201 2. Price list of Ram Janaki Fruit Shop Janakpur, Dhanusha Price list Apple Banana Orange Mango Rs. 250 per kg Rs. 120 per dozen Rs. 125 per kg Rs. 190 per kg Ram Bharosh Yadhav bought 3 kg apples, 2 kg oranges, 3 kg mangoes and 1 dozen banana, on the date 2078 Falgun 28. Prepare the bill. Ram Janaki Fruit Shop Janakpur, Dhanusha PAN No: 10215234 Bill No. 226 Mr/Mrs/Ms ........................... Date ..................... ........................... Salesman S.N Description Quantity Rate Amount 1. 2. 3 4. 5. Total In words: ................................................ ...............................................................
202 3. Menu in 'Gaule Bhanchhaghar" is shown as below. Sapana, Rajani and Sushma went for dinner in Gaule Bhanchhaghar. They ordered 2 Dalbhat, 1 Roti Tarkari, 2 Tea, 1 coffee and 1 Aalu fry Dalbhat : Rs. 450 Tea : Rs. 50 Coffee : Rs. 190 Roti Tarkari : Rs. 250 per plate Aalu fry : Rs. 90 per plate Gaule Bhanchhaghar Jawalakhel Lalitpur, Nepal. Bill No. ................ Mr/Mrs/Ms ........................................................................ Date ..................... ........................... Salesman S.N Description Quantity Rate Amount 1. 2. 3 4. 5. Total In words: ................................................ ............................................................... PAN No: ......... Abudget shows information aboutthe income and expenditure under different headings. Budget
203 The monthly expenditure of a household is given below. Income Expenditure Title Amount Title Amount Salary Rs 15,000 Food Rs 12000 Business Rs 25,000 Education Rs 6000 Clothes Rs 14000 Miscellaneous Rs 4000 Total Rs 40,000 Total Rs 36,000 From the budget information given above, we should be able to answer the following questions. • What are sources of income in a household? • What is the income of the household from the salary? • What is the total income of the household? • What is the total expenditure of the household? • What is the expenditure of the household on food? • On which title there is maximum expenditure in the household? • Monthly income of a household is Rs. 80,000. • Which is only from salary. • The different title of expenditure are Food, Cloths, Education, Miscellaneous, etc. • The expense on food is Rs. 25,000, eduction Rs. 15,00, clothes Rs. 10,000 and miscellaneous Rs. 10,000. Prepare the Budget. • The total saving per month of a household is Rs. 20,000. I can answer all the questions. Income from salary = Rs. 15000 Total income = Rs. 40,000 Total expenditure = Rs. 36,000 Expenditure on food = Rs. 12000
204 Income Expenditure Title Rs. Title Rs. Salary 80,000 Food Cloths Education Miscellaneous Rs. 25,000 Rs. 10,000 Rs. 15,000 Rs. 10,000 Total Rs. 80,000 Total Rs. 60,000 1. Raju Sthapit purchased 6 pencils, 2 erasers and a pen from Akriti Stationery, Sukedhara, Kathmandu. Copy the sample of given bill and prepare a bill given to her by the shopkeeper. Rs 15 Rs 25 Rs 5 Rs 50 Akriti Statonery Sukedhara, Kathmandu PAN No: 1234567 Bill No. .............. Mr/Mrs/Ms: ....................... Date ................. ........................... Salesman S.N Description Quantity Rate Amount 1. 2. 3 4. Total In word: ....................................... ....................................................... Exercise 13.1
205 2. Khusilal Rajbamshi bought the following items from Mai Store, Surunga Jhapa. a. 5 kg of rice at the rate Rs. 55 per kg b. 2 kg of sugar at the rate of Rs 65 per kg c. 3 kg of tea at the of Rs 220 per kg d. 1 litre sunflower oil at the rate of Rs. 120 per litre PAN of the store is 1123175 and Bill number is 345. Prepare the bill. [Follow the model of Q.No. 1] 3. Price list of Ghale store, Besishahar, Lamjung is given below: Price List Dal Potato Tomato Basmati rice Sugar Beaten rice Rs 150 per kg Rs. 30 per kg Rs. 60 per kg Rs. 120 per kg Rs. 90 per kg Rs. 110 per kg Sunita Durabought 5 kg Dal, 2 kg Potato,3 kg Tomato, 1 kg Basmati rice and 5 kg Sugar. Prepare a bill. 4. Bikalpa purchased the following items from Manisha Books and Stationery Butawal. Prepare the bill given to him by the shopkeeper: Rs. 10 Rs. 35 Rs. 150 Rs. 50
206 5. Aadhya bought the following items from Arun Stores Nepalgunj. Prepare the bill issued by the shopkeeper. a. 6 kg rice at the rate of Rs 60 per kg. b. 3 kg sugar at the rate of Rs 65 per kg. c. 1 kg tea at the rate of Rs 350 per kg. d. 2 litres oil at the rate of Rs 120 per litre. e. 5 soaps at the rate of Rs 35 per soap. KIDS FUN CORNER Menu Vegetable Mo:Mo - Rs 65 per plate Chicken Mo:Mo - Rs 80 per plate Tandoori chicken - Rs 220 Tea - Rs 25 Coffee - Rs 40 6. Ankit , Anishma and Anasuya went to Kids Fun Corner: • Ankit ordered one plate vegetable Mo : Mo and a cup of tea. • Anishma ordered one plate chicken Mo : Mo and a coffee. • Anasuya ordered a tandoori chicken and a tea. Prepare the bill.
207 8. The income of a household is from salary, rent and business. From salary there is an income of Rs. 80, 000. From house rent there is an income of Rs. 25,000 and from business there is an income of Rs. 65,000 per month. If the household spends Rs. 50,000 on food, Rs. 25,000 on cloths, Rs. 30,000 on education and Rs. 20,0000 on miscellaneous . Prepare the budget of the above income and expenditure. Answers Consult your teacher. 7. Monthly budget of a household is given below: Income Expenditure Title Rs Title Rs Salary Rs 22,000 Food Rs 18,000 Rent Rs 25000 Education Rs 15,000 Business Rs 30,000 Colthes Rs 16,000 Miscellaneous Rs 10,000 Total Rs 77,000/- Total Rs 59,000 Study the budget above and answer the following questions: a. What is the total income of the household? b. What is the total expenditure of the household? c. What is the income of the household from the business? d. What is the expenditure of the household on clothes? e. What is the monthly saving of the household? f. On which title does the household spend the least?
208 Chart A chart gives the numerical information. Let's study the given chart. Number of students in different sectors are shown in the given chart. Boarders Day Boarders Day Scholars Boys Girls Total Boys Girls Total Boys Girls Total 124 118 242 105 110 215 340 365 705 Study the chart above properly and answer the following questions. In which sector the number of students is maximum? In which sector the number of students is minimum? What is the number of Day Boarders? What is the total number of girls in the school? What is the total number of boys in the school? What is the number of Day Scholars? • The number of students is maximum in Day Scholars. • The number of students is minimum in Day Boarders. • The number of Day Boarders is 215. • The total number of girls in the school is (118 + 110 + 365) = 593 • The total number of boys in the school is (124 + 105 + 340) = 569 • The number of day scholars is 705. I can answer all the questions. Chart and Bar graph UNIT 14 Review
209 Data can be represented by using different types of figures. A figure gives complete information about the data. A bar graph consists of bars which are vertical with equal width. The gap between two bars should be same. To get numerical information, note the height of the vertical bar corresponding to the reading on the left. Amit Anisha Manish Asmita Ranjan Marks obtained by 5 students of class V is shown below on a bar graph. Study the graph above properly and answer the following questions. • Which student got the highest marks? • Which student got the lowest marks? • How much did Anisha score? • How much did Asmita score? • What is the highest mark of the class? Bar-graph
210 Manish got the highest marks. Asmita got the lowest marks. Anisha got 20 marks. Oh! Its easy to study the bar graph. Asmita got 15 marks. Highest marks in the class is 30. 1. Thehighestmarksindifferentsubjectsinthesecondterminalexaminations are given below. Subject English Nepali Maths Social Studies Science Marks 60 80 90 70 90 Show the information in the given bar graph. Class Assignment
211 Section A B C D E Number of Students 40 35 45 30 32 2 The given table shows the number of students in different sections of class IV. Draw the bar diagram to show this information: T.V Channel Cartoons Sports Comedy News Adventure Number of students 8 6 10 4 6 3. The given table shows the favourite TV channel of students of class IV. Draw the bar graph from the given information. A 45 40 35 30 25 20 15 10 5 0 B C D E
212 16 14 12 10 8 6 4 2 0 Cartoons Sports Comedy News Adventures 1. The given chart shows the number of students in different classes: Class I II III IV V Number of students Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total 15 25 40 27 23 50 18 19 37 23 16 39 31 12 43 Study the chart above properly and answer the questions given below: a. What is the total number of students in i. class I? ii. class II? iii. class III? iv. class IV? v. class V? b. What is the total number of students in the school? c. Which class has the maximum number of boys? d. Which class has the highest number of students? e. What is the total number of boys in the school? f. What is the total number of girls in the school? Exercise 14.1
213 2. The given graph shows the favourite game of class IV students. Study the bar graph and answer the following questions: 45 40 35 30 25 20 15 10 5 0 Basketball Cricket Football Volleyball Tennis 3. Given bar graph represents favourite subjects of students of class IV. Read this bar graph carefully and answer the questions given below: a. Which is the most popular game in the class? b. Which is the least popular game in the class? c. How many students like basketball? d. How many more students like football than cricket? e. How many students are there altogether in class IV? 40 35 30 25 20 15 10 5 0 English Nepali Maths Science Social Studies
214 a. What are the number of boys and girls in class I? b. In which class, there are equal number of boys and girls? c. In class II, how many more boys are there than girls? d. In which classes, the number of girls is more than the number of boys? e. In which classes the number of boys is more than the number of girls? f. In class V, how many more girls are there than the boys? g. Find the total number of students in each class. a. Which is the most favourite subject of the class? b. Which is the least favourite subject of the class? c. How many students' favourite subject is Nepali? d. How many more students prefer Nepali to Maths? 4. The given bar graph shows the number of boys and girls in different classes of a school. Study the given bar graph and answer the questions given below. O 5 10 15 20 25 30 35 40 X Y Class-I Class-II Class-III Class-IV Class-V Class Number of Students
215 Subjects English Nepali Maths Science Social Studies Marks 35 32 28 40 30 Fruits Mango Orange Apple Banana Pomegranate Number 15 25 30 35 10 5. Marks obtained by Appa in an examination in different subjects are given below. Draw the bar graph to represent this information. 6. The given table shows the number of different types of fruits in a fruit shop. Draw a bar graph to represent this information. 7. Thegiventable shows themarksobtainedbyArya indifferent subjects. Draw bar graph to represent the information: Subjects English Nepali Mathematics Science Social Studies Marks 35 30 40 30 20 8. The given table shows the number of students present in class V in a week. Draw the bar graph to show the information: Days Sunday Monday Tuesday Wednesday Thursday Friday Number of present students 28 30 32 25 30 35 Answers Consult your teacher. Project Work 1. Collect the given information from your house. • Income on different titles. • Expenditure of the Budget • Prepare the Budget 2. Collect the data about the number of students in five different classes of your school, prepare the bar graph and present in your classroom.
216 (i) 88 (i) Red (i) 10 (ii) 96 (ii) Green (ii) 9 (iii) 92 (iii) Yellow (iii) 2 2. The number of girls in Yellow House is 3. Which House has the highest number of students? 4. How many more students are there in Red House than in Yellow House? A. Colour the correct alternatives: A. Income and expenditure of a household is given in the table, 1. The total expenditure of the household is 2. The total saving of the household is 3. On which title the household has maximum expenditure? 4. The total income of the household is Income Expenditure Business Rs 80,000 Food - Rs 25,000 Clothes - Rs 20,000 Education - Rs. 20,000 (i) Rs. 45,000 (i) Rs. 15,000 (i) Food (i) Rs. 80,000 (i) 92 (ii) 93 (iii) 86 (ii) Rs. 65,000 (ii) Rs. 20,000 (ii) Cloths (ii) Rs. 25,000 (iii) Rs. 25,000 (iii) Rs. 25,000 (iii) Education (iii) Rs. 25,000 B. The given chart shows the number of students of different Houses. 1. The number of boys in Red House is Red House Blue House Green House Yellow House Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total 92 88 180 79 91 170 93 96 189 86 92 178
217 (i) 350 (ii) 180 (iii) 170 5. The total number of boys in the school is 1. Marks obtained by Dorje in Maths is C. The given bar graph shows the marks obtained by Dorje in four different subjects. 10 0 Nepali English Maths Science 20 30 40 50 (i) 35 (ii) 45 (iii) 40 2. In which subject did he get the lowest marks? 3. How many more marks he got in Maths than in Science? (i) Nepali (i) 5 (ii) Science (ii) 10 (iii) English (iii) 20 (i) 88 (ii) 9.1 (iii) 367 6. The total number of girls in the school is Unit Test Full marks – 20 Attempt all the questions. 1. The given chart shows the number of students of different sections of class IV. Read the given chart carefully and answer the questions given below. 5 Section A Section B Section C Section D Boys Girls Total Boys Girls Total Boys Girls Total Boys Girls Total 22 13 35 18 16 34 14 21 35 15 21 36 (a) Which section has the highest number of students? (b) Which two sections have equal number of students? (c) What is the total number of boys in class V?
218 4. The monthly income and expenditure of Indra Kumari Tamang is given below. 5 Income Expenditure Salary : Rs. 70,000 House rent : Rs. 40,000 Business : Rs. 80,000 Food : Rs. 25,000 Cloths : Rs. 35,000 Eduction: Rs. 30,000 Miscellaneous : Rs. 20,000 Total : Rs. 1,60,000 Total : 1,10,000. a. What is the total income of Indra Kumari Tamang? b. What is the her total expenditure? c. What is her income from salary? d. What is her income from Business? e. On which item she spends more? (d) What is the total number of girls in class V? (e) What is the total number of students in class V? 2. Show the following informations in the Bar graph. 5 Subjects English Nepali Mathematics Science Social studies Obtained marks 75 70 90 85 80 3. Sumi bought some goods from Arun store Busundhara Kathmandu. Prepare the bill. 5 She bought • 5 tooth paste tubes • 4 soaps • 2 breads • 10 kg rice • 4 kg sugar. Price List Tooth paste Soap Bread Rice Sugar Rs. 140 Rs. 60 Rs. 30 Rs. 130 per kg. Rs. 120 per kg.
219 Materials Required : Flash cards, chart paper, A4 size paper, etc. • Algebraic Expression Contents • Equation • To identify algebraic terms and expression • To separate like and unlike terms • To add or subtract the like terms • To solve the equation of one variable using principal of balance Expected Learning Outcomes Upon completion of the unit, students will be able to develop the following competencies: 12 3 6 9 25 algebra 5
220 Algebra is the generalised form of arithmetic.In arithmetic, we use the numbers 1, 2, 3, 4..., etc. which have fixed value. In algebra, quantities are represented by the symbols a, b, c, d,.... x, y, z etc. which may have any value. Constants and variables: Constant: The numbers 1, 2, 3, 4, 5..., etc. can be used in algebra to represent different values of quantities. These numbers are called constants. Variables: The letters like a, b, c, .....x, y, z can be used in algebra to represent different values of quantities. These letters are called variables. Height of Mount Everest is constant. Temperature of Kathmandu is a variable. Algebraic terms and expressions: x, y, 2x, 5y, 6z are algebraic terms. If two or more algebraic terms are connected by arithmetic operations, such expression is called an algebraic expression. x + y, 3x + 5, 6y + z, etc. are algebraic expressions. Formation of algebraic expressions: The arithmetic operations can be used to connect variables and constants. See some examples of algebraic expressions. (5x² + 3xy) term term expression Algebraic Expression UNIT 15 Review
221 Example 1 5 more than x means x + 5 y is added to z means y + z Increase x by 8 means x + 8 Again, 6 less than x means x - 6 Decrease z by 2 means z - 2 7 is subtracted from x means x - 7 2 is decreased by y means y -2 Use (+) sign in case of ‘more than’, ‘is added to’ and ‘increase by’. Use (-) sign in the case of ‘less than’ ‘is subtracted from’ and is ‘decreased by’. Again, 5 times x means 5x The product of 6 and y means 6y z is multiplied by 8 means 8z Again, Quotient of x by 4 means x 4 y is divided by 5 means y 5 Use (×) sign in case of product and ‘multiplied by’. Use (÷) sign in case ‘quotient of’ and ‘is divided by’. Types of Algebraic Expressions Monomial : An algebraic expression having only one term is called monomial. 5, 4x, 6y, etc. are monomials. Binomial : An algebraic expression having two terms is called binomial. Eg. (x + 2), (x + y), (3x + 4), etc. are binomials. Trinomial : An algebraic expression having three terms is called trinomial. For example; (x + y + z), (2x + y + 2), (3x - y + 2), etc. are trinomials.
222 Evaluation of terms or algebraic expression: If there is numerical value of each of the literal, the total value of algebraic expression or a term can be calculated. Example, If x = 2, y = 3 and z = 4, then the value of 2x - 3y + 4z is 2x - 3y + 4z = 2 × 2 - 3 × 3 + 4 × 4 = 4 - 9 + 16 = 20 - 9 = 11 Factors and co-efficients: We know that 5 × 7 = 35 5 and 7 are the factors of 35. x × y = xy ‘x’ and ‘y’ are the factors of xy. a × b × c = abc ‘a’, ‘b’ and ‘c’ are the factors of abc. 2 × x × y = 2xy 2, x and y are the factors of 2xy. Again, 2 × x = 2x 2 and x are the factors of 2x. 2 is the co-efficient of x. ‘x’ is the literal co-efficient of 2. Again, x × y × z = xyz ‘x’, ‘y’ and ‘z’ are the factors of xyz. Literal co-efficient of yz = x Co-efficient of xyz = 1 Base and power: In x², power of x is 2. 2 is the power and x is the base. In y5 , 5 is the power and y is the base. In,
223 1. Complete the given table Term Factors Co-efficient Literal co-efficient of x 3xy 5xy²z³ xy5 z6 2. Complete the given table Term Base Power Co-efficient 6x³ 7x8 10x² 5x7 Class Assignment 1. State whether the following letters are constants or variables: a. ‘x’ denotes the number of fingers in human body. b. ‘y’ denotes the temperature of Kathmandu. c. ‘a’ denotes the number of sides of a quadrilateral. d. ‘b’ denotes the number of public vehicles in Kathmandu. 2. Identify whether the given are an algebraic term or an expression: a. 2x + 3y b. 5x c. x + y + z d. x e. 3 f. 4x + y + z g. x -y + z h. 2a + 3b + c i. y 3. Write the expression for each of the following statements: a. 1 more than x b. z is decreased by 5 Review Exercise 15.1
224 4. State whether the following expressions are monomial, binomial or trinomial: a. 5xy b. 3x + 2y c. a + b + c d. 2x + 3y e. x + y + 2z f. 3a g. 5x² - 2xy h. x² + 2xy + y² i. 2p + 3q - 4r j. 4c k. a - 4c l. 2m + 3n + 5c 5. If a = 1, b = 2, c = 3, d = 0, x = 4, y = 5 and z = 6, find the value of: a. 3ab b. 2a + 5b c. a + b +c d. 2x - y + z e. 5x + 6y - 2z f. 3x - y + z g. 2a + b - c h. 3a + b - 4d i. a + b + c + d j. 4xyz - 2abcd k. (2x + y - 2z) × a l. (2a + b) ÷ x m. (3c + 2d) ÷ (x + y) n. a² + b² o. x² + xy + y² 6. a If l = 3 cm, b = 2 cm, find the value of P, if P = 2(l + b). b. If l = 5 cm, b = 4 cm, find the value of A, if A = l × b. c. If l = 3 cm, b = 4 cm and h = 5 cm, find the value of V, if V = l × b × h. d. If l = 3 cm, b = 2 cm and h = 4 cm, find the value of S, if S = 2(lb + lh + bh). c. 2 less than y d. 3 times z e. x is increased by 3 f. z is decreased by 5 g. x is decreased by 10 h. 5 times of y i. z is multiplied by 3 j. twice of x is subtracted by 2 k. 7 is added to three times of x l. 5 is added to six times of y m. 3 times of x is added to 2 times of y n. 4 is added to 5 times of x o. the quotient of x to 3 is added to 6
225 7. If x = 2 cm,y= 4 cmandz = 5 cm,findthe totallengthofthegivenline segments: 8. Name the co-efficient, base and power of: a. 3x² b. 5x7 c. 10x³ d. 7y6 e. 6z³ 9. If x × y × z = xyz: a. What are the factors of xyz? b. What is the numerical co-efficient of xyz? c. What is the literal co-efficient of x? d. What is the literal co-efficient of z? x², 5x², 7x² are like terms. 3ab, 7ab, 10ab are like terms. 3 oranges, 5 oranges, 6 oranges are like terms 4y, 8y, 10y are like terms. Hence, the terms with same variables with the same power or exponent are called like terms. 5a², 2b², 3x² are unlike terms 2ab, 4y², 6xy are unlike terms Hence, the terms which have different variables or same variables with different power or exponent are called unlike terms. Answers 1. Consult your teacher, 2. Consult your teacher, 3. Consult your teacher, 4. Consult your teacher 5. a) 6 b) 12 c) 6 d) 9 e) 38 f) 13 g) 1 h) 5 i) 6 j) 480 k) 1 l) 1 m) 1 n) 5 o) 61 6. a) 10 cm b) 20 cm2 c) 60 cm2 d) 52cm2 7. a) 10 cm b) 21 cm 7. Consult your teacher, 8. Consult your teacher, 9. Consult your teacher. Like and Unlike Terms
226 Way of addition: Addition of algebraic expression can be done in two methods a. Vertical arrangement method b. Horizontal arrangement method Vertical arrangement method: Example 1 Example 2 6a + 4a + 3a = (6 + 4 + 3)a = 13a Add (3a + 4b + 5c) and (2a + b + 6c) Again, - 5x - 4x - 2x = -(5 + 4 + 2)x = -11x Again, 2y - 7y + 9y = (2 - 7 + 9)y = 4y Solution: Here, a + 4b + 5c + 2a + b + 6c 5a + 5b + 11c Horizontal arrangement method: Let's see an example, Add: (5x - 2y + 6z) and (2x + 5y - 2z) Solution: (5x - 2y + 6z) + (2x + 5y - 2z) = 5x - 2y + 6z + 2x + 5y - 2z = 7x + 3y + 4z • Write each expression in separate rows in such a way that like terms are arranged one below the other in column. • Add the like terms in each column. • Write the given expressions horizontally. • Collect the like terms and simplify. Addition and subtraction of algebraic expressions can be done only if the like terms are like. While adding or subtracting algebraic expressions, we should just add or subtract the co-efficient of like terms. Addition and subtraction of algebraic expressions
227 Ways of subtraction: As in addition, subtraction of two algebraic expressions can also be done in two methods a. Vertical arrangement method b. Horizontal arrangement method Horizontal arrangement method: Let's see an example, Subtract: (5a - 2b - 3c) from (6a + 2b - 2c) Here, (6a + 2b - 2c) - (5a - 2b - 3c) = 6a + 2b - 2c - 5a + 2b + 3c = a + 4b + c 5x + y - 3z 4x - 3y + 6z x + 4y - 9z (–) (+) (–) • Rewrite the given expressions in two lines such away that lower lines are the expression to be subtracted and like terms of both expressions are one below the other. • Change the sign of each term of the lower line. • Combine the terms column wise. • Enclose the expression to be subtracted in the brackets with a minus (-) sign. • Open the bracket with minus sign. • Combine the like terms. Vertical arrangement method: Subtract: (4x - 3y + 6z), from (5x + y - 3z) Here, 1. Add or Subtract: a. 5x + 3x b. 7a - 4a c. 8b + 12b d. 8b - 5b e. 3x + 2x + 5x f. 5a² + 2a² g. 3mn + 6mn + 2mn h. 11mn - 7mn i. 7a² - 5a² j. 4p - 8p k. 2xy + 5xy + xy l. 5x² + 2x² + x² m. n² + 3n² + 2n² Exercise 15.2
228 4. Add the following: 5. Add: a. 2a + 3b + 5c and a - 2b + 3c b. x² - xy + 2y² abd 2x² - 5xy + 3y² c. 2x² + 5 - 3x and 3x - 2x² + 8 d. 5a + 4b + 6c and - 3c + 2b - 8a e. -2m + 3n + 4p and 2m + 3n - 2p f. 4xy - 3yz + 2zx, 2xy + 5yz - 3zx and xy + yz + zx g. a² - ab + 2b², 4a² + 3ab + b² and 5a² - 2ab + b² h. 3m² + 2n² - 4mn, m² - mn + n² and m² + mn + n² i. 4xy - 7yz + 8zx, 3xy + 2yz - 4zx and 5xy - 16yz - 12zx j. 8x - 6y - 8z, 5x + 3y + 6z and 5x - 3y + 5z a. 5a + 2b + 8a - 6b b. 2a + 7b + 6a + 3b c. 4x - 3y + 8x + 6y d. x + 2y + 3z + 2x + y + 2z 6. Subtract: a. 2x + 5 from 3x + 7 b. 2a + b + c from 3a + 2b + 2c c. 2x + y from 5x + 6y d. m + n - 2p from 4m - 2n + 3p e. 8x + 6y - 4z from 9x + 8y + 2z f. 3a² - 2a + 4 from 4a² - a + 6 2. Simplify: a. 6x - 3x - 5x b. 7a² - 6a² - 2a² c. 5ab - 2ab + 3ab d. 3xy + 5xy - 2xy e. 4mn - 2mn + 6mn f. n² - 3n² + 4n² g. - 5y - 6y + 8y - y + 2y h. 3x - x - 2x + 8x - 7x i. 3a²b + 4a²b + 5a²b 3. Find AB + BC + AC. Find PQ + QR + RS + PS. a. b.
229 7. Simplify: a. 3x + 4y - 2x - 3y b. 5x² + 6xy + 2y² - 3x² + 2xy - y² c. 5x² - 2xy + 3y² - x² + 6xy - 2y² d. a²b + b²c + c²a - 2a²b - 3c²a + 4b²c 8. a. What should be added to 3x - 4y + 6z to make 7x - y + 4z? b. By how much 4x - y + z is greater than x - 2y + 3z? c. From what 2a + 3b - c be subtracted to get 3a - b -c? d. By how much x² + xy + y² is less than 2x² - xy? e. What should be subtracted from 3m² - 4n² to make 2m² - mn + 6n²? f. Take a²b - b²c + 2c²a from 2a²b - 6b²c + c²a. 10. Find the perimeter of the given figures: 9. If x = a + 3b - 4c, y = 2a - 5b + c and z = a + b + c, find the value of x + y + z. a. b. Answers 1. Consult your teacher 2. Consult your teacher 3. Consult your teacher 4. Consult your teacher 5. Consult your teacher 6. Consult your teacher 7. Consult your teacher 8. a) 4x+3y-2z b) 3x+y-2z c) 5a + 2b – 2c d) x2 –2xy–y2 e) m2 +mn–10n2 f) a2 b–5b2 c–c2 a 9. 4a–b–2c 10. a) 10a+b+6c b) 10x+3z g. 5a² + 3ab + 5b² from 3a² + 2ab + 8b² h. 4x³ - 6x² - 2xy from 2x³ + 5x² i. 3x² + 2xy + 5y² from x² - xy - 2y² j. 4m² + 6mn - n² from 2m² + 3mn - n² Project Work Write some algebraic terms from your textbook. Separate them into like and unlike terms.
230 An equation is a mathematical statement equating two quantities. x + 5 = 9 is an equation. (x + 5) and 9 are two quantities. In an equation, the value of the terms on the left hand side is equal to the terms on the right hand side. An equation is like a weighing balance having equal weight on each pan. Solution of an equation: Let's take an example, x + 2 = 6. We have to find the value of x, which will satisfy the equation. When, x = 1, 1 + 2 = 6 3 = 6 (which is not true) When, x = 2, 2 + 2 = 6 4 = 6 (which is not true) ‘=’ stands for equal to To solve an equation, we have to determine the value of variable that will make the equation true. Algebraic Equation UNIT 16 Algebraic Equation
231 1. Determine by substitution, whether: a. 3 is the solution to 2x - 1 = 5 b. 2 is the solution to 3x + 1 = 5 c. 5 is the solution to x + 2 = 7 2. Replace the variable by 1, 2 and 3 to solve the given equations: a. x + 3 = 6 b. 2x + 3 = 7 c. 3x - 1 = 2 d. 4x + 3 = 11 e. 3x - 5 = 1 Example 1 Determine, if 5 is the solution to the equation 2x + 3 = 13 Solution: Here, Given equation, 2x + 3 = 13 When x = 5, Left hand side = 2 × 5 + 3 = 10 + 3 = 13 Right hand side = 13 When x = 5, Left hand side = Right hand side \ x = 5 is the solution to the given equation. Exercise 16.1 If x + 2 = 6, x = ? means, what should be added to 2 to make 6? When, x = 3 3 + 2 = 6, 5 = 6 (which is not true) When, x = 4, 4 + 2 = 6 6 = 6 (which is true) \ x = 4 is the solution to this equation. This method of solving equation is known as Trial and Error method.
232 3. Replace the by suitable number to make the given statement true: a. + 3 = 4 b. 2 × - 3 = 3 c. 3 × + 5 = 8 d. 2 × + 1 = 3 e. 4 × + 1 = 5 While solving any equation, we have to use the following facts. i. If same numberis added to both sides of an equation, the equation remains true. Example 1 x - 2 = 5 or x - 2 + 2 = 5 + 2 or x = 7 ii. If the same number is subtracted from both sides of an equation, the equation remains true. Adding 2 on both sides Solving of the equation using the principle of balance Shortcut way x – 2 = 5 or, x = 5 + 2 or, x = 7 Example 2 y + 5 = 7 or y + 5 - 5 = 7 - 5 \ y = 2 Subtracting 5 from both sides iii. If the same number is multiplied on both sides of an equation, the equation remains true. Shortcut way y + 5 = 7 or, y = 7 – 5 or, y = 2 iv. If the same number divides both sides of an equation (except zero), the equation remains true. Example 3 x 3 = 2 or, x 3 × 3 = 2 × 3 or, x = 6 Multiplying both sides by 3 Shortcut way x 3 = 2 or, x = 2 × 3 or, x = 6
233 Example 4 3x = 12 or, 3x 3 = 12 3 or, x = 4 Dividing both sides by3 Shortcut way 3x = 12 or, x = 12 3 or, x = 4 Example 5 Solve: 3x 4 - 3 = 6 Solution: 3x 4 – 3 = 6 or, 3x 4 – 3 + 3 = 6 + 3 or, 3x 4 = 9 or, 3x 4 × 4 = 9 × 4 or, 3x = 36 or, 3x 3 = 36 3 or, x = 12 Adding 3 on both sides Multiplying both sides by4 Dividing both sides by 3 Shortcut way 3x 4 - 3 = 6 3x 4 = 6 + 3 3x 4 = 9 or, 3x = 9 × 4 or, x = 9 × 4 3 or, x = 12 I understand! 3x 4 - 3 = 6 can be written as 3x 4 = 6 + 3 directly. 1. Solve the following: a. y - 2 = 6 b. x - 1 = 5 c. z - 2 = 7 d. x - 4 = 7 e. x - 8 = 9 2. Solve: a. x + 2 = 4 b. y + 3 = 7 c. y + 5 = 8 d. x + 1 = 6 e. z + 2 = 5 3. Solve: a. x 4 = 2 b. y 3 = 2 c. z 2 = 1 d. x 5 = 1 e. y 3 = 1 4. Solve: a. 3x = 15 b. 2x = 6 c. 5y = 20 d. 4y = 12 e. 7z = 14 Exercise 16.2
234 Example 1 A number added to 7 equals to 11. Solution: Here, Let the number be ‘x’ From the given condition, x + 7 = 11 While forming an algebraic equation, we have to suppose the unknown quantity as variables like x, y, z etc. and we have to translate the statement into algebraic equation. A number means ‘x’ a number + 7 = 11 x + 7 = 11 5. Solve: a. 2x + 5 = 9 b. 3x - 1 = 8 c. 2x + 9 = 13 d. 2z + 5 = 13 e. 3y - 2 = 10 f. 4z + 1 = 13 g. 5 + 3y = 20 h. 5x + 3 = 23 6. Solve: a. 3x + 2 = 2x + 1 b. 5x + 3 = 4x + 5 c. 2x - 3 = x + 2 d. 3x - 1 = x + 1 e. 6x - 7 = 4x + 5 7. Solve: or, x + 7 - 7 = 11 - 7 or, x = 4 \ The required number is 4. Formation of Algebraic Equation and Their Solution
235 Example 2 5 less than a number y is 13. Solution: Here, y - 5 = 13 y - 5 + 5 = 13 + 5 y = 18 5 less than y means (y - 5) Example 3 2 times a number added to 5 is 17. Solution: Here, Let the number be ‘x’. From the given condition, 2x + 5 = 17 Two times of x = 2x two times of x added to 5 is 2x + 5 Example 4 or, 2x + 5 - 5 = 17 - 5 or, 2x = 12 or, 2x 2 = 12 2 or, x = 6 Subtracting 5 from both sides. Dividing both sides by 2 Area of a rectangle is 24 cm². If its length is 8 cm, find its breadth. Solution: Let, the breadth of the rectangle be ‘x’. l = 8 cm, b = x cm, A = 24 cm² We have, A = l × x or, 24 = 8x or, 24 8 cm = 8x 8 or 3cm = x or, x = 3 cm \ Breadth of the rectangle = 3 cm. 8 cm x cm
236 1. Form the equations and solve them: a. Sum of y and 6 is 11 b. Difference of x and 2 is 9 c. 25 added to y is 30 d. 15 taken away from z is 25 e. y is taken away from 10 is 7 f. Product of x and 5 is 25 g. 3 multiplied to x gives 18 h. x divided by 3 is 6 2. Make the equation and solve the following problems: a. 2 times a number added to 6 is 20. b. 3 times a number added to 4 is 25. c. If 7 is subtracted from 3 times a number the result is 8. d. If 4 is subtracted from 2 times a number, the result is 2. e. If a number is divided by 4, the result is 3. 4. Write the algebraic expression for the following: a. Select a number: x Multiply it by 4: ..................... Add 7 to the result: ..................... b. Select a number: y Multiply it by 4: ..................... Subtract 7 to the result: ..................... 3. a. Area of a rectangle is 15 cm². If the length is 5 cm, find the breadth. b. The product of two numbers is 21. If one of them is 7, find the other number. Exercise16.3
237 Mathematical fun - Think of a number x 5 10 15 20 100 - Add 50 to it x + 50 55 - Double it 2x + 100 110 - Add 48 to it 2x + 148 158 - Divide it by 2 x + 74 79 - Take away the number you thought of 74 74 Try this with other numbers and compare the result. It's interesting! Every time the result is 74. Project Work • Make five different equations, convert that equation into verbal form and make the solution in the chart paper. • Using addition, subtraction, multiplication and division, collect some verbal problems. Convert them into the equation and solve the equation in the chart paper.
238 Let's check Aadhya's copy. Tick the correct answer. Cross the wrong one and do the correction. Name : Aadhya Sharma Class : V Activity 2x 5 30 2x 2 5 2x 5 2x 5
239 Colour the correct alternatives: 3. The equation of the statement '2 times of x added to 7 is equal to 19' is 4. Formula of volume of a cuboid is l × b × h. Volume of the cuboid having length, breadth and height (x + 2), (x – 2) and x respectively is 5. The value of x in the equation 3x 4 + 3 = 6 is 1. The literal co-efficient of x in 2xy is 2. Perimeter of the given DABC is 3x - y x + 2y 2x + y C A B 6. If 2x + 9 = 13, then the value of x is equal to (i) 4 (ii) 2 (iii)3 7. 3 times a number added to 10 is 40 The mathematical sentence of this statement is (i) 3x + 10 = 40 (ii) 3x + 40 = 10 (iii)3x + 10 + 40 = 0. (i) 2 (ii) 1 (iii) y (i) 6x (ii) 6x + 2y (iii) 2y (i) 2x + 7 = 19 (ii) 2x + 19 = 7 (iii) 7 – 2x = 19 (i) x2 – 4 (ii) x2 + 2x (iii) x3 – 4x (i) 12 (ii) 4 (iii) 15
240 1. Write the expression for each of the following statements: 2 a. x is increased by 3 b. z is decreased by 5 2. If a = 1, b = 2, c = 3, d = 0, x = 4, y = 5 and z = 6, find the value of: 1.5 × 2 = 3 a. 3a + b - 4d b. (2x + y - 2z) × a 3. Add: 1.5 × 2 = 3 a. 2a + 3b + 5c and a - 2b + 3c b. a² - ab + 2b², 4a² + 3ab + b² and 5a² - 2ab + b² 4. Subtract: 1.5 × 2 = 3 a. m + n - 2p from 4m - 2n + 3p b. 8x + 6y - 4z from 9x + 8y + 2z 5. Multiply: 2 a. 6b×2b×4b b. (-2x³y7 ) × (-7x4y³) 7. Divide: 1.5 × 2 = 3 a. 12y5 ÷ 3y² b. (12x4 y³ - 8xy + 4x²y³) ÷ 2xy 8. Solve: 1.5 × 2 = 3 a. x + 1 = 6 b. 5x + 3 = 23 9. Make the equation and solve the following problems: 2 × 2 = 4 a. If 7 is subtracted from 3 times a number the result is 8. b. If a number is divided by 4, the result is 3. 6. Using the formula A = l × b, find the area of the following figures: 2 l = 2x + 3 b = x + 1 Unit Test Full marks : 25
241 Time: 2 hrs. Attempt all the questions. 1. Study the given figure and answer the questions given below. a. What is the name of given figure? Measure each angle by Protractor. 2 b. Mention the type of each angle whether they are acute, obtuse or right angle. Again, identify a pair of parallel lines and a pair of perpendicular lines. 2 2. Study the given figure and answer the questions given below. a. What is the name of given object? 1 b. What are its vertices? 1 c. What are its edges? 1 d. What does the parts ABCD and AH represent? 1 3. The yearly budget of a municipality is Rs.157368200. a. Put comma in the number using Local place value system and write its name according to Local place value system and in Devanagari. 2 b. Identify whether the digits 5, 7, 6 and 8 are prime or composite. Is 31 a prime number? Given reason. 2 4. a. Find the value of: (i) –12 + 17 (ii) –15 –35 (iii) –30 + 25 (iv) (+20) ÷ (–5) 1 b. Simplify: 56 ÷8 – 17 × 7 – 24 2 c. Convert the statement "25 divided by 5 is added to the product of 3 and 7" and simplify it. 2 5. Study the given figure and answer the questions given below. a. Writethefractionrepresented by the shaded part of given figure in mixed number and Improper fraction. 1 Model Test Paper A D B C A B C G E F D H
242 b. Write the fraction represented by the given figure and find the sum of the fractions of (a) and (b). 1 c. Why the fraction 3 10, a proper fraction? Convert this fraction into decimal. 1 d. Out 20 full marks, Roshan gets 17 marks. Convert this statement into fraction and percentage. Compare the percentage represented by (c) and (d). 2 6. a. Round off the number 34632 to its nearest hundred 1 b. Three sides of a triangle are 3.72cm, 6.98cm and 5.65 cm, find the total length of 3 sides. 1 c. A boy cut a cake into 12 equal pieces. He ate 2 pieces himself and distributed 4 pieces among his friends, find what fraction of cake left there? 2 7. Study the questions given below and answer the given questions. a. It is 5 o'clock in the evening. Write this time in both 12 hours system and 24 hours system. 1 b. A teacher teaches 3 hours 25 minutes in a day. For how long will he teach in a week if school runs 6 days a week. 1 c. Divided 5l 700ml by 4. 1 d. The length and breadth of rectangular piece of land is 25ft and 15ft respectively. Find its perimeter and area. 2 8. Study the given table and answer the questions given below. S.N Item Number of packet Quantity in each packet 1. Milk 8 650ml. 2. Rice 9 7kg 500gm. 3. Oil 1 750ml. a. What is the total quantity of 8 packet milk? 1 b. What is the total weight of 9 packet of rice? 2 c. If there is 5200ml milk and it is to be packed into the capacity of packet given above, how many such packets are needed? 2
243 9. The total income and expense of the picnic organized by the students of class V in given below. Income Expenditure Title Amount Title Amount Collection from students Help from the side of school Rs. 50,000 |– Rs. 30,000 |– Food Transportation Music system Drinks Miscellaneous 34,000|– 16,000|– 8,000|– 5,400|– 3,600|– Total Rs. 80,000 Total 67,000|– Study the above table and answer the questions given below. a. What amount of budget is collected for picnic? 1 b. What is the total expenditure and the saving? 1 c. If school has donated only Rs. 15,000 for the picnic. Is the amount sufficient for the programme? If not how much money is not enough? 10. The given graph shows the favourite game of class IV students. Study the bar graph and answer the following questions: 45 40 35 30 25 20 15 10 5 0 Basketball Cricket Football Volleyball Tennis
244 a. Which is the most popular game in the class? b. Which is the least popular game in the class? c. How many students like basketball? d. How many more students like football than cricket? e. How many students are there altogether in class IV? 11. Identify whether the given terms are like or unlike a. 2x and 5y, 7xy, 2xy, 4xy. 1 b. What is the sum of 7 the terms of (a)? 1 c. Subtract x – 2y + 10xy from the sum of (a) and (b). 2 12. The sum of 3 times a number and 4 is 19. a. Convert this statement into mathematical equation. 1 b. Find the number. 2