्पादन (90°), प्रशासन (36°), और बिजली (18°)। यह आरेख प्रत्येक क्षेत्र में श्रमिकों की संख्या का दृश्य प्रतिनिधित्व करता है, जिससे उनकी सापेक्षिक संख्या को समझना आसान हो जाता है।
Measure of central angle (\(\theta\)) = \(\frac{\text{Number of scores in the components}}{\text{Total number of scores}} \times 360^\circ\)
i. Central angle for construction field (\(\theta\)) = \(72^\circ\)
\(72^\circ = \frac{\text{Number of workers in construction}}{10000} \times 360^\circ\)
\( \implies \) Number of workers in construction = \(\frac{72^\circ \times 10000}{360^\circ} = 2000\)
Therefore, there are 2000 workers working in the field of construction.
ii. Central angle for administration field (\(\theta\)) = \(36^\circ\)
\(36^\circ = \frac{\text{Number of workers in administration}}{10000} \times 360^\circ\)
\( \implies \) Number of workers in administration = \(\frac{36^\circ \times 10000}{360^\circ} = 1000\)
Therefore, there are 1000 workers working in the administration.
iii. Central angle for production field (\(\theta\)) = \(90^\circ\)
\(90^\circ = \frac{\text{Number of workers in production}}{10000} \times 360^\circ\)
\( \implies \) Number of workers in production = \(\frac{90^\circ \times 10000}{360^\circ} = 2500\)
Percentage of number of workers in production = \(\frac{2500}{10000} \times 100 = 25\%\)
Therefore, 25% of workers are working in the production field. In simple words: To find the number of workers in a specific field from a pie chart, divide the central angle of that field by 360 degrees and multiply by the total number of workers. To find the percentage, first find the number of workers, then divide by the total workers and multiply by 100.

🎯 Exam Tip: Always pay attention to whether the question asks for the number of workers or the percentage. Ensure your final answer directly addresses the question asked for each sub-part. Be careful with unit consistency.

Question 6. The annual investments of a family are shown in the given pie diagram. Answer the following questions based on it.
(i) If the investment in shares is Rs. 2000, find the total investment.
(ii) How much amount is deposited in bank?
(iii) How much more money is invested in immovable property than in mutual fund?
(iv) How much amount is invested in post?
Answer: Solution:
ℹ️ चित्र व्याख्या (Diagram Explanation): यह एक पाई आरेख है जो एक परिवार के वार्षिक निवेश को विभिन्न क्षेत्रों में दर्शाता है। निवेश के खंड हैं: अचल संपत्ति (120°), शेयर (60°), म्यूचुअल फंड (60°), पोस्ट (30°), और बैंक में जमा (90°)। यह आरेख परिवार की निवेश प्राथमिकताओं का एक दृश्य विश्लेषण प्रदान करता है, जिससे पता चलता है कि वे अपने धन को कहाँ आवंटित कर रहे हैं।
Measure of central angle (\(\theta\)) = \(\frac{\text{Number of scores in the components}}{\text{Total number of scores}} \times 360^\circ\)
i. Central angle for shares (\(\theta\)) = \(60^\circ\)
\(60^\circ = \frac{\text{Amount invested in shares}}{\text{Total investment}} \times 360^\circ\)
\(60^\circ = \frac{2000}{\text{Total investment}} \times 360^\circ\)
\( \implies \) Total investment = \(\frac{2000 \times 360^\circ}{60^\circ} = \text{Rs. }12000\)
The total investment is Rs. 12000.
ii. Central angle for deposit in bank (\(\theta\)) = \(90^\circ\)
\(90^\circ = \frac{\text{Amount deposited in bank}}{\text{Total investment}} \times 360^\circ\)
\(90^\circ = \frac{\text{Amount deposited in bank}}{12000} \times 360^\circ\)
\( \implies \) Amount deposited in bank = \(\frac{90^\circ \times 12000}{360^\circ} = \text{Rs. }3000\)
The amount deposited in bank is Rs. 3000.
iii. Difference in central angle for immovable property and mutual fund (\(\theta\)) = \(120^\circ - 60^\circ = 60^\circ\)
\(60^\circ = \frac{\text{Difference in investments}}{12000} \times 360^\circ\)
\( \implies \) Difference in investments = \(\frac{60^\circ \times 12000}{360^\circ} = \text{Rs. }2000\)
Rs. 2000 more is invested in immovable property than in mutual fund.
iv. Central angle for post (\(\theta\)) = \(30^\circ\)
\(30^\circ = \frac{\text{Amount invested in post}}{12000} \times 360^\circ\)
\( \implies \) Amount invested in post = \(\frac{30^\circ \times 12000}{360^\circ} = \text{Rs. }1000\)
The amount invested in post is Rs. 1000. In simple words: To calculate investment amounts from a pie chart, use the given total investment (or calculate it first using one category) and the central angle for each investment type. The amount for any category is its central angle divided by 360 degrees, then multiplied by the total investment. For differences, subtract the central angles first.

🎯 Exam Tip: When a total is not directly given, use the information provided for one segment (like shares here) to first find the total investment. This total then becomes crucial for solving all subsequent parts of the question. Remember to convert all currency symbols to "Rs.".