Vedanta Excel in Mathematics Teachers' Manual Book 9 PDF

vedanta EXCEL in MATHEMATICS Book 9 Authors Hukum Pd. Dahal Tara Bahadur Magar vedanta Vedanta Publication (P) Ltd. Vanasthali, Kathmandu, Nepal +977-01-4382404, 01-4362082 vedantapublicat...

vedanta EXCEL in MATHEMATICS Book 9 Authors Hukum Pd. Dahal Tara Bahadur Magar vedanta Vedanta Publication (P) Ltd. Vanasthali, Kathmandu, Nepal +977-01-4382404, 01-4362082 [email protected] www.vedantapublication.com.np vedanta EXCEL in MATHEMATICS Book 9 Authors Hukum Pd. Dahal Tara Bahadur Magar All rights reserved. No part of this publication may be reproduced, copied or transmitted in any way, without the prior written permission of the publisher. Published by: Vedanta Publication (P) Ltd. Vanasthali, Kathmandu, Nepal +977-01-4382404, 01-4362082 [email protected] www.vedantapublication.com.np  Preface This “Teachers’ Manual of Vedanta EXCEL in MATHEMATICS BOOK-9” is prepared for teachers to aiming at assistance in pedagogical teaching learning activities. Its special focus intends to fulfillment the motto of text books EXCEL in MATHEMATICS approved by the Government of Nepal, Ministry of Education, CDC, Sanothimi, Bhaktapur. EXCEL in MATHEMATICS has incorporated the applied constructivism which focuses on collaborative learning so that the learners actively participate in the learning process and construct the new knowledge. The project works given at the end of each chapter provides the ideas to connect mathematics to the real life situations. Similarly, the text book contains enough exercises for uplifting critical thinking and creation as per the optimum goal of Bloom’s Taxonomy. The objective questions at the end of each area of subject content strengthen the students’ knowledge level. This manual helps the teachers to have the chapter-wise learning competencies, learning outcomes and level-wise learning objectives. Also, it helps the teachers in selecting the effective instructional materials, adopting the productive teaching activities, solving the creative problems and getting more extra objective and subjective questions which can be useful for the summative assessments. Grateful thanks are due to all Mathematics Teachers throughout the country who encouraged and provided the feedback to me in order to prepare the new series. Last but not least, any constructive comments, suggestions and criticisms from the teachers for the further improvements of the manual will be highly appreciated. Authors  Contents Topics Page No. 1. Set 1 2. Profit and Loss 5 3. Commission and Taxation 15 4. Household Arithmetic 23 5. Mensuration 33 6. Algebraic Expressions 48 7. Indices 52 8. Simultaneous Linear Equations 58 9. Quadratic Equations 62 10. Ratio and Proportions 70 11. Geometry - Triangle 78 12. Geometry - Similarity 106 13. Geometry - Parallelogram 113 14. Geometry - Circle 127 15. Geometry - Construction 140 16. Trigonometry 143 17. Statistics 148 18. Probability 154  Unit Set 1 Allocated teaching periods 8 Competency - To find the relation between the sets and solve the related problems by demonstrating the relations on the basis of the properties of the sets Learning Outcomes - To solve the verbal problems related to the cardinality of sets by using Venn diagrams and solve the behavioural problems by using the properties of relations of sets. Level-wise learning objectives S.N. Levels Objectives - To define a set - To tell the types of sets 1. Knowledge (K) - To state the relations between sets - To list the operations on sets - To define cardinality of sets - To tell the formulae involving two sets - To find the cardinality of a set. - To identify the set notation with special terminologies 2. Understanding (U) - To write the word problems based on cardinality relations in set notations - To solve the verbal problems on operations (union, intersection, difference and complement) of sets by 3. Application (A) using Venn-diagram - To solve the verbal problems on operations of sets by using formulae - To relate the problem related to set with other areas of 4. High Ability (HA) learning like percentage, ratio and so on. - To link various real life/ contemporary problems with sets and solve Required Teaching Materials/ Resources Definitions and formulae in colourful chart-paper, scissors, cello tape, different coloured markers, highlighter, models of Venn-diagrams, ICT tools etc. Pre-knowledge: Check the Pre-knowledge on cardinality of sets, relation of sets, operations of sets and Venn-diagram Teaching Activities 1. Make a group discussion on the definition of sets, set notation, types of sets, cardinality of sets and operations of sets by using Venn-diagram. 2. Ask individually the basic concepts from set as revision. 1 Vedanta Excel in Mathematics Teachers' Manual - 9 3. Make the group of students and give the questions on set operations from exercise. 4. Solve the problems and verify the relations from the exercise with discussion. 5. Discuss about the cardinality of sets, list the following formula by using Venn-diagrams along with examples. Give the work to the students to write the formulae in chart paper after discussion and paste the best one in math corner of the classroom or math lab. Case-I: When A is a subset of B (i) n (AªB) = A B A U (ii) n (A«B) = n (B) (iii) n o (B) = n (B) – n (A) Case- II: When A and B are disjoint sets. (i) n (AªB) = 0 A B U (ii) n (A«B) = n (A) + n (B) Case- III: When A and B overlapping sets (i) n (A« B) = n (A) + n (B) – n (AªB) (ii) n (only A) = no (A) = n (A) – n (AªB) (iii) n (only B) = no (B) = n (A) – n (AªB) (iv) n (only one) or n (exactly one) = no (A) + no (B) (v) n (A«B) = no (A) + no (B) + n (AªB) (vi) n(A « B ) = n(U) – n (A « B) (vii) (U) = n (A) + n (B) – n (AªB) + n(A « B ) 6. Explain the useful terminologies in solving verbal problems. (i) No. of people who like at least one fruits / either apple or banana= n (A«B) (ii) No. of people who like both apple and banana / who like apple as well as banana = n (AªB) (iii) No. of people who like only apple = no (A) and only banana = no (B) (iv) No. of people who like only one fruit = no (A) + no (B) (v) No. of people who don’t like both the fruits/like neither apple nor banana = n(A « B ) (vi) No. of people who don’t like apple only = no (B) and no. of people who don’t like banana only = no (A) Note: 1. If the information are given in percentage, consider that n (U) = 100 or x. 2. If the data are given in fraction then suppose that n (U) = x. 3. If each people participates in at least one activity then n (U) = n (A«B) or n(A « B )= 0 Solution of selected questions from Vedanta Excel in Mathematics 1. In a survey of 900 students in a school, it was found that 600 students liked tea, 500 liked coffee and 125 did not like both drinks. (i) Draw a Venn-diagram to illustrate the above information. (ii) Find the number of students who like both drinks (iii) Find the number of students who didn’t like tea only. Solution: Let T and C denote the sets of students who liked tea and coffee respectively. Vedanta Excel in Mathematics Teachers' Manual - 9 2 Then, n (U) = 900, n (T) = 600, n (C) = 500 and n(T « C )= 125 U Let the no. Of students who liked both the drinks be x. T C (i) Drawing a Venn-diagram to show the above information. 600-x x 500 – x (ii) From Venn-diagram, n (U) = (600 – x) + x + (500 – x) + 125 or, 900 = 1225 – x = 325 Hence, 325 students liked both the drinks. (iii) No. of students who didn’t like tea only i.e., n o (C) = 500 – x = 500 – 325 = 175 2. There are 900 students in a school. They are allowed to cast vote either only for A or for B as their school prefect. 36 of them cast vote for both A and B, 483 cast vote for A and 367 cast vote for B. (i) How many students did not cast the vote? (ii) Find the number of valid votes. (iii) Show the information in Venn-diagram. Solution: Here, n (U) = 900, no(A) = 483, no(B) = 367 and n (Aª∩ B) = 36 Now, we have, n (A « B) = n (U) - n(A « B )= 900 – 36 = 864. Again, n (A«B) = no (A) + no (B) + n (AªB) or, 864 = 483 + 367 + n (AªB) ? n (AªB) = 14 U Hence, 14 students didn’t cast the vote. A B (i) The number of valid votes (ii) Illustration in Venn-diagram 483 36 367 = no (A) + no (B) = 483 + 367 = 850 14 3. 54 students of class IX are taking part in sports or in music or in both activities. Out of them 9 students are taking part in both activities. The ratio of the number of the students who are taking part in sports to those who are taking part in music is 5:4. (i) How many students are taking part in sport? (ii) How many students are taking part in music only? (iii) Illustrate this information in a Venn-diagram. Solution: Let S and M denote the sets of students who are participating in sports and music respectively. Then, n (U) = n (S«M) = 54, n (S) = 5x, n (M) = 4x (say) and n (SªM) = 9 Now, n (S«M) = n (S) + n (M) – n (SªM) or, 54 = 5x + 4x – 9 ?x = 7 (i) The number of students who are taking part in sport n (S) = 5x = 35 (ii) The number of students who are taking part in music n (M) = 4x = 28 No. of students who are taking part in sport only, no (S) = n (S) - n (SªM) = 28 – 9 = 19 (iii) Illustration in Venn-diagram U S M 26 9 19 3 Vedanta Excel in Mathematics Teachers' Manual - 9 4. In a survey of some students, it was found that 60% of students were studying commerce and 40% were studying science. If 40 students were studying both the subjects and 10% did not study any of two subjects, by drawing a Venn-diagram, (i) Find the total number of students. (ii) Find the number of students who were studying science only. Solution: U Let C and S denote the sets of students who were studying commerce C S (60 – X)% (40 – x) and science respectively. x% Then n (U) = 100%, n (C) = 60%, n (S) = 40%, n(C « S)= 10% and n (CªS) = 40 10% Let n (CªS) = x% Now, representing the above data in a Venn-diagram Again, From the Venn-diagram, n (U) = n (C) + n (S) – n (CªS) + n(C « S) or, 100% = (60 – x)% + (40 – x)% + x% + 10% ?x % = 10% (i) Let the total number of students n (U) = x Then n (CªS) =10% of x or, 40 = 0.1x ?x = 400 Hence, there were 400 students. (ii) The number of students who were studying science only no(S) = (40-10)% of 400 = 30% of 400 = 120 Extra Questions 1. Out of 30 students of class IX, 15 students like to play volleyball, 20 students like to play basketball and each student like to play at least one of the game. (i) How many students like to play volleyball and basketball both? (ii) Show the above information in the Venn-diagram. [Ans: 5] 2. Out of 77 districts of Nepal, 27 districts have shared their boarder with India, 15 districts have shared their boarder with China and 37 districts have not shared their boarder with India and China both. (i) How many districts have shared their boarder with China only? (ii) How many districts have shared their boarder with India only? (iii) Draw a Venn-diagram to show the above information. [Ans: (i) 13, (ii) 25] 3. In a group of 30 children, 13 favored apple only, 8 favored guava only and 3 favored none of these fruits, by showing in Venn-diagram (i) Find the ratio of number of children who like both and don’t like both the fruits. (ii) What percent of the students like guava? [Ans: (i) 2:1, (ii) 60%] 4. In a survey of community, it was found that 50% of the people preferred yoga, 60% preferred jugging and 10% preferred neither yoga nor jugging. If 200 people preferred both yoga and jugging, by using a Venn-diagram find: (i) How many people were participated in the survey? (ii) How many people preferred only one of these? [Ans: (i)1000, (ii) 900] Vedanta Excel in Mathematics Teachers' Manual - 9 4  Unit Profit and Loss 2 Allocated teaching periods 6 Competency - To solve the daily life problems on profit and loss by using fundamental rules of profit/ loss and formulae. Learning Outcomes - To collect the real life problems on profit and loss and solve them. Level-wise learning objectives S.N. LEVELS OBJECTIVES - To define cost price, selling price - To tell the relation of C.P., S.P. and profit % or loss% 1. Knowledge (K) - To define marked price - To tell the formula of finding the discount - To define VAT - To find the profit/ loss amount 2. Understanding (U) - To calculate the profit and profit - To find the discount/ VAT amount - To calculate the rates of discount and VAT - To solve the verbal problems on profit and loss 3. Application (A) - To solve the verbal problems on discount and VAT - To mathematize the daily life problems related to 4. High Ability (HA) profit and loss and solve them. - To link various real life/ contemporary problems with discount and VAT Required Teaching Materials/ Resources Colourful chart-paper with required definitions and formulae, bills, VAT bills, audio/video clips related to profit and loss, projector etc. Pre-knowledge: cost price, selling price, profit and loss A. Profit and Loss Teaching Activities 1. Discuss upon cost price, selling price, profit and loss of the articles like watch, mobile, books, copies, bags etc. 2. Divide the class into 5/6 groups and ask the formulae of profit amount, loss amount, profit percentage and loss percentage. 3. List out the following formulae with examples (i) Profit amount = S.P. – C.P. (ii) Loss amount = C.P. – S. P. (iii) Profit amount = profit % of C.P. (iv) Loss amount = loss % of C.P. (v) S.P. = C.P. + P% of C.P. 5 Vedanta Excel in Mathematics Teachers' Manual - 9  (vi) S.P. = C.P. – L% of C.P. (vii) Profit percent = Profit × 100% o S. P. – C. P. × 100% C.P. C. P. (viii) loss percent = Loss × 100% o C. P. – S. P. × 100% C.P. C. P. Solution of selected problems from Vedanta Excel in Mathematics 1. A grocer bought 20 kg of sugar at Rs 70 per kg. He sold 15 kg of sugar at Rs 75 per kg and the remaining quantity at Rs 69 per kg. Find his profit or loss percent in the transaction. Solution: Here, C.P. of 20 kg of sugar = 20 ×Rs 70 = Rs 1400 S.P. of 15 kg of sugar = 15 ×Rs 75 = Rs 1125 Remaining quantity of sugar = 20 kg – 15 kg = 5 kg S.P. of 5 kg of sugar = 5 ×Rs 69 = Rs 345 S.P. of 20 kg of sugar = Rs 1125 + Rs 345 = Rs 1470 Since, S.P. > C.P., he made a profit Profit amount = S.P. – C.P. = Rs 1470 – Rs 1400 = Rs 70 Now, profit percent = Profit amount × 100% = Rs 70 × 100% = 5% Cost price (C.P.) Rs 1400 Hence, his profit percent is 5%. 2. Mrs. Rokaya bought 2 quintals of apples in Jumla for Rs 17000. She paid Rs 15 per kg for the transportation from Jumla to Nepalgunj. 10 kg of apples is damage and she sold the remaining quantity of apples at Rs 120 per kg. Calculate her profit or loss percent in the total transaction. Solution: Here, C.P. of 2 quintals i.e., 200 kg of apples = Rs 17000 C.P. of the apples with the transportation cost = Rs 17000 + 200×Rs 15 = Rs 20000 Saleable quantity of apples = 200kg – 10 kg = 190 kg S.P. of 190 kg of apples = 190 ×Rs 120 = Rs 22800 Since, S.P. > C.P., she made a profit Profit amount = S.P. – C.P. = Rs 22800 – Rs 20000 = Rs 2800 Now, profit percent = Profit amount × 100% = Rs 2800 × 100% = 14% Cost price (C.P.) Rs 20000 Hence, her profit percent is 14% in the total transaction. 3. A stationer sells 8 pencils for the cost of 10 pencils, find his gain percent. Solution: Let the C.P. of 1 pencil be Rs x. Then, C.P. of 10 pencils = Rs 10x and C.P. of 8 pencils = Rs 8x According to the question, S.P. of 8 pencils = C.P. of 10 pencils = Rs 10x But C.P. of 8 pencils = Rs 8x. So, gain amount = Rs 10x – Rs 8x = Rs 2x Now, gain percent = Gain amount × 100% = Rs 2x × 100% = 20% Cost price (C.P.) Rs 10x Hence, his gain percent is 20%. 4. A stationer bought 2000 exercise books. He distributed 200 exercise books to the students from poor economical background as donation. He sold each of the remaining exercise books at Rs 10 more than the cost price of each and gained 8%, find the cost price of each exercise book. Vedanta Excel in Mathematics Teachers' Manual - 9 6 Solution: Let the C.P. of each exercise book be Rs x. Then S.P. of each exercise book is Rs (x + 10) Now, C.P. of 2000 exercise books = Rs 2000x Saleable quantity of exercise books = 2000 – 200 = 1800 S.P. of 1800 exercise books = Rs 1800 (x + 10) = Rs (1800x + 18000) Gain percent = 8% Now, S.P. = C.P. + G% of C.P. or, 1800x + 18000 = 2000x + 8% of 2000x or, 1800x + 18000 = 2160x or, 18000 = 360x ?x = Rs 50 Hence, the cost price of each exercise book is Rs 50. 5. Bikash purchased 10 pens. He sold 5 pens at 25% profit and the remaining 5 pen at 162 % loss. If he received Rs 625 in total, find the cost price of each pen. 3 Solution: Let the C.P. of each pen be Rs x. Then C.P. of 5 pens = Rs 5x and C.P. of 10 pens = Rs 10x S.P. of 5 pens = C.P. + P% of C.P. = 5x + 25% of 5x = 5x + 25 × 5x = Rs 25x 100 4 S.P. of remaining 5 pens = C.P. – L% of C.P. = 5x – 162% of 5x = 5x – 50 × 5x = Rs 25x 3 3 × 100 6 According to question, total S.P. = Rs 625 or, 25x + 25x = Rs 625 or, 125x = Rs 625 ?x = 60 4 6 12 Hence, the cost price of each pen is Rs 60. 6. A dealer bought a pen-drive for Rs 500. He sold it at 10% loss. If he wanted to make a profit of 12.5% without increasing the selling price, by how much should the cost price of the pen-drive be reduced? Solution: Here, C.P. of a pen-drive = Rs 500 and loss percent = 10% Now, S.P. of pen-drive = C.P. – L% of C.P. = Rs 500 – 10% of Rs 500 = Rs 450 Again, S.P. of pen-drive = Rs 450 and profit percent = 12.5% We have, S.P. = C.P. + P% of C.P. or, Rs 450 = C.P. + 12.5% of C.P. or, Rs 450 = 1.125 C.P. ?C.P. = Rs 400 Difference between cost price = Rs 500 – Rs 400 = Rs 100 Hence, the cost price of the pen-drive should be reduced by Rs 100 to make 12.5% profit. 7. A trader sold an article at 10% profit. If he sold it at 10% loss, it would yield Rs 140 less than the previous selling price; find the cost price of the article. Solution: Let the cost price of an article be Rs x. Now, S.P. of article = C.P. + P% of C.P. = Rs x + 10% of Rs x = Rs 1.1x According to the question, new S.P. of the article = Rs (1.1x – 140) We have, S.P. = C.P. – L% of C.P. or, 1.1x – 140 = x – 10% of x or, 1.1x – 140 = 0.9x or, 0.2x = 140 ? x = 700 Hence, the cost price of the article was Rs 700. 7 Vedanta Excel in Mathematics Teachers' Manual - 9 8. Rajesh Das bought two calculators for Rs 1000. He sold one of them at 20% profit and the other at 20% loss. If the selling prices of both the calculators are same, find the cost price of each calculator. Also, calculate his gain or loss in the total transaction. Solution: Let the cost price of the one calculator be Rs x then that of the another one is Rs (2000-x) From the first calculator: C.P.1 = Rs x, profit percent = 20% ?S.P.1 = C.P.1 + P% of C.P.1 = Rs x + 20% of Rs x = Rs 1.2x From the second calculator: C.P.2 = Rs (1000 – x), loss percent = 20% ?S.P.2 = C.P.2 - L% of C.P.1 = Rs (1000 – x) – 20% of Rs (1000 – x) = Rs (1000 – x – 200 + 0.2x) = Rs (800 – 0.8x) According to the question, S.P.1 = S.P.2 or, 1.2x = 800 – 0.8x or, 2x = 800 ?x = Rs 400 Now, the cost price of the first calculator was Rs 400 and that of the other was Rs (1000 – 400) = Rs 600 Again, S.P.1 = 1.2×Rs 400 = Rs 480 S.P. of two calculators = 2×Rs 480 = Rs 960 and C.P. = Rs 1000 Since, C.P. > S.P., there is a loss Loss amount = C.P. – S.P. = Rs 1000 – Rs 960= Rs 40 Now, loss percent = Loss amount × 100% = Rs 40 × 100% = 4% Cost price (C.P.) Rs 1000 Hence, his loss percent was 4% in the total transaction. 9. Ajaya bought a fan and a heater for Rs 4000. He sold the fan at a profit of 25% and the heater at a loss of 5%. If he gained 7% on his total outlay, at what price did he buy each item? Solution: Let the cost price of the fan be Rs x then that of the heater is Rs (4000-x) From the fan: C.P.1 = Rs x, profit percent = 25% ?S.P.1 = C.P.1 + P% of C.P.1 = Rs x + 25% of Rs x = Rs 1.25x From the heater: C.P.2 = Rs (4000 – x), loss percent = 5% ?S.P.2 = C.P.2 - L% of C.P.1 = Rs (4000 – x) – 5% of Rs (4000 – x)= Rs (4000 – x – 200 + 0.05x)= Rs (3800 – 0.95x) Total C.P. = Rs 4000 Total S.P. = Rs (1.25x + 3800 – 0.95x)= Rs (3800 + 0.3x) and profit percent = 7% S.P. = C.P. + P% of C.P. or, 3800 + 0.3x = Rs4000 + 7% of 4000 or, 3800 + 0.3x = Rs4280 or, 0.3x = Rs480 ?x = Rs 1600 and Rs (4000 – x) = Rs (4000 – 1600) = Rs 2400 Hence, the cost price of the fan was Rs 1600 and that of heater was Rs 2400. Vedanta Excel in Mathematics Teachers' Manual - 9 8 10. A shopkeeper decided to make equal rate of profit in each fancy item. If he sold a jacket costing Rs 4400 for Rs 5060, at what price did he sell the shoes which was purchased for Rs 3420? Solution: Here, The cost price (C.P.) of a jacket = Rs 4400 and selling price (S.P.) = Rs 5060 Now, profit percent = S. P. – C.P. × 100% = Rs 5060 – Rs 4400 × 100% = 15% C. P. Rs 4400 Again, cost price of the shoes (C.P.) = Rs 3420 and profit percent = 15% We have, S.P. = S.P. = C.P. + P% of C.P. = Rs 3420 + 15% of Rs 3420 =Rs 3933 Hence, he sold the shoes for Rs 3933. 11. A dishonest shopkeeper has two false balances. One balance weighs 10% more while buying the goods and other weighs 10% less while selling the goods. Find the gain percent just by weighing. Solution: Here, While buying the goods, he weighs 10% more. So, C.P. of goods of worth Rs 110 = Rs 100 While selling the goods, he weighs 10% less. So, S.P. of goods of worth Rs 100 = Rs 110 S.P. of goods costing Re 1 = Rs 110 100 Since, he has the goods of worth Rs 110. ?S.P. of goods of worth Re 110 = Rs 110 × 110 = Rs 121 100 Now, profit percent = S. P. – C.P. × 100% = Rs 121 – Rs 100 × 100% = 21% C. P. Rs 100 12. A crooked shopkeeper sells goods at the cost price. But his 1 kg weight weighs 900 g only. Find his gain percent. Solution: Let the C.P. of 1 g of goods be Rs x. Then, C.P. of 1 kg of goods = Rs 1000x and S.P. of 900 g of goods = Rs 900x According to the question, S.P. of 900 g = C.P. of 1 kg of goods = Rs1000x But C.P. of 900 g of goods = Rs 900x. So, gain amount = Rs 1000x – Rs 900x = Rs 100x Now, gain percent = Gain amount × 100% = Rs 100x × 100% = 111% Cost price (C.P.) Rs 900x 9 Hence, his gain percent is11 %. 1 9 13 A grocer has some rice of worth Rs 3000. He sold 1 of it with 10% loss. By how many 3 percent must the selling price be increased for making 10% profit on the outlay? Solution: Here, C.P. of certain quantity of rice = Rs 3000 Total S.P. for making 10% profit = C.P. + P% of C.P. = Rs 3000 + 10% of Rs 3000 = Rs 3300 C.P. of 1 of the rice = 1 of Rs 3000 = Rs 1000 and loss percent = 10% 3 3 9 Vedanta Excel in Mathematics Teachers' Manual - 9 ?S.P. of 1of the rice = C.P. – L% of C.P. = Rs 1000 – 10% of Rs 1000 = Rs 900 3 For making no any gain, S.P. of the remaining 2 of the rice = Rs 3000 – Rs 900 = Rs 2100 3 For making 10% gain, S.P. of the remaining 2 of the rice = Rs 3300 – Rs 900 = Rs 2400 3 Difference in S.P. = Rs 2400 – Rs 2100 = Rs 300 Now, increased percent in S.P. of remaining quantity = Rs 300 × 100% = 142 Rs 2100 7 Hence, the selling price of the remaining quantity of rice should be increased by 142 for 7 making 10% profit on the outlay. Extra Questions 1. Mr. Lama bought a second hand bike for Rs 1,10,000 and immediately he spent Rs 5,000 to repair it. Then he sold it for Rs 1,26,500. Find his profit or loss percent. [Ans: Profit percent = 10%] 2. A grocer sold 5 kg of wheat flour at Rs 55 per kg and gained 10%. If he sold all the flour for Rs 260, what would be his gain or loss percent? [Ans:Profit 4%] 3. A man bought a hen and a duck for Rs 3,400. He sold the hen at 25% profit and then duck at 10% loss. If he gained 10% on his total outlay. At what price did he buy the hen and the duck each? [Ans: Rs 2,000, Rs 1500] 4. Rajendra is a stationer. Once, he bought 2,000 books. Out of them he donated 200 books to a school library. He sold the remaining books with 8% profit at the rate of Rs. 120 per book. What will be the cost price of a book? Find it. [Ans:Rs 100] 5. A shopkeeper sold a sewing machine for Rs 3,600 and made a loss of 10%. For what price should he sell it to gain 10%? [Ans:Rs 4,400] B. Marked price (M.P.) and Discount Teaching Activities 1. Recall cost price (C.P.) and selling price (S.P.) of an article. 2. With examples, discuss on marked price (M.P.) and discount. 3. Explain discount as the amount of reduction in the marked price of an article. 4. Paste/show the different types of taxes in the colourful chart paper and explain with appropriate examples. 5. Make clear VAT as tax levied on purchase of goods or service 6. List the following formulae after discussion (i) Discount amount = M.P. – S. P. (ii) Discount amount = Discount % of M.P. (iii) Rate of discount = Discount amount × 100% M.P. (iv) S.P. = M.P. – Discount amount (v) S.P. = M.P. – Discount% of M.P. = M.P. (1 – Discount %) (vi) S.P. = M.P. (1 – D1%)(1 – D2%) when two successive discount rates D1% and D2 % are given. (vii) VAT amount = S.P. including VAT – S. P. excluding VAT (viii) VAT amount = VAT% of.P. Vedanta Excel in Mathematics Teachers' Manual - 9 10  (ix) Rate of VAT = VAT amount × 100% S.P. (x) S.P. with VAT = S.P. + VAT amount (xi) S.P. with VAT = S.P. + VAT% of S.P. = S.P. (1 + VAT %) Solution of selected problems from Vedanta Excel in Mathematics 1. A retailer allows 15% discount on the marked price of an electric fan. If a customer pays Rs 2,244 with 10% VAT, find the marked price of the fan. Solution: Let the marked price of the electric fan be Rs x Then, S.P. after 15% discount = M.P. – D% of M.P. = x – 15% of x = Rs 0.85x Again, S.P. with VAT = S.P. + VAT% of S.P. or, 0.85x + 10% of 0.85x = Rs 2,244 or, 0.935x = Rs 2,244 ? x = Rs 2,400 Hence, the marked price of the electric fan is Rs 2,400. 2. A tourist paid Rs 5,610 for a carved window made up of wood with discount of 15% including 10% value added tax (VAT). How much does he get back while leaving Nepal? Solution: Let the marked price of the carve window be Rs x Then, S.P. after 15% discount = M.P. – D% of M.P. = x – 15% of x = Rs 0.85x Also, S.P. with VAT = S.P. + VAT% of S.P. or, Rs 5,610 = 0.85x + 10% of 0.85x or, Rs 5,610 = 0.935x ? x = Rs 6,000 Again, S.P. = Rs 0.85x = Rs 0.85× 6000 =Rs 5,100 And VAT amount = 10% of Rs 5,100 = Rs 510 Hence, the tourist gets Rs 510 back while leaving Nepal. 3. The marked price of a cycle is Rs 5,500. After allowing certain percent of discount with 10% VAT levied, the cycle is sold at Rs 5,445, find the discount percent. Solution: The marked price of a cycle = Rs 5,500, VAT percent = 10% and S.P. with VAT = Rs 5,445, discount percent =? Let S.P. after discount of the cycle be Rs x. Then,S.P. with VAT = S.P. + VAT% of S.P. or, Rs 5,445 = x + 10% of 0.85x or, Rs 5,445 = 1.1x x = Rs 4,950 Also, discount amount = M.P. – S.P. = Rs 5,500 – Rs 4,950 = Rs 550 Again, discount percent = Discount amount × 100% = Rs 550 × 100% = 10% M.P. Rs 5500 Hence, the required discount percent is 10%. 4. The mobile price is tagged Rs 5,000. If a customer gets 12% discount and adding certain percent VAT reaches as Rs 4,972, find out the VAT percent. 11 Vedanta Excel in Mathematics Teachers' Manual - 9 Solution: The marked price of the mobile = Rs 5,000, discount percent = 12% and S.P. with VAT = Rs 4,972, VAT percent =? Now, Then, S.P. after discount = M.P. – D% of M.P. = Rs 5,000 – 12% of Rs 5,000 = Rs 4,400 Also, VAT amount = S.P. with VAT – S.P. = Rs 4,972 – Rs 4,400 = Rs 572 Again, VAT percent = VAT amount × 100% = Rs 572 × 100% = 13% S.P. Rs 4400 Hence, the required VAT percent is 13%. 5. After allowing 15% discount on the marked price of a camera, 15% VAT was levied and sold it. If the difference between the selling price with VAT and selling price after discount is Rs 1,122, find the marked price of the camera. Solution: Let the marked price of the camera be Rs x Then, S.P. after 15% discount = M.P. – D% of M.P. = x – 15% of x = Rs 0.85x Again, S.P. with 15% VAT = S.P. + VAT% of S.P. = Rs 0.85x + 15% of Rs 0.85x = 0.9775x According to the question, S.P. with VAT – S.P. after discount = Rs 1,122 or, 0.9775x – 0.85x = Rs 1,122 or, 0.1275x = Rs 1,122 ? x = Rs 8,800 Hence, the marked price of the camera is Rs 8,800. 6. The marked price of an article is 25% above the cost price. When it is sold at a discount of 15%, there is a gain of Rs 200. Find. (i) The cost price of the article. (ii) The marked price of the article. Solution: Let the cost price (C.P.) of the article be Rs x Then, M.P. of the article = C.P. + 25% of C.P. = Rs x + 25% of x = Rs 1.25x Now, S.P. after 15% discount = M.P. – D% of M.P. =1.25x – 15% of 1.25 x = Rs 1.0625x Again, profit amount = S.P. – C.P. or, Rs 200 = 1.0625x – x or, Rs 200 = 0.0625x ? x = Rs 3,200 And, M.P. = Rs 1.25x = Rs 1.25×3200 = Rs 4,000 Hence, the cost price of the article is Rs 3,200 and its marked price is Rs 4,000. 7. When a pen is sold at a discount of 15%, there is a gain of Rs 10. But if it is sold at 25% discount, there is a loss of Rs 2. Find the marked price of the pen. Solution: Let the marked price (M.P.) of the pen be Rs x. According to the given first condition, discount percent = 15%, profit = Rs 10 We have, S.P. after 15% discount = M.P. – D% of M.P. = x – 15% of x = Rs 0.85x Again, profit amount = S.P. – C.P. or, Rs 10 = 0.85x – C.P. ? C.P. = Rs (0.85x – 10) … (i) According to the given second condition, discount percent = 25%, loss = Rs 2 Vedanta Excel in Mathematics Teachers' Manual - 9 12 We have, S.P. after 25% discount = M.P. – D% of M.P. = x – 25% of x = Rs 0.75x Again, loss amount = C.P. – S.P. or, Rs 2 = Rs ( 0.85x – 10) – Rs 0.75x [From (i)] or, Rs 12 = 0.1x ? x = Rs 120 Hence, the marked price of the pen is Rs 120. 8. A shopkeeper marked the price of an article a certain percent above the cost price and he allowed 16% discount to make 5% profit. If a customer paid Rs 9,492 with 13% VAT to buy the article, by what percent is the marked price above the cost price of the article? Solution: Let the marked price of the article be Rs x Then, S.P. after 16% discount = M.P. – D% of M.P. = x – 16% of x = Rs 0.84x Again, S.P. with VAT = S.P. + VAT% of S.P. or, 0.84x + 13% of 0.84x = Rs 9,492 or, 0.9492x = Rs 9,492 ? x = Rs 10,000 Hence, the marked price of the article is Rs 10,000. Also, S.P. = Rs 0.84x = Rs 0.84×10,000 = Rs 8,400 Let C.P. of the article be Rs y We have, S.P. = C.P. + P% of C.P. or, Rs 8,400 = y + 5% of y or, Rs 8,400 = 1.05y ? y = Rs 8,000 Again, difference between M.P. and C.P. = Rs 10,000 – Rs 8,000 = Rs 2,000 ?The marked price of the article is above the cost price by Rs 2,000 × 100% = 25% Rs 8,000 9. The marked price of an item is Rs 1,500 and 10% discount is given to make 20% profit. By what percent is the discount to be increased to get only 12% profit? Solution: Here, the marked price (M.P.) of the item = Rs 1,500, discount = 10 % and profit percent= 20% Then, S.P. after discount = M.P. – D% of M.P. = Rs 1,500 – 10% of Rs 1,500 = Rs 1,350 Let the cost price (C.P.) of the item be Rs x. We have, S.P. = C.P. + P% of C.P. or, Rs 1,350 = x + 20% of x or, Rs 1,350 =1.2x ? x = Rs 1,125 Thus, the cost price (C.P.) of the item is Rs 1,125 Again, to get only 12% profit S.P. = C.P. + P% of C.P. = Rs 1,125 + 12% of Rs 1,125 = Rs 1,260 Discount amount = M.P. – S.P. = Rs 1,500 – Rs 1,260 = Rs 240 Discount percent = Discount amount × 100% = Rs 240 × 100% = 16% M. P. Rs 1500 Hence, the discount should be increased by 16% - 10% = 6% to make only 12% profit. 13 Vedanta Excel in Mathematics Teachers' Manual - 9 10. The price of a watch is marked Rs 11,250. When it is sold allowing 20% discount, 20% profit is made. By what percent is the discount to be reduced to increase the profit by 3%? Solution: Here, the marked price (M.P.) of the item = Rs 11,250, discount = 20 % and profit percent= 20% Then, S.P. after discount = M.P. – D% of M.P. = Rs 11,250 – 20% of Rs 11,250 = Rs 9,000 Let the cost price (C.P.) of the item be Rs x. We have, S.P. = C.P. + P% of C.P. or, Rs 9,00 = x + 20% of x or, Rs 9,000 =1.2x ? x = Rs 7,500 Thus, the cost price (C.P.) of the item is Rs 7,500 Again, to get only (20 + 3)% = 23% profit S.P. = C.P. + P% of C.P. = Rs 7,500+ 23% of Rs 7,500 = Rs 9,225 Discount amount = M.P. – S.P. = Rs 11,250 – Rs 9,225= Rs 2.025 Discount percent = Discount amount × 100% = Rs 20,250 × 100% = 18% M. P. Rs 11,250 Hence, the discount should be decreased by 20% - 18% = 2% to increase the profit by 3%. 11. The marked price of a digital watch is Rs 6,000. Allowing 10% discount and including same percent of value added tax, the watch is sold. By how much percent is the VAT amount less than discount amount? Solution: Here, the marked price (M.P.) of the digital watch = Rs 6,000, discount = 10 % and VAT = 10% Now, Discount amount = D% of M.P. = 10% of Rs 6,000 = Rs 600 Also, S.P. after discount = M.P. – D = Rs 6,000 – Rs 600 = Rs 5,400 Again, VAT amount = VAT% of S.P. = 10% of Rs 5,400 = Rs 540 Difference between discount and VAT amounts = Rs 600 – Rs 540 = Rs 60 VAT amount is less than discount amount by Rs 60 × 100% = 10% Rs 600 Hence, the VAT amount is less than discount amount by 10%. Extra Questions 1. What is the price of a bag costing Rs 2000 after allowing 15% discount? Find it. [Ans: Rs 1700] 2. Mr. Ajay sold a watch for Rs 880 after allowing 20% discount, what was the marked price of the watch? [Ans:Rs 1,100] 3. A shopkeeper in Nepalgunj fixed the price of a suitcase in such a way that the he could gain 10% after allowing 10% discount on it. If the customer paid Rs 9900 for the suitcase, find the marked price and the selling price of the suitcase. [Ans: Rs 11,000; Rs 9000] 4. What will be the price of a calculator costing Rs 600 with 13% value added tax (VAT)? [Ans: Rs 678] 5. The marked price of a scooter is Rs 2,40,000. If the shopkeeper allows 15% discount and levies 13% value added tax, how much should a customer have to buy the scooter? Find it. [Ans: Rs 2,30,520] 6. A cycle was sold after allowing 20% discount on the marked price and levying 10% VAT. If the customer got Rs 555 as the discount, how much VAT amount was levied on the cost of the cycle? Find it. [Ans: Rs 222] Vedanta Excel in Mathematics Teachers' Manual - 9 14  Unit Commission and Taxation 3 Allocated teaching periods 6 Competency - To solve the daily life problems on profit and loss by using fundamental rules of profit/ loss and formulae. Competency - To solve the daily life problems on by using fundamental rules of commercial mathematics and formulae Learning Outcomes - To solve the daily life problems related to Commission, Tax and Dividend. Level-wise learning objectives S.N. LEVELS OBJECTIVES - To define commission - To tell the formula of finding the commission 1. Knowledge (K) amount - To define bonus - To tell the formula of finding the dividend - To define income tax - To calculate the commission amount 2. Understanding (U) - To calculate the bonus amount received by each employee - To find the commission or total sales 3. Application (A) - To find the income tax. - To solve the verbal problems related to dividend - To compare the income taxes paid by an individual 4. High Ability (HA) and the married couple. - To mathematize the daily life problems about the shares of some companies or banks or business solve them. Required Teaching Materials/ Resources Colourful chart-paper with definitions of buyer, seller, agent, commission and formulae, bills, income tax rates published by IRD etc Pre-knowledge: cost price, selling price, profit and loss, discount and VAT A. Bonus Teaching Activities 1. Start the classroom discussion with the following questions: (i) Have you heard about bonus? (ii) Have anyone of your family members received bonus yet? (iii) What is Net profit? (iv) How do you define bonus? 15 Vedanta Excel in Mathematics Teachers' Manual - 9 2. Explain bonus as the extra amount of money that is distributed to the employees by the company for their goods performance from the certain percent of net profit of the fiscal year. 3. Focus on group work to let the students formulate the concepts through examples. 4. Ask the formula of finding the bonus amount , bonus amount received by each employee, bonus , explain the following formulae (i) Bonus amount = Bonus % of net profit (ii) Bonus amount received by each employee = Total bonus amount No. of employee (iii) Bonus percent = Bonus amount × 100% Net profit (vi) Annual income = annual salary + bonus Solution of selected problems from Vedanta Excel in Mathematics 1. A publication housed announced to distribute 10% bonus equally to its 20 employees from the net profit of Rs 18,36,000 at the end of a fiscal year, find the bonus received by each employee. Solution: Here, net profit of the company = Rs 18,36,000, bonus percent = 10%, number of employees = 20 Now, bonus amount =10% of Rs 18,36,000 = Rs 1,83,600 Bonus amount received by each employee = Rs 1,83,600 = Rs 9,180 20 Hence, each employee received Rs 9,180. 2. A garment factory announced 20% bonus to its 25 workers from the net profit at the end of last fiscal year. If every worker received Rs 18,500, how much was the profit of the factory? Solution: Here, bonus percent= 20%, number of employees = 25, bonus amount received by each worker = Rs 18,500 Now, bonus amount =25× Rs 18,500 = Rs 4,62,500 Let net profit of the factory be Rs x Then, bonus amount = bonus % of net profit or, Rs 4,62,500 = 20% of x = 0.2x ?x = Rs 23,12,500 Hence, the profit of the factory is Rs 23,12,500. 3. A business company distributed bonus to its 24 employees from the net profit of Rs 16,48,000. If every employee received Rs 8,240, what was the bonus percent? Solution: Here, net profit of the company =Rs 16,48,000, number of employees = 24, bonus amount received by each worker = Rs 8,240, bonus percent = ? Bonus amount =24× Rs 8,240 = Rs 1,97,760 Now, bonus percent = Bonus amount × 100% = Rs 1,97,760 × 100% = 12% Net profit Rs 16,48,000 Hence, the required bonus percent was 12%. 4. A garment factory made a net profit of Rs 48,00,000 in the last year. The management of the factory decided to distribute 18% bonus from the profit to its 25 employees. (i) Find the bonus amount received by each employee. (ii) By what percent should the bonus be increased so that each employee can receive Rs 38,400? (iii) What should be the profit of the company so that it can provide Rs 40,000 to each at 15% bonus? Vedanta Excel in Mathematics Teachers' Manual - 9 16 Solution: Here, net profit of the company = Rs 48,00,000, bonus percent = 18% number of employees = 25 (i) Bonus amount =18% of Rs 48,00,000 = Rs 8,64,000 Bonus amount received by each employee = Rs 8,64,000 = Rs 34,500 25 Hence, each employee received Rs 34,500. (ii) Net profit of the company =Rs 48,00,000, number of employees = 25, bonus amount received by each employee= Rs 38,400, bonus percent = ? Bonus amount =25× Rs 38,400 = Rs 9,60,000 Bonus percent = Bonus amount × 100% = Rs 9,60,000 × 100% = 20% Net profit Rs 48,00,000 Hence, the bonus should be increased by 20% - 18% = 2% so that each employee can receive Rs 38,400. (iii) Bonus percent= 15%, number of employees = 25, bonus amount received by each worker = Rs 40,000 Now, bonus amount =25× Rs 40,000 = Rs 10,00,000 Let net profit of the factory be Rs x Then, bonus amount = bonus % of net profit or, Rs 10,00,000 = 15% of x = 0.15x ? x = Rs 66,66,666.67 Hence, the profit of the factory is Rs 66,66,666.67 5. When a publication house increased its profit from 20% to 25%, the amount of profit increased to Rs 52,08,000. If the company decided to distribute 60% bonus to its 30 employees equally from the increased amount of profit, how much bonus will each employee receive? Solution: Let the yearly income of the publication house be Rs x Then, 25% profit of yearly income = Rs 52,08,000 or, 0.25x = Rs 52,08,000 ? x = Rs 2,08,32,000 Also, 20% profit of the yearly income = 20% of Rs 2,08,32,000 = Rs 41,66,400 Increased amount of profit = Rs 52,08,000 – Rs 41,66,400 = Rs 10,41,600 Now, bonus amount = 60% of Rs 10,41,600 = Rs 6,24,960 ?Bonus amount received by each employee = Rs 6,24,960 = Rs 20,832 30 Hence, each employee received Rs 20,832. Extra Questions 1. A noodle factory announced to distribute 10% bonus equally to its 45 employees from the net profit of Rs 34,65,000 at the end of a fiscal year, find the bonus received by each employee. [Ans:Rs 7700] 2. A garment factory announced 20% bonus to its 70 workers from the net profit at the end of last fiscal year. If every worker received Rs 15,000, how much was the profit of the factory? [Ans:Rs 52,50,000] 3. The management of a supermarket decided to distribute a bonus to its 180 employee from the net profit of Rs 3,00,00,000 at the end of a fiscal year. If every employee received Rs 25,000, what was the bonus percent? [Ans:15%] 17 Vedanta Excel in Mathematics Teachers' Manual - 9 B. Commission Teaching Activities 1. With story/ related examples on commission, explain the words sales, agent and commission. 2. Define commission as the amount of money paid to the agent for performing the business service such as buying and selling goods, property (land, building, car etc.) or collection of money. 3. Under discussion, explain the following formulae (i) Commission amount = commission % of total sales (ii) Commission percent = Commission amount × 100% Total sales Monthly income = salary+ commission Solution of selected problems from Vedanta Excel in Mathematics 1. A real estate company gives 5% commission on selling a piece of land for Rs 10,00,000 and 7% commission for the additional amount of selling price above the fixed price. If the agent sold the land for Rs 12,99,000, how much commission did he/she receive from the company? Solution: Here, the fixed selling price of the land = Rs 10,00,000 The selling price of the land = Rs 12,99,000 Now, the commission received by the agent = 5% of Rs 10,00,000 + 7% of (Rs 12,99,000 – Rs 10,00,0000) = Rs 50,000 + Rs20,930 = Rs70,930 Hence, the agent received the commission of Rs 70,930. 2. The monthly salary of a sales person of a subway restaurant is Rs 21,600 and an additional incentive of 1.5% on the total monthly sale is provided as commission. (i) Calculate his/her total income in a month if he/she makes a total sale of Rs 5,80,000 in the month. (ii) What should be his/her total sale in the next month so that he/she can receive a total income of Rs 31,350 in the month? Solution: Here, the monthly salary of a sales person = Rs 21,600 (i) The total sales of the month = Rs 5,80,000 and commission percent = 1.5% Now, the commission received by the sales person = 1.5% of Rs 5,80,0000 = Rs 8,700 Hence, the income of the sales person in the month = salary + commission = Rs 21,600 + Rs 8,700 = Rs 30,300 (ii) The income of the next month = Rs 31,350 ? Commission amount received in the next month = Rs 31,350 – Rs 21,600 = Rs 9,750 Let the total sales of the next month be Rs x. Then, the commission amount = commission % of total sales or, Rs 9,750 = 1.5% of x or, Rs 9,750 = 0.015x ? x = Rs 6,50,000 Hence, the total sales of the next month should be Rs 6,50,000 so that he/she can receive a total income of Rs 31,350 in the month. 3. Mr. Bibek is an online salesperson in an online shopping store. His monthly salary Rs 18,700 and 2% commission is given to him when the monthly sales is more than 5 lakh rupees. If the sales of the store in a month is Rs 7,20,000, calculate his total income of the month. Vedanta Excel in Mathematics Teachers' Manual - 9 18 Solution: Here, the monthly salary of a salesperson = Rs 18,700 The fixed sales of the month = 5 lakh rupees = Rs 5,00,000 The total sales of the month = Rs 7,20,000 The sales eligible for commission = Rs 7,20,000 – Rs 5,00,000 = Rs 2,20,000 Now, the commission received by the sales person = 2% of Rs 2,20,0000 = Rs 4,400 Hence, the income of the sales person in the month = salary + commission = Rs 18,700 + Rs 4,400 = Rs 23,100 4. Mrs. Nepali draws Rs 19,800 as her monthly salary in a wholesale cosmetic shop and a certain commission is given as per the monthly sales. If the sales of the month is Rs 12,00,000 and her total income of the month including commission is Rs 31,800, find the rate of commission. Solution: Here, monthly salary = Rs 19,800 and the income of the month = Rs 31,800 ?Commission amount received in the next month = Rs 31,800 – Rs 19,800 = Rs 12,000 The total sale of the month = Rs 12,00,0000 Now, commission percent = Commission amount × 100% = Rs 12,000 × 100% = 1% Total sales Rs 12,00,000 Hence, the rate of the commission is 1%. Extra Questions 1. Mr. Shrestha is a salesperson in a hardware shop. His monthly salary Rs 22,500 and 1.5% commission is given to him when the monthly sales is more than 10 lakh rupees. If the sales of the hardware shop in a month is Rs 17,50,000, calculate his total income of the month. [Ans: Rs 33,750] 2. Aravi draws Rs 15,000 as her monthly salary in a wholesale fancy shop and a certain commission is given as per the monthly sales. If the sales of the month is Rs 10,00,000 and her total income of the month including commission is Rs 27,500, find the rate of commission. [Ans:1.25%] 3. The monthly salary of Sanjay, a salesman of a departmental store, is Rs 25,000 and an additional incentive of 1.5% on the total monthly sale is provided as commission. (i) Find his total income in a month if he makes a total sale of Rs 5,55,000 in the month. (ii) What should be his total sale in the next month so that he can receive a total income of Rs 38,320 in the month? [Ans:Rs 33,250;Rs 8,88,000] C. Income Tax Teaching Activities 1. Ask about the yearly income of parents of the student and discuss upon the tax is to be paid to the government. 2. Divide the students into groups and give them to study the printed form of the present rates of income taxes fixed by Inland Revenue Department (IRD) and discuss upon the following questions (i) What is income tax? (ii) Which authentic body is responsible to collect the tax? (iii) Why should we pay tax to the government? (iv) What do you mean by taxable income? (v) Which incomes are entitles for tax rebate? (vi) What is the rate of social security tax? 19 Vedanta Excel in Mathematics Teachers' Manual - 9 3. Present the income tax slabs for individual and married couple separately on a chart paper. 4. Tell the students to write the important notes and formulae on the colourful chart paper as project work 5. With some related examples, let the students identify the following formulae then explain them one by one. (i) Taxable income = Yearly income – tax free income (ii) Income tax = rate of tax of taxable income (iii) Tax rate = Income tax × 100% Taxable income Solution of selected problems from Vedanta Excel in Mathematics 1. Mr Sunil Jha is a Secomdary level Mathematics Teacher in a community school. His monthly salary is Rs 38,700 and one month’s salary as Dashain Bonus. 10% of his salary is deducted to deposit in his provident fund. If his marital status is single, calculate the annual income tax paid by him. Solution: Here, his monthly income after deducting the provident fund = Rs 38,700 – 10% of Rs 38,700 = Rs 34,830 His annual income with Dashain bonus = 12—Rs 34,830 + Rs 38,700 = Rs 4,56,660 Social security tax up to Rs 4,00,000 = 1% of Rs 4,00,660 = Rs 4,000 Taxable income = Rs 4,56,660 – Rs 4,00,000 = Rs 56,660 Income tax for Rs 56,660 = 10% of Rs 56,660 = Rs 5,666 Hence, the total annual income tax paid by him = Rs 4,000 + Rs 5,666 = Rs 9666 2. After deducting 10% provident fund a married person draws Rs 40,500 salary per month and one month’s salary as festival bonus. He/she pays Rs 14,500 annually as premium of his/her insurance. Calculate the annual income tax paid by the person. Solution: Here, his monthly income after deducting the provident fund = Rs 40,500 Let the monthly income of the person be Rs x. Then, x – 10% of x = Rs 40,500 ?x = Rs 45000 His annual income with Dashain bonus = 12—Rs 40,500 + Rs 45,000 = Rs 5,31,000 Taxable income after premium of insurance = Rs 5,31,000 – Rs 14,500 = Rs 5,16,500 Social security tax up to Rs 4,50,000 = 1% of Rs 4,50,660 = Rs 4,500 Taxable income for 10% tax = Rs 5,16,500 – Rs 4,50,000 = Rs 66,500 Income tax for Rs 66,500 = 10% of Rs 66,500 = Rs 6,650 Hence, the total annual income tax = Rs 4,500 + Rs 6,650 = Rs 11,150 3. Mr. Sayad Sharma an unmarried employee of a UN Project draws monthly salary of Rs 51,000 after deducting 10% salary in his provident fund and 5% in citizen investment trust. He also receives a Dashain Bonus of one month’s salary. He pays Rs 22,000 annually as the premium of his life insurance. How much income tax does he pay in a year? Solution: Here, his monthly income after deducting 10% provident fund and 5% citizen investment trust = Rs 51,000 Let his monthly income before deducting the provident fund and citizen investment trust be Rs x. Vedanta Excel in Mathematics Teachers' Manual - 9 20 Then, x – (10 + 5) % of x = Rs 40,500 ?x = Rs 60,000 His annual income with Dashain bonus = 12—Rs 51,000 + Rs 60,000 = Rs 6,72,000 Taxable income after premium of insurance = Rs 6,72,000 – Rs 22,000 = Rs 6,50,000 Social security tax up to Rs 4,00,000 = 1% of Rs 4,00,660 = Rs 4,000 Taxable income for 10% tax = Rs 5,00,000 – Rs 4,00,000 = Rs 1,00,000 Income tax for Rs 1,00,000 = 10% of Rs 1,00,000 = Rs 10,000 Taxable income for next 20% tax = Rs 6,50,000 – Rs 5,00,000 = Rs 1,50,000 Income tax for Rs 1,50,000 = 20% of Rs 1,50,000 = Rs 30,000 Hence, the total annual income tax = Rs 4,000 + Rs 10,000 + Rs 30,000 = Rs 44,000. Extra Questions 1. The monthly income of an unmarried individual is Rs 45,000. If 1% social security tax is charged up to Rs 4,00,000. Then 10% and 20% are charged for the next Rs1,00,000 and Rs 2,00,000 respectively. Calculate the annual income tax paid by the individual. [Ans: Rs 22,000] 2. Rumakanta Jha is a married professor. His monthly income is Rs 55,000. If 1% social security tax is charged up to Rs 4,50,000; 10% tax for the income from Rs 4,50,000 to Rs 5,50,000 and 20% tax from Rs 5,50,000 to Rs 7,50,0000 are to be paid. Calculate the annual income tax paid by the individual. [Ans: Rs 36,500] 3. Mrs Pandey is a Branch Manager of a commercial bank. Here monthly salary is Rs 95,400 and 10% of her salary is deducted as provident fund. She pays Rs 24,520 as the premium of her life insurance. If 1% social security tax is levied upon the income of Rs 4,50,000, 10%,20% and 30% taxes are levied upon the next incomes of Rs 1,00,000, Rs2,00,000 and up to Rs 12,50,000 respectively, how much income tax should she pay in a year? [Ans: Rs 1,31,240] D. Dividend Teaching Activities 1. Recall the bonus. 2. Create a short story about dividend and tell in the class. 3. Define dividend as the certain amount distributed among the shareholders of a corporation as per the number of shares from the net profit. 4. Ask the following questions during classroom discussion (i) What is dividend? (ii) Among whom the dividend is distributed? (iii) Tell the difference between the bonus and the dividend. 5. Make a discussion upon the following formulae with examples. (i) Dividend = Rate of dividend (in %) — Net profit (ii) Dividend = Value of dividend per share — Total number of shares (iii) Value of dividend for each share = Dividend amount No. of shares (iv) Rate of dividend (in %) = Dividend amount × 100% No. of shares 21 Vedanta Excel in Mathematics Teachers' Manual - 9  Solution of selected problems from Vedanta Excel in Mathematics 1. Mrs Rai bought 250 shares out of 10,000 shares from a financial company. The company earned a net profit of Rs 85,20,000 and declared 17% dividend to its shareholders. Calculate the amount of dividend received by Mrs Rai. Solution: Here, dividend amount = 17% of Rs 85,20,000 = Rs 14,48,400 Value of dividend for each share = Dividend amount = Rs 14,48,400 = Rs 144.84 No. of shares 10,000 ?Dividend for 250 shares = 250 —Rs 144.84 = Rs 36,210 Hence, Mrs Rai received Rs 36,210 as dividend. 2. A Cable Car Company sold 3000 shares to the local people. The company earned a profit of Rs 1,20,00,000 in a year and distributed a certain percent of profit as dividend. If a shareholder who has bought 125 shares received Rs 1,10,000 dividend, what percent of profit was distributed as dividend? Solution: Here, dividend amount distributed for 1 share = Rs 1,10,000 = Rs 880 125 ?Dividend amount distributed for 3000 shares = 3000—Rs880 = Rs 26,40,000 Net profit = Rs 1,20,00,000 Now, rate of dividend = Dividend amount × 100% = Rs 26,40,000 × 100% = 22% Net profit Rs 1,20,00,000 Hence, 22 of the profit was distributed as dividend. 3. Mr. Dhurmus bought 500 shares out of 10,000 shares sold by a commercial bank. The bank earned some profit and distributed 14% of the net profit as the dividend in a year. If Dhurmus received Rs 1,03,600 in the year, find the net profit of the bank. Solution: Here, dividend received for 500 shares = Rs 1,03,600 or, dividend received for 1 share = Rs 1,03,600 = Rs 207.20 500 ?Dividend distributed for 10,000 shares = 10,000 — Rs 207.20 = Rs 20,72,000 Let the net profit of a year of the commercial bank be x. Then, dividend amount = Rate of divided — Net profit or, Rs 20,72,000 = 14% of x ?x = Rs 1,48,00,000 Hence, the net profit of the bank was Rs 1,48,00,000 Extra Questions 1. Mr Dahal bought 400 shares out of 20,000 shares from a Business Company. The company earned a net profit of Rs 2,25,00,000 and declared 15% dividend to its shareholders. Calculate the amount of dividend received by Mr Dahal. [Ans: Rs67,500] 2. Mr Jeevan bought 225 shares out of 10,000 shares from commercial bank. If the company earned a profit of Rs 1,50,00,000 in a year and distributed a certain percent of profit as dividend. If Mr Jeevan received Rs 54,000 dividend, what percent of profit was distributed as dividend? [Ans:16%] 3. Rajesh bought 300 shares out of 5,000 shares sold by an insurance company. The company earned some profit and distributed 20% of the net profit as the dividend in a year. If Rajeshs received Rs 66,600 in the year, find the net profit of the bank. [Ans:Rs 55,50,000] Vedanta Excel in Mathematics Teachers' Manual - 9 22  Unit Household Arithmetic 4 Allocated teaching periods 10 Competency - To solve the daily life problems by using the basic rules of household arithmetic and financial arithmetic. Learning Outcomes - To solve the problems on household arithmetic such as electricity bills, telephone bills, water bills, taxi meter reading (including discount and VAT) Level-wise learning objectives S.N. LEVELS OBJECTIVES - To define 1 unit of electricity - To tell the formula of finding the consumed units - To tell the formula of finding the tariff (Sub-total) - To write the relation of total charge, tariff, TSC and VAT 1. Knowledge (K) - To define 1 unit of water consumption - To name the concerned authority of ministry of water supply - To name the concerned authority for implement the rules and regulations regarding taximeter. - To recall the minimum taxi fares during 6:00 am to 9:00 pm or 9:00 pm to 6:00 am. - To find the electricity charge for simple problems 2. Understanding (U) - To calculate the telephone charge - To calculate the taxi fare - To find the electricity charge with rebate/fine - To find the number of consumed units of electricity 3. Application (A) - To apply the rules and find the water charge - To find the telephone charge with TSC and VAT - To estimate the consumed unit of electricity of a 4. High Ability (HA) month and find the bill amount as per present rate of electricity - To prepare report on few ways of reducing unnecessary use of electricity/telephone/water and use of ICT used for payment of the bills Required Teaching Materials/ Resources Electricity bills, models of meter box, telephone bills, water bills, calculator, recharge card, chart paper mentioning the taxi fare rates etc. 23 Vedanta Excel in Mathematics Teachers' Manual - 9 Pre-knowledge: discount, VAT A. Electricity bills Teaching Activities 1. Ask about the electric appliances used in the students’ houses. 2. With some samples/models of meter readings of two successive months, ask the following questions: (i) What is the reading of recent month? (ii) What is the reading of preceding month? (iii) What is the number of consumed units of electricity? (iv) What do you mean by 1 unit of consumed electricity? 3. Present the recent rates, rules and regulations of electricity fixed by Nepal Electricity Authority (NEA) on chart paper or power point presentation or though available website. 4. Encourage the students to find the electricity charge with electricity service charge and rebate/fine. Solution of selected problems from Vedanta Excel in Mathematics 1. The rate of electricity charge up to 20 units is Rs 3 per unit and Rs 7 per unit from 21 to 30 units. Find the charge of consumption of 28 units with Rs 50 service charge. Solution: Here, consumption of electricity = 28 units Rate of charge up to 20 units = Rs 3 per unit ?Charge up to 20 units = 20 ×Rs 3 = Rs 60 The excessive number of units = 28 – 20 = 8 units ?Charge of excessive 8 units = 8 ×Rs 7 = Rs 56 Total charge of electricity with service charge = Rs 60 + Rs 56 + Rs 50 = RS 166 2. Electricity tariff rates and rebate/fine rules are given below kWh 5 Ampere 15 Ampere 30 Ampere 60 Ampere (Monthly) Service Energy Service Energy Service Energy Service Energy Units Charge Charge Charge Charge Charge Charge Charge Charge per unit per unit per unit per unit 0 – 20 Rs30.00 Rs 3.00 Rs50.00 Rs4.00 Rs75.00 Rs5.00 Rs125.00 Rs6.00 The rules of rebate/fine Meter Within 7 Within 8-22 Within 23-30 Within 31-40 Within 41-60 reading days days days days days Rebate/fine 3% rebate - 5% fine 10% fine 25% fine From the tables given above, workout the following problems a) A household having 5A electricity meter consumed 18 units of electricity in one month. Find the amount of payment made by the household within 7 days. b) A household having a 15A meter consumed 16 units of electricity in one month. Find the amount of payment made by the household on 25th day of meter reading. c) A household having a 30A meter consumed 19 units of electricity in one month and if the payment was made on 20th day of meter reading, find the amount of payment. Vedanta Excel in Mathematics Teachers' Manual - 9 24 d) A household having a 60A meter consumed 20 units of electricity in one month and the payment was made on 50th day of meter reading. Calculate the amount of payment made by the house. Solution: (a) Here, capacity of meter box = 5A, consumption of electricity = 18 units Rate of charge up to 20 units = Rs 3 per unit Now, charge for 18 units = 18×Rs 3 = Rs 54 ? Total charge of electricity with service charge = Rs 54 + Rs 30 = Rs 84 Since, the payment was made within 7 days of meter reading. So, 3% rebate was allowed. Hence, the required payment was Rs 84 – 3% of Rs 84 = Rs 81.48 (b) Here, capacity of meter box = 15A, consumption of electricity = 16 units Rate of charge up to 20 units = Rs 4 per unit Now, charge for 16 units = 16×Rs 4 = Rs 64 ? Total charge of electricity with service charge = Rs 64 + Rs 50 = Rs 114 Since, the payment was made on 25th day of meter reading. So, 5% fine was charged. Hence, the required payment was Rs 114 + 5% of Rs114 = Rs 119.70 (c) Here, capacity of meter box = 30A, consumption of electricity = 19 units Rate of charge up to 20 units = Rs5 per unit Now, charge for 19 units = 19×Rs5 = Rs95 ? Total charge of electricity with service charge = Rs95 + Rs75 = Rs170 Since, the payment was made on 20th day of meter reading. So, there was no rebate no fine. Hence, the required payment was Rs170. (d) Here, capacity of meter box = 60A, consumption of electricity = 20 units Rate of charge up to 20 units = Rs 6 per unit Now, charge for 20 units = 20 ×Rs 6 = Rs 120 ? Total charge of electricity with service charge = Rs 120 + Rs 125 = Rs 245 Since, the payment was made on 50th day of meter reading. So, 25% fine was charged. Hence, the required payment was Rs 245 + 25% of Rs 245 = Rs 306.25 3. Mr. Sharma has a 5A meter in his house. He uses 5 CFL bulbs of 15 watt each for 4 hours and an electric heater of 1200 watt for 1 hour every day. Find the cost of payment of the bill of the month at the rate of Rs 3 per unit up to 20 units, Rs 7 per unit from 21 to 30 units and Rs 8.50 from 31-50 units with Rs 75 service charge, if the payment is made on 10th day of the meter reading. Solution: Here, consumption of electricity in 1 day = (5 × 15×4 + 1200×1) watts = 1500 watts Consumption of electricity in 1 month = 30×1500 watts = 45000 watts = 45 kW ? Number of consumed units = 45 units because the time duration of use of electric appliances were measured in hours. Now, Consumption block No. of units Rate of charge Electricity charge 0 – 20 20 – 0=20 Rs 3 20×Rs 3 = Rs 60 21 – 30 30 – 20 = 10 Rs 7 10×Rs 7 = Rs 70 31 – 50 45 – 30 = 15 Rs 8.50 15×Rs 8.50 = Rs 127.50 25 Vedanta Excel in Mathematics Teachers' Manual - 9 Total charge of electricity with service charge = energy charge + service charge = Rs 60 + Rs70 + Rs 127.50 + Rs 75 = Rs 332.50 Since, the payment was made on 10th day of meter reading. So, there was neither rebate nor fine. Hence, the required payment was Rs332.50 4. Mrs.Bajracharya’s house has a 15A meter. She uses 5 LED bulbs of 10 watt each for 4 hours, 2 televisions of 60 watt each for 5 hours and a refrigerator of250 watt for 2 hours every day. Find the cost of payment of the bill of the month at the rate of Rs4 per unit up to 20 units, Rs 7 per unit from 21 to 30 units and Rs 8.50 from 31-50 units with Rs100 service charge, if the payment is made on 35th day of the meter reading. Solution: Here, consumption of electricity in 1 day = (5×10×4 + 2×60×5 + 1×250×2) watts = 1300 watts Consumption of electricity in 1 months = 30×1300 watts = 39000 watts = 39 kW ? Number of consumed units = 39 units because the time duration of use of electric appliances were measured in hours. Now, Consumption No. of units Rate of charge Electricity charge block 0 – 20 20 – 0 = 20 Rs4 20×Rs4 = Rs 80 21 – 30 30 – 20 = 10 Rs 7 10×Rs 7 = Rs 70 31 – 50 39– 30 = 9 Rs 8.50 9×Rs 8.50 = Rs76.50 Total charge of electricity with service charge = energy charge + service charge = Rs 80 + Rs 70 + Rs76.50 + Rs100 = Rs 326.50 Since, the payment was made on 35th day of meter reading. So, 10% fine was added Hence, the required payment was Rs326.50 + 10% of Rs 326.50 = Rs 359.15 5. The meter box of a family house is 15 A. If the family made the payment of Rs 1336.50 with service charge of Rs125 on 36th day of meter reading, how many units of electricity was consumed in the month? Calculate it under the following rates. Units 0 – 20 21 – 30 31 – 50 51 – 150 Rate of charge per unit Rs 4 Rs 7 Rs 8.50 Rs 11 Payment up to 40 day from the meter reading – 10% fine th Solution: Let the number of consumed units in the month be x units Now, Consumption block No. of units Rate of charge Electricity charge 0 – 20 20 – 0 = 20 Rs 4 20×Rs 4 = Rs 80 21 – 30 30 – 20 = 10 Rs 7 10×Rs 7 = Rs 70 31 – 50 50 – 30 = 20 Rs 8.50 20×Rs 8.50 = Rs170 51 – 150 x – 50 Rs 11 (x – 50)×Rs 11 = Rs (11x – 550) Total charge of electricity with service charge = energy charge + service charge Vedanta Excel in Mathematics Teachers' Manual - 9 26  = Rs 80 + Rs 70 + Rs170 + + 11x – Rs 550 + Rs 125 = Rs(11x – 105) Also, extra fine = 10% of Rs (11x – 105) = Rs (1.1x – 10.5) According to question, Payment of bill with fine = Rs 1336.50 or,Rs (11x – 105) + Rs (1.1x – 10.5) = Rs 1336.50 ?x = 120 Hence, the required payment number of consumed units is 120. Extra Questions 1. The rate of electricity charge up to 20 units is Rs 3 per unit and Rs 7 per unit from 21 to 30 units. Find the charge of consumption of 24 units with Rs 50 service charge. [Ans:Rs138] 2. The rate of electricity charge up to 20 units is Rs 3 per unit and Rs 7 per unit from 21 to 30 units. If a family paid the bill of Rs 166 with Rs 50 service charge, how many units of electricity was consumed in the month? [Ans: 28] 3. The meter readings of Krishna’s house in 1 Asar was 02967 and 1 Sharwan was 03015. st st Find the electricity charge for the month of Asar according to the given information if the bill was made on 5th day from meter reading. KWh (units) Service charge Energy charge per unit 0 – 20 Rs 30 Rs3 21 – 30 Rs 50 Rs 7 31 – 50 Rs 75 Rs 8.50 Payment within 7 days of meter reading – 3% rebate [Ans: Rs347.26] 4. The meter box of a family house is 15 A. If the family made the payment of Rs1331 with service charge of Rs125 on 36th day of meter reading, how many units of electricity was consumed in the month? Calculate it under the following rates. Units 0 – 20 21 – 30 31 – 50 51 – 150 Rate of charge per unit Rs 4 Rs 7 Rs 8.50 Rs 11 Payment up to 40th day from the meter reading – 10% fine [Ans: 110 units] B. Telephone bills Teaching Activities 1. With a recharge card, discuss about the mobile network service 2. Provide some telephone bills to the students, and ask the following questions: (i) What is concerned authority for the implementation of this bill? (ii) To whom is the telephone bill issued? (iii) What is the address of the telephone line? (iv) What type of telephone is it? (v) What status of telephone is mentioned? (vi) What type of telephone is it? (vii) What is the previous reading of telephone given in the bill? (viii) What is the current reading of telephone given in the bill? (ix) How many telephone calls is made in the month? (x) What is the rental amount? (xi) How many extra calls are made in the month? 27 Vedanta Excel in Mathematics Teachers' Manual - 9  (xii) What is the sub-total amount given in the bill? (xiii) What amount is to be paid for TSC? (xiv) What amount of VAT is to be paid? (xv) What is the grand total given in the bill? 3. Explain the telephone billing system implemented by Nepal Doorsanchar Company Ltd. 4. List the following formulae after discussion (i) Tariff = Sub-total = Minimum charge + Extra charge (ii) Telecom service charge (TSC) = 10% of Sub-total (iii) Total = Sub-total + TSC (iv) VAT amount = VAT% of Total = 13% of Sub-total (v) Grand total = Total + VAT amount Solution of selected problems from Vedanta Excel in Mathematics 1. The minimum charge up to 175 calls is Rs 200. If the charge for each additional call is Re 1, how much will be the charge for 475 calls with 10% TSC and 13% VAT? Solution: Here, minimum charge up to 175 calls = Rs 200 The additional number of calls = 475 – 175 = 300 The charge for additional calls = 300 × Re 1 = Rs 300 Now, sub-total = Minimum charge + additional charge = Rs 200 + Rs 300 = Rs 500 TSC = 10% of Rs 500 = Rs 50 Also, total = sub-total + TSC = Rs 500 + Rs 50 = Rs 550 Again, grand-total = Total + VAT% of Total = Rs 550 + 13% of Rs 550 = Rs 621.50 Hence, the total charge for 375 calls is Rs 621.50 2. The minimum charge of telephone calls up to 175 calls is Rs 200. The charge for each extra call of 2 minutes duration is Re 1. If the household paid Rs 633.93 with 10% TSC and 13% VAT to clear the bill of the month, find the total number of calls made in the month? Solution: Let the charge of telephone calls without VAT be Rs x. Then, grand total = Total + VAT% of total or, Rs 633.93 = x + 13% of x ? x = Rs 561 Also, let the charge of telephone calls without TSC be Rs y. Then, total = sub-total + TSC or, Rs 561 = y + 10% of y ? y = Rs 510 Again, the charge for extra calls = Rs 510 – Rs 200 = Rs 310 Now, the number of extra calls = Rs 310 = 310 Re 1 Hence, the total number of calls of the month is 175 + 310 = 485 calls Extra Questions 1. The minimum charge up to 175 calls is Rs 200. If the charge for each additional call is Re 1, how much will be the charge for 375 calls with 10% TSC and 13% VAT? [Ans: Rs 497.20] 2. The reading of the Baishakh-1 of local calls in Supriya’s house is 5270 and that of the Jestha-1 is 5605.The minimum charge up to 175 calls is Rs 200 and charge for each additional call is Re 1. Vedanta Excel in Mathematics Teachers' Manual - 9 28  (i) How many calls are made in the month of Baishakh? (ii) What is the bill amount of telephone callswith 10% TSC and 13% VAT? [Ans: (i) 635 calls, (ii) Rs 820.38] 3. The minimum charge of telephone calls up to 175 calls is Rs 200. The charge for each extra call of 2 minutes duration is Re 1. If the household paid Rs 559.35 with 10% TSC and 13% VAT to clear the bill of the month, find the total number of calls made in the month? [Ans: 425 calls] C. Water bills Teaching Activities 1. Provide some water bills to the students, and ask the following questions: (i) What is concerned authority for the implementation of this bill? (ii) How many units of water is consumed? (iii) What is the size of pipe? (iv) What is the previous reading of water consumption? (v) What is the present reading of water consumption? (vi) What is the minimum charge? (vii) What is the additional charge? (viii) What is the total charge? 2. Discuss about 1 unit of water consumption 3. Show the water tariff rules in the chart paper implemented by Nepal Water Supply Corporation. S.N. Size of pipe Tap with meter Taps without meter Minimum Minimum Additional Main Tap Branch Tap consumption Charge consumption per Charge(Rs) Charge (Rs) (litre) 1000 litre (Rs) 1. 1" 10,000 110 25 560 200 2 2. 3" 27, 000 1490 40 3360 1600 4 3. 1" 56,000 3420 40 9200 2700 The compulsory provision of sewerage service charge = 50% of the water consumption charge. Payment schedule of the bill S.N. Payment is made after the bill issued Rebate/ Fine 1. Within the 1st and the 2nd month 3% rebate 2. Within the 3rd month No rebate and no fine 3. Within the 4th month 10% fine 4. Within the 5th month 20% fine 5. After 5th month 50% fine 4. Present the water tariff rules implemented by Kathmandu Upatyaka Khanepani Limited (KUKL)in the chart paper. 29 Vedanta Excel in Mathematics Teachers' Manual - 9  Size of Taps without S.N. Tap with meter pipe meter Minimum Minimum Additional Branch Tap consumption Charge consumption per Charge (Rs) (litre) 1000 litre (Rs) 1. 1" 10,000 100 32 785 2 2. 3" 27, 000 1910 71 4595 4 3. 1" 56,000 3960 71 9540 Solution of selected problems from Vedanta Excel in Mathematics 1. 127 units of water is consumed by using 3 "pipe in a hotel. If the payment of the bill 4 is made within the fifth month after the bill issued, how money is required to clear the bill with 50% sewerage service charge? Solution: Here, According to the water tariff provisions of NWS for the use of pipe of the size 3 " 4 The minimum charge up to 27000 litres i.e., 27 units = Rs1490 The charge of additional units = Rs 40 per unit The additional number of units = 127 – 27 = 100 units The charge for additional units = 100 × Re 40 = Rs4000 Now, total charge = Rs 1490 + Rs 4000 = Rs 5,490 Again, the charge including 50% sewerage service = Rs 5490 + 50% of Rs 5490 = Rs 8235 2. A household uses 1 " of water pipe. The meter reading of the household on 1st of Asar 2 was 1260 units and on 1st of Shrawan was 1330 units. Calculate the charge to be paid including 50% sewerage service charge if the payment of the bill is made in the following schedule. (i) Within the first month after the bill issued (ii) Within the third month after the bill issued (iii) Within the fifth month after the bill issued (iv) Within the seventh month after the bill issued Solution: Here, The meter reading of 1st Asar = 1260 and that of 1st Shrawan = 1330 ? Consumed units of water = 1330 – 1260 = 70 According to the water tariff provisions of NWS for the use of pipe of the size 1 " 2 The minimum charge up to 10000 litres i.e., 10 units = Rs110 The charge of additional units = Rs 25 per unit The additional number of units = 70 – 10 = 60 units The charge for additional units = 60 × Re 25 = Rs1500 Now, Vedanta Excel in Mathematics Teachers' Manual - 9 30 Total charge = Rs110 + Rs 1500 = Rs 1610 Again, the charge including 50% sewerage service = Rs 1610 + 50% of Rs 1610 = Rs 2415 (i) When the payment of the bill is made within the first month after the bill issued, 3% rebate is allowed ? Required payment = Rs 2415 – 3% of Rs 2415 = Rs 2342.55 (ii) When the payment of the bill is made within the third month after the bill issued, there is no rebate no fine. ? Required payment = Rs 2415 (iii) When the payment of the bill is made within the fifth month after the bill issued 20% fine is charged ? Required payment = Rs 2415 + 20% of Rs 2415 = Rs 2898 (iv) When the payment of the bill is made within the seventh month after the bill issued, 50% fine is charged ?Required payment = Rs 2415 + 50% of Rs 2415 = 3622.50 Extra Questions 1. 18 units of water is consumed by using 1 "pipe in Rameshwor’s house. If the payment of 2 the bill is made within the second month after the bill issued, how money is required to clear the bill with 50% sewerage service charge? [Ans:Rs 451.05] 3 2. 147 units of water is consumed by using inch pipe in Everest Hotel. If the payment 4 of the bill is made within the fourth month after the bill issued, how much amount is required to pay as bill? [Ans: Rs 10,378.50] 3. A household uses 1 " of water pipe. The meter reading of the household on 1st of Kartik 2 was 1420 units and on 1st of Mansir was 1480 units. Calculate the charge to be paid including 50% sewerage service charge if the payment of the bill is made in the following schedule. (i) Within the second month after the bill issued (ii) Within the third month after the bill issued (iii) Within the fifth month after the bill issued (iv) Within the sixth month after the bill issued [Ans: (i) Rs 1076.70, (ii) Rs 1110, (iii) 1332 (iv) 1665] D. Taxifare Teaching Activities 1. Discuss upon taxi fare paid by parents or students themselves 2. Present the rules and regulations implemented and monitored by Nepal Bureau of Standards and Metrology (NBSM) regarding the taxi fare as shown in the following table. Time Minimum fare Fare of per Waiting charge per 200 meters 2 minutes 6:00 am to 9:00 pm Rs 14 Rs 7.20 Rs 7.20 9:00 pm to 6:00 am Rs 21 Rs 10.80 Rs 10.80 31 Vedanta Excel in Mathematics Teachers' Manual - 9  Solution of selected problems from Vedanta Excel in Mathematics 1. Mr. Kattel travelled 15 km by a hired taxi at 5:00 am. The minimum fare of Rs 21 appeared immediately after the meter was flagged down. Then, the fare went on at the rate of Rs 10.80 per 200 metres. An additional waiting charge of Rs 10.80 per 2 minutes was charged for waiting of 10 minutes. Calculate the total of the taxi fare paid by him. Solution: Here, the minimum fare = Rs 21, distance travelled = 15 km = 15000 m Now, The fare of 200 metres = Rs 10.80 or, the fare of 1 m = Rs 0.054 ? The fare of 15000 m = 15000×Rs 0.054 = Rs 810 Also, Waiting charge of 2 minutes = Rs 10.80 ? Waiting charge of 10 minutes = Rs 54 Hence, the total fare = Rs 21 + 810 + Rs 54 = Rs 885 2. Rita hired taxi and travelled a certain distance at 8:00 a.m. She paid the total fare of Rs 194. If the minimum fare is Rs 14 and the fare per 200 metres is Rs 7.20, find the distance travelled by her. Solution: Here, minimum charge = Rs 14 ? The taxi fare excluding minimum fare = Rs 194 – 14 = Rs 180 Now, Rs 7.20 is the fare of 200 metres or, Re 1 is the fare of 200 metres 7.20 ? Rs 180 is the fare of 200 × 180 = 5000 meters = 5 km 7.20 Hence, she travelled 5 km. Extra Questions 1. Mr. Koirala travelled 7 km by a hired taxi at 1:30 p.m. The minimum fare of Rs 14 appeared immediately after the meter was flagged down. Then, the fare went on at the rate of Rs 7:20 per 200 metres. Calculate the total of the taxi fare paid by him. [Ans: Rs 266] 2. Mrs. Maharjan travelled 10 km by a hired taxi at 4:45 am. The minimum fare of Rs 21 appeared immediately after the meter was flagged down. Then, the fare went on at the rate of Rs 10.80 per 200 metres. An additional waiting charge of Rs 10.80 per 2 minutes was charged for waiting of 10 minutes. Calculate the total of the taxi fare paid by him. [Ans: Rs 615] 3. Smriti hired taxi and travelled a certain distance at 7:15 a.m. She paid the total fare of Rs 158. If the minimum fare is Rs 14 and the fare per 200 metres is Rs 7.20, find the distance travelled by her. [Ans: 4 km] 4. Rupesh hired taxi and travelled a certain distance at 10:00 p.m. He paid the total fare of Rs 1155 including the waiting charge of 10 minutes. Find the distance travelled by him. [Ans: 4 km] Vedanta Excel in Mathematics Teachers' Manual - 9 32  Unit Mensuration 5 Competency Allocated teaching periods 22 - To find the area of plane surface and surface area and volume of solids then solving real life problems based on cost estimation. Learning Outcomes - To estimate the cost of carpeting, constructing and gravelling the paths, painting, papering, plastering etc. related to real life situations - To find the problems based on area of walls, floor and ceiling of the room - To find the cross sectional area, LSA , TSA and volume of prisms and solve the related problems - To prepare reports and project works on problems of area related to daily life situations Level-wise learning objectives S.N. LEVELS OBJECTIVES - To define the area of plane surface - To tell the formula of finding the area of cross, inner and outer paths - To relate the area, rate of cost per sq. units and the total cost of working (carpeting, plastering etc.) - To tell the formula of finding the area of four walls of 1. Knowledge (K) room - To define cross-sectional area of prism - To recall the general formulae of finding the LSA, TSA and volume of the prism - To area of plane figures (rectangle, square, parallelogram, 2. Understanding triangle, quadrilateral, trapezium and circle) (U) - To calculate the area of paths (cross, inner and outer paths) - To find the area of floor, walls and ceiling of room - To find the volume of prism - To find the surface area of prisms - To find the area of paths and cost of graveling, covering with stones - To find the area of floor and cost of carpeting 3. Application (A) - To find the area of walls and estimate the cost of painting, papering, carpeting - To solve the problems related to the volume and surface area of prisms - To estimate the number of bricks and cost required to construct the walls etc. - To mathematize the daily life problems related to area 4. High Ability (HA) and volume then solve them. - To prepare the projects works and reports and present in the class 33 Vedanta Excel in Mathematics Teachers' Manual - 9 Required Teaching Materials/ Resources Colourful chart-paper with definitions and formulae, scale, scissors, pencil, geo-board, tangram, graph paper, models of various prisms, box, cartoon, measuring tape, ICT tools etc. Pre-knowledge: Perimeter and area of rectangle, parallelogram, volume of cube, cuboid etc. A. Area of plane surfaces and paths Teaching Activities 1. Divide the students into 4/5 groups and give them the figure of rectangle, square, parallelogram, rhombus, circle, quadrilateral, trapezium etc. on the sheet of paper and tell them to recall the perimeter and area formula of related figures and call for presentation. 2. Discuss upon the following formulae with related figures and models. (i) Area of rectangle = length (l) × breadth (b) 1 (ii) Area of square = (side)2 or (diagonal)2 2 (iii) Area of parallelogram = base (b) × height (h) 1 (iv) Area of triangle = ×base (b) × height (h) 2 1 (v) Area of right angled triangle = × base (b) × perpendicular (p) 2 3 (vi) Area of equilateral triangle = a2 where ‘a’ is the side length 4 (vii) Area of triangle = s(s – a) (s – b) (s – c) where a, b and c are the lengths of sides and s is the semi-perimeter. 1 (viii) Area of rhombus= — d1 — d2 where d1 and d2 are the diagonals of the rhombus. 1 2 (ix) Area of kite = — d1 — d2 where d1 and d2 are the diagonals of the kite. 2 1 (x) Area of trapezium = — h (a + b) where ‘a’ and ‘b’ are the lengths of parallel sides 2 and h is the height of the trapezium. 1 (xi) Area of quadrilateral = — d (h1 + h2) ‘h1’ and ‘h2’ are the heights of two triangles 2 on the same base which the diagonal (d) of quadrilateral 1 (xii) Area of circle = Sd2 or Sr2 2 1 (xiii) Area of semi-circle = Sd2 or Sr2 8 3. With discussion, derive the following formula of finding the area of pathways (i) Area of path running outside the rectangle = 2d (l + b + 2d) (ii) Area of path running inside the rectangle = 2d (l + b - 2d) (iii) Area of crossing paths = d (l + b – d) (iv) Area of path running outside the circle = Sd (2r + d) or S (R2 – r2) (