Statistics 251 (b) The yearly budget of a school: Income Expenditure Topics Amount Topics Amount Admission Rs. 728300 Salary Rs. 525200 Fee Rs. 835400 Rent Rs. 120500 Maintenance Rs. 17800 Maintenance Rs. 5780 Stationery Rs. 25400 Stationery Rs. 12890 Other Rs. 12420 Other Rs. 13550 Total Rs. ............... Total Rs. .................. i. What is the total monthly income of the school? ii. What is the total monthly expenditure of the school? iii. Which topic has least expenditure? iv. How much amount does it gains from the selling of books? v. How much more expenditure is there in stationery than maintenance? vi. How much money is gained by the school in a month? 2. (a) The annual budget of Kumari Primary School, Patan is given below. See the budget and answer the following questions. Income Expenditure Source Amount (Rs.) Topic Amount (Rs.) Donation from District Education Office 4,50,000 Salary 3,50,000 Donation from VE 1,50,000 Maintenance 35,000
252 The Leading Mathematics - 5 Local Donation 40,000 Miscellaneous 20,000 Other 35,000 Education Materials 45,000 Total 6,75,000 Total 4,53,000 (i) What is the biggest source of income? (ii) On which source is the highest expenditure there? (iii) Which is the more expenditure or income of the school? (iv) What is the annual saving of the school? (v) How much income is on the topic 'Local Donation'? (b) Karma is a farmer. His yearly income and expenditure are shown in the table below: Income Expenditure Source Amount (Rs.) Topic Amount (Rs.) Vegetables 2,50,000 Education 1,25,000 Grain 35,000 Food 13,000 Goat 1,22,000 Medicine 25,000 Other 50,000 Rent 35,000 Other 12,000 Total Total Answer the following questions from the above budget: i. Is Karma's income sufficient for expenditure? ii. From which source does Karma earn more? iii. In which heading does Karma expend more? iv. What is the yearly saving of Karma? v. How much amount does he spend on food?
Statistics 253 13.2 Preparing Budget At the end of this topic, the students will be able to: ¾ construct the family budget sheet. Learning Objectives Read, Think and Learn The income and expenditure of a family in a month is as given here. Income : Salary = Rs. 40000, Rent = Rs. 20000, Business = Rs. 60000 Others = Rs. 15000 Expenditure : Food = Rs. 35000, Education = Rs. 22500, Clothes = Rs. 9000 Medicine = Rs. 5000, Others = Rs. 4500 Now, preparing the budget for the above income and expenditure, Income Expenditure Topics Amount Topics Amount Salary Rs. 40000 Food Rs. 35000 Rent Rs. 20000 Education Rs. 22500 Business Rs. 60000 Clothes Rs. 9000 Others Rs. 15000 Others Rs. 4500 Total Rs. 135000 Total Rs. 71000 In the above budget the income of the family is Rs. 135000 and expenditure is Rs. 71000. Here, the family earns more money than expenditure. The more money than the expenditure is the more saving amount of the family. So, saving amount = Income – Expenditure = Rs. 135000 – Rs. 71000 = Rs. 64000 Therefore, the saving amount of the family is Rs. 64000 in a month.
254 The Leading Mathematics - 5 A budget is a spending plan based on income and expenses. In other words, it's an estimate of how much money you'll make and spend over a certain period of time, such as a month or year. PRACTICE 13.2 Your mastery depends on practice. Practice as you play. 1. Study the income and expenditure of a family in a month below and answer the following questions. Income : Salary = Rs. 30000, Rent = Rs. 15000, Agriculture = Rs. 6000 Others = Rs. 5000 Expenditure : Food = Rs. 15000 Education = Rs. 10500 Clothes = Rs. 6000 Medicine = Rs. 4500 Others = Rs. 3200 i. Prepare a budge of the family. ii. Find the saving or loss amount of the family. 2. Study the income and expenditure of a fruit shop in a month below then answer the following questions. Income : Sales of Apple = Rs. 59000 Sales of Banana = Rs. 19000 Sales of Pineapple = Rs. 42000 Fresh Juice = Rs. 25300 Miscellaneous = Rs. 43000 Expenditure : Rent = Rs. 22500 Salary = Rs. 24000 Fruits Purchase = Rs. 105000 Shop Maintenance = Rs. 3500 Taxes = Rs. 4500 Others = Rs. 3200 i. Prepare a budge of the office. ii. Find the saving or loss amount of the office.
Statistics 255 13.3 Reading Chart At the end of this topic, the students will be able to: ¾ read the chart. Learning Objectives Read, Think and Learn # Study the given schedule of buses going to different places from New Bus Park, Kathmandu and discuss the following questions: Bus Time Table Places Bus 1 Bus 2 Bus 3 Pokhara 6:15 am 7:30 am 8:45 am Gorkha 8:30 am - - Bhairahawa 6:00 am 8:30 am - Birgunj 5:30 am - 4:15 pm Dharan 2:30 pm 5:30 pm - i. How many buses are going to Pokhara? ii. How many buses are going to Dharan? iii. For which place, only one bus is going? iv. At what times the buses are going to Bhairahawa? v. If you have missed your bus for Birgunj going at 5:30 am, how long time must you wait? Ö The given chart shows the rates of bus fare from Ratnapark to Lagankhel in rupees. Study it and answer the following questions. Maitighar Thapathali Kupandol Pulchok Jawalakhel Lagankhel Ratnapark 13 14 15 17 18 20 Maitighar 13 14 16 17 19 Thapathali 13 15 16 18 Kupandol 13 14 16
256 The Leading Mathematics - 5 Maitighar Thapathali Kupandol Pulchok Jawalakhel Lagankhel Pulchok 13 15 Jawalakhel 13 (a) How much amount will a passenger pay to arrive Jawalakhel from Ratnapark? (b) How much amount will a passenger pay to arrive Lagankhel from Maitighar? (c) How much amount will a passenger pay to arrive Thapathali from Ratnapark? (d) How much amount will a passenger pay to arrive Pulchok from Maitighar? (e) How much amount will a passenger pay to arrive Maitighar from Jawalakhel? PRACTICE 13.3 Your mastery depends on practice. Practice as you play. 1. Look at the following table of the bus fare from Koteshwor to Kalanki in rupees and answer the following questions: Koteshwor Baneshwor Babarmahal Thapathali Tripureshwor Teku Kalanki Baneshwor 13 Babarmahal 14 13 Thapathali 15 14 13 Tripureshwor 16 15 14 13 Teku 17 16 15 14 13 Kalimati 18 17 16 15 14 13 Kalanki 19 18 17 16 15 14 13
Statistics 257 (a) How much amount will a passenger pay to arrive Baneshwor from Kalimati? (b) How much amount will a passenger pay to arrive Babarmahal from Kalanki? (c) How much amount will a passenger pay to arrive Thapathali from Kalanki? (d) How much amount will a passenger pay to arrive Baneshwor from Teku? (e) How much amount will a passenger pay to arrive Babarmahal from Teku? 2. The given table shows the grades of students secured in an exam: E D D+ C C+ B B+ A A+ 1 0 3 5 6 7 10 15 12 8 2 1 5 7 14 15 20 25 15 10 3 7 3 10 7 20 23 18 10 5 4 4 6 8 19 21 26 35 4 2 5 3 4 6 8 18 24 20 17 4 (a) How many students are there who secured grade A+? (b) How many students are there in the class 4? (c) In which class did the more students secure the grade A? (d) In which class there is no student who secured grade E? (e) In which class a more students passed except the grades D and E ? Grade Class
258 The Leading Mathematics - 5 13.4 Reading Bill At the end of this topic, the students will be able to: ¾ real the bill of the purchasing goods. Learning Objectives Read, Think and Learn Suman Sharma bought some books and stationery goods. After paying the amount for the items he purchased, the shopkeeper handed the bill as in the following. Bhuwan Books and Stationery Center Pokhara, Nepal Bill no. 2345 Customer’s Name : Suman Sharma Address : Balaju, Kathmandu In words: One thousand three hundred sixty five rupees only. ............................ Date: 2080/ 4 / 6 S.No. Particulars Quantity Rate (Rs.) Price (Rs.) 1. The Leading Maths 3 335 1005 2. Note copy 5 40 200 3. Pen 2 65 130 4. Pencil 3 10 30 Total 1365 This is a paper bill which contains description, name, quantities, date, amount, etc. Generally, the bill has the following titles: Ö Name and address of the shop. Ö Bill number that helps to find out the details. Ö Name and address of customer. Ö Date that helps to find record of the sold information. Ö Name of items, their quantities and rates. Ö Sum of price of each item with grand total. Ö Other charges (e.g. fax, service charge etc.) (sometimes)
Statistics 259 Ö Grand total amount in words. Ö Signature of the shopkeeper at last. PRACTICE 13.4 Your mastery depends on practice. Practice as you play. 1. Read the following bill carefully. Allid Publication Kathmandu, Nepal Bill no. 78352 Customer’s Name : Sweta Mahato Address : Samakhusi, Ktm In words: Twelve thousand eight hundred seventy rupees only. ............................ Date: 2080 / 6 / 8 S.No. Particulars Quantity Rate (Rs.) Price (Rs.) 1. The Leading Maths (1) 9 250 2250 2. The Leading Maths (2) 13 265 3445 3. The Leading Maths (3) 15 275 4125 4. The Leading Maths (4) 10 305 3050 Total 12870 Salesman : Chandra Maharjan Now, answer the following questions: (a) What types of books did Sweta buy? (b) Who was the salesman? (c) Who bought the books? (d) Who sold the books? (e) Where did Sweta bought the books from? (f) What amount did she pay in all? (g) What change did she receive if she handed on the amount of Rs. 13000?
260 The Leading Mathematics - 5 2. Read the following bill carefully. Yadav Grocers Gwarko, Lalitpur Bill no. 34567 Customer’s Name : Aariya Dangol Address : Koteshwor, Ktm In words: Six hundred thirty only. ............................ Date: 2080 / 6 / 7 S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Orange 2.5 kg 40 100.00 2. Apple 4 kg 60 240.00 3. Grape 2.5 kg 80 200.00 4. Banana 3 dozen 30 90.00 Total 630.00 Salesman : Gautam Khadka Now, answer the following questions: (a) Where is the Grocer’s shop? (b) Who bought the fruits? (c) What items of fruits did Aariya buy? (d) Who sold the fruits? (e) What items of fruits did Aariya buy? (f) What was the rate of grapes per kg? (g) What was the rate of bananas? (h) How much in all did she pay for the fruits? (i) What is the benefit of taking bill after purchasing?
Statistics 261 13.5 Preparing Bill At the end of this topic, the students will be able to: ¾ prepare the bill of the goods. Learning Objectives Read, Think and Learn Look at the price list for the fruits in the Grocers shop. Prepare a bill for Raju Kafle if he bought 4 kg apples, 2 kg oranges and 3 dozen of bananas and think about the following questions. (a) What is the bill number? (b) Who purchases the fruits? (c) What is the address of the customer? (d) In which date does Raju Kafle purchase the fruits? (e) How can the shopkeeper calculate the amount of each kind of fruits? (f) What is the total amount of the bill? (g) Who sold the fruits? Let’s prepare a bill for purchasing fruits. CK Grocers Kathmandu, Nepal Bill no. 0258 Customer’s Name : Raju Kafle Address : Bhotahiti, Ktm # In words: Nine hundred seventy five rupees only. ............................ Date: 2080/ 4 / 6 S.No. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Apples 2 kg 140 280 2. Orange 5 kg 85 425 3. Banana 3 doz 90 270 Total 975 General Store Apples Rs. 140 per kg Oranges Rs. 85 per kg Bananas Rs. 90 per doz Grapes Rs. 125 per kg
262 The Leading Mathematics - 5 PRACTICE 13.5 Your mastery depends on practice. Practice as you play. 1. The price list of each item and the sample bill of a store are shown below: Price list Sugar Rs. 75 per kg Butter Rs. 450 per kg Coffee Rs. 950 per kg Banana Rs. 120 per dozen Tomato Rs. Rs. 35 per kg Apple Rs. 120 per kg Rice Rs. 85 per kg Unique Grocers Mangalbazar, Lalitpur Bill no. 26785 Customer’s Name : ........................... Address : .......................... In words: .......................................................................................... Date: .......................... S.No. Items Quantity Rate (Rs.) Price (Rs.) 1. 2. 3. 4. 5. 6. 7. Total Salesman : ....................... Prepare the bills for the purchase of the given items for: 2. (a) Ram Tamang, Kavre - 4 (b) Sabin Shakya, Patan, Lalitpur Tomato 4 kg Banana 2 Dozen Apple 5 pieces Apple 8 pieces Rice 5 kg Potato 3 kg 3. (a) Marpha Sherpa, Bouddha-2 (b) Bhakta Tamang, Banepa - 4 Tomato 4 kg Banana 2 Dozen Apple 5 pieces Apple 8 pieces Rice 5 kg Potato 3 kg 4. (a) Krishna K.C., Palpa-4 (b) Sohani Giri, Gulmi-4 Butter 1 kg Potato 4 kg Banana 1 dozen Sugar 3 kg Rice 6 kg Tomato 1 kg
Statistics 263 CHAPTER 14 BAR DIAGRAM How many books are there in your bag ? Can you count the total pages of all books ? It takes more time in the class. How can you find the total pages of all your books easily ? List out the total pages of all your books in the above table. Can you read the height of three students in the above picture ? Who is taller and who is shorter ? WARM-UP Lesson Topics Pages 14.1 Reading Tables 264 14.2 Reading Bar Graph 266
264 The Leading Mathematics - 5 14.1 Reading Tables At the end of this topic, the students will be able to: ¾ read the simple table. Learning Objectives Read, Think and Learn Look at the following table. STUDENTS' DISTRIBUTION Borders students Day scholar students Scholarship students Boys Girls Total Boys Girls Total Boys Girls Total 35 25 60 300 250 550 10 20 30 Now, answer the following questions: Ö Scholarship students are either day scholars or borders. If 30% of the scholarship students are border, how many scholarship students are border? Ö To find the total number of students, you don’t have to add scholarship students. Why? Ö How many girls are there in total? Ö What percentage of the total students are the day scholar students? PRACTICE 14.1 Your mastery depends on practice. Practice as you play. 1. A survey carried on the vehicle types that pass within 1 hour from Thapathali bridge is given below: Taxi Public Bus Minibus Microbus Private car Government vehicle Motorbike 8 17 12 20 35 22 60 (a) Which means of transportation are used the most? (b) Which means of transportation are used the least? (c) What percentage of the total vehicles are the taxi users?
Statistics 265 (d) If 1 3 of the vehicles go to Ratna Park, how many vehicles are going to Ratna Park in all? (e) In average 40 persons travel in a public bus. What was the number of travelling in the public buses? 2. Study the following and answer the given questions: (a) Weight of animals Name Bear Yak Giraffe Dolphin Lion Weight (kg) 475 667 800 13 175 i. Which animal has more weight? ii. Which animal has least weight? iii. What is the weight of Yak? iv. How much more weight does lion have than Dolphin? v. How much less weight does Bear have than Giraffe? (b) Weather report of our five cities: Cities Maximum Temperature Minimum Temperature Rainfall Dhankuta 24° C 10° C 12 mm Kathmandu 28° C 7° C 25 mm Pokhara 31° C 12° C 52 mm Surkhet 33° C 15° C 40 mm Dipayal 29° C 10° C 32 mm i. Which is the hottest city? ii. Which city has minimum temperature? iii. Which cities have the same temperature? iv. Which city has maximum temperature 28° C? v. Which city has more rainfall? vi. How much more rainfall does Dipayal have than Dhankuta? vii. Which city has the least rainfall? viii. How much less rainfall does Surkhet have than Pokhara?
266 The Leading Mathematics - 5 14.2 Reading Bar Graph At the end of this topic, the students will be able to: ¾ read the simple bar graph. Learning Objectives Read, Think and Learn The given graph shows the number of toys sold on Monday at Shanti’s Toy shop. Use the graph to answer the following questions: Number of toys Car 1 2 3 4 5 6 7 8 9 0 Elephant Duck Drums Toys (a) How many cars were sold? (b) How many more elephants than duck were sold? (c) Which toy was sold the most? (d) How many elephants and drums were sold in all? (e) Which toy did the shop sell more, car or duck?
Statistics 267 PRACTICE 14.2 Your mastery depends on practice. Practice as you play. 1. Graph below gives the vehicles passed by Mahendrapul, Pokhara in Sunday morning. Read the bar graph carefully and answer the questions given below: Vehicles Number of Vehicles Bus 5 10 15 20 25 30 35 0 Bus Car Jeep Taxi Van (a) Which vehicle is passed more from thew Mahendrapul? (b) Bus carries 40 people in average. How many people travelled in the bus through Mehendrapul? (c) Which vehicle is through the Mehendrapul? (d) Which two types of vehicles passed in the same number this morning? (e) What is the total number of vehicles recorded in that morning? (f) How many more car is passed through the Mahendrapul than bus? (g) How many less jeep is passed through the Mahendrapul than taxi?
268 The Leading Mathematics - 5 2. Answer the following questions based on the bar graph. No. of Pets Sold (in dozen) Months 1 2 3 4 5 6 7 8 9 10 0 January February March April May June July (a) How many pets were sold in July and April altogether? (b) How many more pets were sold in June than in March? (c) How many pets were sold in March, January and June? (d) In August, twice the number of pets were sold than in May. How many pets were sold in August? (e) Were more pets sold in January or in April? 3. The given bar graph shows the career preferences of group of middle school students. Study the bar graph and answer the following questions. Number of Students Carrier chosen 10 20 30 40 50 60 70 80 90 0 Teacher Lawyer Doctor Sport man Actor Scientist (a) What is the most popular career? (b) What is the least popular career? (c) Which career do 30 students prefer? (d) Name all the career of choices for students.
Statistics 269 Read, Understand, Think and Do 1. Sunil bought some books and stationery goods. After paying Rs. 2000 and he tells to prepare the bill. 3 The Leading Maths with Rs. 300 each, 3 Pen with Rs. 15 each, 5 copies with Rs. 40 each and 1 calculator with Rs. 500. (a) Prepare the bill for the above list. (b) How much amount did he refund? 2. The income and expenditure of a family in a month is as given below. Income: Salary = Rs. 50000, Rent = Rs. 20000 Expenditure: Food = Rs. 25000, Education = Rs. 10000, Clothes = Rs. 8000 Medicine = Rs. 5000, Others = Rs. 7000 (a) Prepare the house bill for the above list. (b) How much amount did the family save in a year? 3. The price list of each item and the sample bill of a store are shown below: Price list Sugar Rs. 75 per kg Butter Rs. 450 per kg Coffee Rs. 950 per kg Banana Rs. 120 per dozen Tomato Rs. Rs. 35 per kg Apple Rs. 120 per kg Rice Rs. 85 per kg Unique Grocers Mangalbazar, Lalitpur Bill no. 26785 Customer’s Name : ........................... Address : .......................... In words: .......................................................................................... Date: .......................... S.No. Items Quantity Rate (Rs.) Price (Rs.) 1. 2. 3. 4. 5. Total Salesman : ....................... Dhaniram, (Godawari - 4 Lalitpur) bought 2 dozens of bananas, 3 kg of tomatoes, 5 kg of sugar and 5 kg of rice at 2080/9/30. (a) Prepare the bill for the above list. (b) Is Rs 1000 sufficient for the above list? Justify. MIXED PRACTICE–IV
270 The Leading Mathematics - 5 4. Nandaram (Mahalaxmi – 4, Laliltpur) buys 6 kg of flour at the rate of Rs. 100 per kg, 9 kg of sugar at the rate of Rs. 95 per kg and 5 kg of rice at the rate of Rs. 85 per kg at 2080/10/12. (a) Make a bill for the purchases. (b) Is Rs 2000 sufficient for the above list? Justify. 5. The list of items with quantity and their cost are presented below. Items Quantity (kg) Rate (Rs.) Mango 3 150 Apple 5 200 Grapes (Black) 4 250 Papaya 3 100 (a) Make a bill for the purchases. (b) How much amount returned from Rs. 3000? 6. The fruits and their cost are presented below. Items (per kg) Rate (Rs.) Mango 150 Apple 200 Grapes (Black) 250 Papaya 100 Orange 120 Pomegranate 280 (a) Draw a simple bar graph. (b) Which of the above fruits is more expensive than others?
Statistics 271 FM : 12 Time : 40 Min. CONFIDENCE LEVEL TEST IV STATISTICS 1. Observe the family budget below and answer the following questions. (a) How many income is there from salary and rent? [1] (b) What is the total monthly expenditure? [1] (c) How much does it save in a month? [1] 2. Study the income and expenditure of a stationary shop in a month below and answer the following questions. Income : Sales of books = Rs. 1,25,000, Sales of pencils = Rs. 10,000, Sales of pens = Rs. 15,000, Sales of erasers = Rs. 8,800, Miscellaneous = Rs. 14,000. Expenditure : Rent = Rs. 20,500, Salary = Rs. 34,000, Books = Rs. 85,000, Taxes = Rs. 5,000, Others = Rs. 10,500 (a) Prepare a budget of the shop. [2] (b) Find the saving or loss amount of the shop by how much? [1] 3. Observe the adjoining bill and answer the following questions. (a) Who is the customer? [1] (b) How many roti did the customer eat? [1] (c) What is the total bill with service charge and VAT that is paid by customer? [1] 4. Pramod bought some books from Allied Publication. The rates of the books and their quantities are mentioned below. 3 pieces of The Leading Maths-5 at Rs. 446, 4 pieces of The Leading Maths-4 at Rs. 424, 5 pieces of The Leading Maths-3 at Rs. 450, 6 pieces of The Leading Maths-2 at Rs. 438, (a) Prepare the bill for the above list. [2] (b) How much amount did he refund if he gives Rs. 8000? [1] Hotel Prince Plaza Mangalbazar, Lalitpur Customer’s Name : Hukka Man Address : Kathmandu Date: 17/05/2080 S.N. Particular Qty. Rate Price (Rs.) 1. Roti 8 45 360 2. Plain rice 1 150 150 3. Dal fried 2 150 300 4. Mix. Veg. curry 1 260 260 Sub Total 1070/– 10% S.C. 107/– Total 1177/– 13% VAT 153/– Grand Total Rs. 1330/– Salesman : ....................... Bill no. 0585 In words: One thousand three hundred thirty only Attempt all the questions. Income Expenditure Topics Amount Topics Amount Salary Rs. 32000 Food Rs. 25500 Rent Rs. 20000 Education Rs. 35000 Business Rs. 40300 Clothes Rs. 16400 Agriculture Rs. 22400 Vehicles Rs. 25430 Other Rs. 12200 Other Rs. 7540 Total Rs. 126900 Total Rs. 109870
272 The Leading Mathematics - 5 ALGEBRA UNIT V COMPETENCY Solving the simple problems related to algebraic expression and equation by using the algebraic skills. CHAPTERS 15 Algebraic Expressions 16 Equation of One Variable LEARNING OUTCOMES After completion of this content area, the learner is expected to be able to: write the algebraic expression to the simple verbal problem (including addition and subtraction of two terms only). solve the equation of one variable.
Algebra 273 CHAPTER 15 ALGEBRAIC EXPRESSIONS What is the income of the lady ? How much amount does she give her daughter ? How much amount does she expense on shopping ? How much amount does her husband give to her ? Again, she buys some foodstuff. How much amount does she pay to the shopkeeper ? How much money is left with her at last? WARM-UP Lesson Topics Pages 15.1 Constant and Variable 274 15.2 Algebraic Expression 277 15.3 Values of Algebraic Expression 280 15.4 Adding Algebraic Expressions 283 15.5 Subtracting Algebraic Expressions 286
274 The Leading Mathematics - 5 15.1 Constant and Variable At the end of this topic, the students will be able to: ¾ identify constant and variable. ¾ perform basic operations on constants and variables. Learning Objectives Read, Think and Learn In a town, one one shoe shop sells its goods in half price. If x is the regular price of a good, then its price in that bazar is expressed as x ÷ 2. Since the price of the goods may be different, so x has different values. Here x is a variable and x ÷ 2 is an expression. The number 2 is a constant. Similarly, next stall decreases Rs.5 on the price of all goods. Suppose y is the price of goods then its selling price is expressed as y – 5. Here, y is variable and y – 5 is an expression. Variable : A variable is a letter or symbol used to represent more values. For example; If x is a factor of 12, then the letter x represents all of the factors of 12. ∴∴ x = 1, 2, 3, 4, 6, 12 Constant : The number or symbol used to represent the fixed number is a constant. For example, the number 2 or the length of the line segment 6 cm etc. are the examples of constant. Basic Operations with Variables and Constants Addition : a + b means the sum of a and b. If a = 3 and b = 2, then a + b = 3 + 2 = 5. Subtraction : a – b means the difference of a and b or a minus b. If a = 3 and b = 2, then a – b = 3 – 2 = 1
Algebra 275 Multiplication : a × b or a . b or (a) (b) or ab means the product of a and b. If a = 3 and b = 2, then a × b = 3 × 2 = 6. Division : a÷ b or a b , b ≠ 0 means the quotient when a is divided by b. If a = 4 and b = 2 then a ÷ b = a b = 4 2 = 2 is the quotient. Points to be Remembered 1. A variable is a letter or symbol used to represent more values. 2. The number or symbol used to represent the fixed number is a constant. 3. The sum of at lest two numbers is called addition. 4. The difference of two numbers is called subtraction. 5. The repeated addition of the same objects or numbers is called multiplication. 6. The sharing of an object or number in equal parts is called division. PRACTICE 15.1 Your mastery depends on practice. Practice as you play. 1. Which of the symbols or letters used in the following statements denote variable or constant? Write their values. (a) x represents the whole numbers between 4 and 7. (b) a represents the whole numbers less than 3. (c) y represents the whole numbers less than 7 and greater than 5. (d) p represents the prime numbers between 15 and 20. (e) b represents the composite numbers greater or equal to 7. (f) q is the factor of 1. (g) z is the sum of 2 and 7. (h) c is the difference between 10 and 19. (i) is the prime factor of 15. (j) ∆ is the element of the set of square numbers less than 10.
276 The Leading Mathematics - 5 2. Find the values of the unknown letters for the given condition and identify which of them is variable or constant? (a) x is the multiple of 5 less than 20. (b) a is the prime factor of 5. (c) b represents remainder when 25 is divided by 4. (d) y is the quotient when 20 is divided by 5. (e) c is the product of the numbers 6 and 3. 3. (a) If a = 4 and b = 3, what is the sum of a and b? Find it. (b) If x = 3 and y = 7, what is the sum of x and y? Find it. (c) If p = 7 and q = 4, subtract q from p. (d) If m = 9 and n = 8, what is the difference of m and n? (e) If c = 6 and d = 3, find the product of c and d. (f) If k = 3 and l = 8, which is the product of k and l? (g) If a = 12 and b = 3, what is the value of the quotient when a is divided by b? (h) If r = 5 and s = 15, what is the value of the quotient when r divides s? (i) If p = 5 cm, q = 6 cm and r = 7 cm, what is the sum of p, q and r? Find it. (j) If a = 4 cm, b = 7 cm and c = 5 cm, find the difference of b from the sum of 'a' and 'c'. (k) If x = 2, y = 4 and z = 6, find product of x, y and z. (l) If m = 4, n = 6 and l = 12, find the quotient when the product of m and n is divided by l.
Algebra 277 15.2 Algebraic Expression At the end of this topic, the students will be able to: ¾ define algebraic expression. Learning Objectives Read, Think and Learn Algebraic Term The price of 1 kg of apples is Rs. 200 or suppose Rs. x. Then the price of 3 kg of apples is Rs. 3x. Similarly, the price of 1 2 kg of apples is Rs. x 2. Here, x, 3x and x 2 give single value or price of the apples. These are terms. The product of constant and variable or a constant or a variable forms an algebraic term. For example, the product of 2 and x is 2x, which is a term. The constant 'a' is itself a term. Similarly, the variable y is also itself a term. Algebraic Expression An expression is the meaningful combination of numbers/variables connected by the operations of addition, subtraction, multiplication or division (+, –, ×, ÷). For example, Ö 2x + 4 is an expression, which is the sum of double of x added to 4. Ö x – + 2 is not an expression because – + together have no meaning. Ö y = 2x + 4 is not an expression. The equality sign = is not an operation sign. Ö Is x is an expression? Yes, because x = x + 0 or 1 × x.
278 The Leading Mathematics - 5 PRACTICE 15.2 Your mastery depends on practice. Practice as you play. 1. Match the following: Column A Column B (a) The sum of 7x and 5 i. 5m 2n (b) The difference of x and 9y ii. z – (3x + 2y) (c) The quotient of 5m divided by 2n iii. 4x – 5 9 (d) The sum of 3x and 2y divided by z iv. 3(4x – 2y) (e) The difference of 4x and 5 divided by 9 v. 7x + 5 (f) The difference of 4x and 2y multiplied by 3 vi. 3x 2y + 5 (g) The quotient of 3x and 2y added by 5 vii. 3x + 2y z (h) The sum of 3x + 2y taken from z viii. x – 9 y 2. Which of the following is an algebraic expression or not? (a) 3x (b) 2x + 1 (c) x = 4 (d) x ± 4 (e) 7x – 1 2 (f) 2x + 4y 3 (g) 3x + 2 2 = 0 (h) 7x = ± 2 (i) 3x – 4y + z (j) 2x + 3(a + b) (k) 4x = 3x – 1 (l) 2x + 1 4 = 4 (m) 2x + 3y = 0 (n) 3x + 4 = 3 (o) 2y – 3 (p) 2x – 1 5
Algebra 279 3. Write the following statements in the form of algebraic expressions: (a) 15 subtracted from 2x. (b) 19x divided by y. (c) 2x added to 5. (d) The product of 5 and 3x. (e) Multiplication of 2y and 3. (f) 3x divided by 4. (g) Sum of 'a' and 'b'. (h) Addition of 2p and 5. (i) Out of Rs. 15 Suman spends Rs. x. (j) Out of 100 students 2x were absent. (k) In a group of y players, another group of x was added. (l) Sunita's age is x years. Her father is 15 years more than double of her age. What is the father's age? 4. Write the following algebraic expressions into words: (a) 2x (b) 3x (c) x + 2 (d) 3x + 1 (e) 5p – 4 (f) 2(a + b) (g) 5(p – 2) (h) 2x + 3y (i) 4a – 5 (j) xy – p (k) ab – 3a (l) 6b – 5a (m) 4x + 3 (n) 3x + y (o) 2x – 1 (p) xyz + 1 (q) xy(z + 1) (r) a(bc + 2)
280 The Leading Mathematics - 5 15.3 Values of Algebraic Expression At the end of this topic, the students will be able to: ¾ find the value of the algebraic expression by substituting the given value of the variable or constant or both. Learning Objectives Read, Think and Learn Replacement If two quantities are equal, one may be replaced in the place of other. For example, consider the expression, a + b and b = 2a Then the replacement of b = 2a in a + b gives, a + b = a + 2a = 3a Value of Algebraic Expression The value of algebraic expression is obtained by substituting/ replacing the value of the variable and operation over the sign/s of operations contained in the expression. For example, x is the variable for the set {1, 2, 3, 4, 5} in the algebraic expression 3x + 2. If x = 1, then 3x + 2 = 3 × 1 + 2 = 5 If x = 2, then 3x + 2 = 3 × 2 + 2 = 8 If x = 3, then 3x + 2 = 3 × 3 + 2 = 11 If x = 4, then 3x + 2 = 3 × 4 + 2 = 14 If x = 5, then 3x + 2 = 3 × 5 + 2 = 17 x 1 2 3 4 5 3x + 2 3 × 1 + 2 3 × 2 + 2 3 × 3 + 2 3 × 4 + 2 3 × 5 + 2 Values 5 8 11 14 17 Here, the numbers 5, 8, 11, 14, 17 are the numerical values of the expression 3x + 2 depending upon different replacement of x from the set of values 1, 2, 3, 4, 5.
Algebra 281 CLASSWORK EXAMPLES Example 1: Write an expression for the perimeter of the triangle ABC and find the actual perimeter if x = 5 and y = 3. Solution: Here, in ∆ABC, AB = 7, BC = 2x, AC = 3y, x = 5, y = 3 Now, Perimeter of the triangle = AB + BC + CA = 7 + 2x + 3y = 7 + 2 × 5 + 3 × 3 [∵ Substituting x = 5 and y = 3] = 26 PRACTICE 15.3 Your mastery depends on practice. Practice as you play. Find the numerical value of the following expressions replacing the given value of the variables: (a) 2m + 3 when m = 3 (b) 2x + 3y when x = 2 and y = – 1 (c) 3p – 4q when p = 4 and q = 3 (d) 3x(2m – n) when x = 1, m = 2 and n = 3 (e) 3x + 2y 5 when x = 6 and y = 1 (f) 3x – 2y 5 when x = 3 and y = 2 (g) πr2 h when π = 22 7 , r = 14 and h = 10 (h) 1 3(l × b × h) when l = 15, b = 17 and h = 2.5 2. Substitute and simplify the following expressions when x = 5, y = 7 and z = 1. (a) 2x + 3y – 4z (b) 4x – 3y + 2z (c) 3x + 2y 4x (d) 5x(x2 + y2 ) 37 A B C 7 3y 2x
282 The Leading Mathematics - 5 3. Write algebraic expression for the length of the following line segments and then find their actual length if a = 3 and b = 2. (a) a 4 (b) a a 3 (c) (3a + 2) 3b (d) (2a – 1) 3b (e) a 2a a (f) a (2a + 1) 2b (g) 3a 2a (a + 5) (h) (a + 2) (2b – 1) (a + b) 4. Write the expression for the perimeter of the following shapes. Find the actual perimeter if x = 2 and y = 3. (a) 3x 2x y (b) 5x (c) 3x 2y (d) x x x y 4x 2y (e) x x y y (f) x x x x xx x x x x y y (g) 2x 2x 2y 3x 3x 2x (h) 3x 5x + y 2x + 1 3y – 2 3x + 2 5x – 3
Algebra 283 15.4 Adding Algebraic Expressions At the end of this topic, the students will be able to: ¾ add the algebraic expressions. Learning Objectives Read, Think and Learn Activity - 1 Observe the fruits on the two plates. How many apples are there in each plate? What are total apples? + = 4 apples 3 apples 7 apples Suppose an apple = x. Then, 4x and 3x are like terms. Like terms can be added. Here, 4x = x + x + x + x (4 times x) 3x = x + x + x (3 times x) ∴ 4x + 3x = x + x + x + x + x + x +x (Putting together) = 7x (7 times x) Alternatively: 4x + 3x = (4 + 3)x = 7x Algebraic expressions can also be added vertically as, 4x + 3x 7x They are the same kinds of apples. This is called like apples or term. Add coefficients only and express with common variable.
284 The Leading Mathematics - 5 Activity - 2 Observe the fruits on the two plates. How many apples and bananas are there on each plate? Can you add apples and bananas? Apples and banana are different fruits. Suppose, an apple = a, a banana = b. On the first plate = 3a and 3b On the second plate = 2a and 4b Fruits on the first plate = 3a + 2b = a + a + a + b +b Fruits on the second plate = 2a + 4b = a + a + b + b + b + b Total fruits on both plate = (3a + 2b) + (2a + 4b) = a + a + a + b + b + a + a + b + b + b + b = 5a + 6b Why we cannot add 'a' and 'b'? The terms 5a and 6b are dislike terms. CLASSWORK EXAMPLES Example 1: What is the sum of 3x, 4x and 5x? Solution: The sum of 3x, 4x and 5x is represented by, 3x + 4x + 5x = (3 + 4 + 5)x = 12x Example 2: Add: (5x + 8) and (7x + 1). Solution: Here, (5x + 8) + (7x + 1) = 5x + 8 + 7x + 1 = 5x + 7x + 8 + 1 = (5 + 7)x + (8 + 1) = 12x + 9 "Alternatively" 3x + 4x + 5x = 7x + 5x = 12x "Alternatively" 5x + 8 + 7x + 1 12x + 9
Algebra 285 PRACTICE 15.4 Your mastery depends on practice. Practice as you play. 1. Add: (a) 2a + 3a (b) 2m + 5n (c) 3x + 6x (d) 7x + 2x + 3 (e) 4p + 2p + 9 (f) 7b + 2b – 3 (g) 8y + 3y + 4 (h) 9z + 3z + 4 + 1 (i) 11m + 3 + 2m + 4 (j) 5a + 2a + 3a (k) 7x + 2x+ 3x (l) m + 4m + 5m (m) 2a + 5b + 3a (n) 4d + 2c + 3d (o) 4y + 7y + 9y (p) 3x + 2y + 2x + y + 4 (q) 5x + 3y + x + 3 + y (r) 6a + b + 2c + 3b + 3c 2. Add: (a) 2a + 5a (b) 3a + 7a (c) 9a + 2a (d) 3a + 2b + 4a + b (e) 5x + 7y + 2x + 3y (f) 5m + np + m + 6np (g) 2a + 3b + 4c + 3a + 4b + 2c (h) 2a + 3b + 4c + 5a + b + 2c (i) 5x + 3y + 2z + x + 2y + 4z 3. Find the sum of: (a) 4x, 3x and x (b) 7a, 3a and 2a (c) 8b, 7b and b (d) 7x, 3y and 2x (e) 8a, 2b and 3b (f) 7p, 6q and 2p (g) 2y, 3x, 3y and 5 (h) 2x, 3y, 4x and 1 (i) 7x, 3y, 2x, 3y and z (j) 2x + 1 and 3x + 5 (k) 6x + 3 and 3x + 1 (l) 7x + y and 2x + 3y (m) 2p + q and 4p + 2q (n) 6x + p and x (o) 7a + 5b and 2a + c (p) 5x + y and y + z (q) 2x + y + 2 and y + 4 (r) 2a + b + 3 and a + 2b
286 The Leading Mathematics - 5 15.5 Subtracting Algrabraic Expressions At the end of this topic, the students will be able to: ¾ subtract the algebraic expressions. Learning Objectives Read, Think and Learn – = 4 apples 3 apples 1 apple Suppose an apple = x, then 4x and 3x are like terms. Like terms can be subtracted. Here, 4x = x + x + x + x and 3x = x + x + x ∴ 4x – 3x = x + x + x + x – (x + x + x) = x + x + x + x – x – x – x = x Alternatively, 4x – 3x = (4 – 3)x = 1x = x CLASSWORK EXAMPLES Example 1: Add: a + 2b – 4c with 2a – 3b + 2c. Solution: Here, (3a + 2b – 4c) + (2a – 3b + 2c) = 3a + 2a + 2b – 3b – 4c + 2c [ collecting the like terms together] = 5a – b – 2c Adding vertically, we get 3a + 2b – 4c + 2a – 3b + 2c 5a – b – 2c 3 + 2 = 5 2 – 3 = –1 – 4 + 2 = –2 Subtract the coefficient only and express with common variable. 4x – 3x x
Algebra 287 Example 2: Subtract : 5m + x – 2p from 7m – n + 4p Solution: Here, 7m – n + 4p – (5m + 3n – 2p) = 7m – n + 4p – 5m – 3n + 2p = 7m – 5m 2 n– 3n + 4p + 2p = 2m – 4n + 6p While adding like term, the terms with the same sign are added but sign is as it, e.g. 4p + 2p = 6p and – n – 3n = – 4n. We can be done vertically as, PRACTICE 15.5 Your mastery depends on practice. Practice as you play. 1. Subtract: (a) 5x – 2x (b) 7a – 3a (c) 4b – b (d) 3a – 2a – 1 (e) 4p – p – q (f) 6m – 3 – 2m (g) 3x – 2y – 2x (h) 5x – 2m – 3n (i) 5x – x – 2x (j) 13m – 5m – 2m (k) 15xy – 2xy – 7xy (l) 4x – 7x + 9x (m) 12pq – 15pq + 9pq (n) 18xy – 20xy + 3xy (o) 24xy – 30xy + 7xy – xy (p) 9xy + 7xy – 2xy – 8xy + 3xy 2. Subtract: (a) 7m – 2m (b) 15n – 12n (c) 13a – 7a (d) 17x + 13y (–) 13x + 10y (e) 12m + 13n (–) 7m – 2n (f) 14pq – 13nt (–) 12pq – 2nt While subtracting the term with '+' changes to '–' and the term with '–' changes to '+'. 7m – n + 4p (–) 5m + 3n – 2p 2m – 4n + 6p – – + Sign changing & operating with new signs.
288 The Leading Mathematics - 5 (g) 22x + 7y (–)15x + 4y (h) 15m + 10n (–) 9m – 3n (i) 12pq – 24nt (–) 7pq – 19nt (j) 7x + 2y + 13z (–)6x – y + 2z (k) 14pq – 13qr + 12rs (–) 10pq – 11qr + 9rs (l) 12abc – 6pqr + 2rst (–) abc – 2pqr + 3rst 3. Find the difference: (a) 3a - (2a + b) (b) 2a –(b – a) (c) 4a – (3a – b) (d) (2a + b) – (a + b) (e) (3a + 4b) – (2a – 3a) (f) (6a – 3b) – a + 4b) 4. Subtract vertically: (a) 3x – 2y from 4x + 7y (b) 2xy + 3yz from 5xy – 2yz (c) pq – 2st + 3uv from 15pq – 12st – 4uv (d) 4stu – 17qpr – 7abc from 16 stu – 12qpr – 5abc (e) 15abc + 14cde – 15bcd from 25cde – 15abc + 21bcd 5. (a) What is to be subtracted from a – 2b to make it 2a + b? (b) What is to be subtracted from 9x + 5y to make it x + y? (c) What is to be added to 11p + 12q to make it 14p – 12q? (d) What is to be added to 5x + 6y – 2z to make it 9x + 2y – 15z? (e) What is to be added with 2ax + 3bx – 4cx to make it 7ax – 5bx – 5cx?
Algebra 289 CHAPTER 16 EQUATION OF ONE VARIABLE What do the above pictures show ? How many potatoes are there in 500 gm in the above Fig. (I)? If we weighs 1 kg of potatoes, how many potato are there ? If we weighs 5 kg of potatoes, how many potatoes are there ? Is the beam balance in the second picture balance ? Is the beam balance of the second picture in balanced? How many apples are there on the left pan of the beam balance? How many apples are there on the right pan of the beam balance? How many bananas are equal to one apple? If one apple is added to the left pan, how many bananas are added to the right pan for balancing it ? WARM-UP Lesson Topics Pages 16.1 Equation in One Variable 290 16.2 Solving Equation in One Variable 293 16.3 Verbal Problems Related to Equations 297 Fig. I Fig. II
290 The Leading Mathematics - 5 16.1 Equation in One Variable At the end of this topic, the students will be able to: ¾ introduce an equation in one variable. Learning Objectives Read, Think and Learn Observe the adjoining beam balance. What do you see? 6 apples = 1 kg and price of 1 kg = Rs.140. Suppose, apple = x, then 6x = 140. Two balancing mathematical expressions is called an equation. For example, 2x = 4, 3x – 1 = 5, 4x = 0, etc. Consider the equation x + 1 = 6 Showing the equation is the beam balance, we have Here both sides of the beam balance are balanced. Now, take out 1 from the left side pan. What happens? Now, the balance shows that. x < 6 or, 6 > x To make the balance balanced, take out 1 from the right side pan. What happens? Again, the beam balance is balanced. Now, this shows the equation x = 5. x + 1 6 x 6 x + 1 – 1 x 5 x + 1 – 1 6 – 1
Algebra 291 Similarly, x + 1+2 6 + 2 3(x+ 1) 3 × 6 x + 1 2 6 2 When any term or equal terms are added or subtracted or multiplied or divided on the both sides of an equation, they are also equal. This is called equal axiom. PRACTICE 16.1 Your mastery depends on practice. Practice as you play. 1. Write down numerical equations of the following: (a) Two plus 6 is equal to 2 multiplied by four. (b) Seven minus five is equal to six divided by three. (c) Four multiplied by 5 is equal to 5 multiplied by four. (d) Twenty divided by 5 is equal to sixteen divided by four. (e) Six plus two minus four is equal to ten minus six. 2. Write down an equation for each picture. (a) x (b) y (c) x x 10 (d) x x x 30 (e) x x x (f) x x x z
292 The Leading Mathematics - 5 3. How many numbers should add in the pan so that the pans are balanced? (a) (b) (c) (d) (e) (f) (g) (h) 4. Rewrite each number sentence by changing the number in the box so that you have a true sentence. (a) 3 + 4 = 10 (b) 10 – 3 = 1 (c) 18 9 = 9 (d) 24 3 = 6 (e) 6 × 2 = 8 + 10 (f) 4 × 5 – 3 =10 – 8 (g) 30 5 = 8 (h) 5 × 3 = 24 + 0 (i) 7 × 6 – 8 =10 × 4
Algebra 293 16.2 Solving Equation in One Variable At the end of this topic, the students will be able to: ¾ solve the linear equation in one variable. Learning Objectives Read, Think and Learn Consider an equation x = 5. How much does one group of x correspond the number to other side? This suggests in the following axioms of equality in solving questions. If equal are added to equal parts, the sum is also equal. i.e. if x = 5 Then x + 2 = 5 + 2 [ Adding both sides by 2], If equal are subtracted from equal, then the difference is also equal. i.e. if x = 5 Then x – 2 = 5 – 2 [ Subtracting 2 both sides], If the equal are multiplied by equal then the product is also equal. i.e. if x = 5 Then 2x = 10 [ Multiplying both sides by 2], If equal are divided by equal, then the quotient is also equal. i.e. if x = 5 Then x 2 = 5 2 [ Dividing both sides by 2] What is the name of these rules on equation? If x = 2, solution of the equation 5x – 12 = 18? Solution: Substitution x = 2 in equation 5x – 12 = 18. We get, 5(2) – 12 = 10 – 12 = – 2 ≠18. Therefore, x = 2 is not the solution of the equation 5x – 12 = 18. These are interesting rules.
294 The Leading Mathematics - 5 CLASSWORK EXAMPLES Example 1: Solve : 3x + 4 = 16 Solution: Here, 3x + 4 = 16 or, 3x + 4 – 4 = 16 – 4 [ Take out 4 from both sides] or, 3x = 12 or, 3x 3 = 12 3 [ Make group of 3's both sides] or, x = 4 When we put the value of x = 4 in the given equation 3x + 4 = 16, it becomes true. x = 4 is called the solution of the equation 3x + 4 = 16. Example 2: Solve : 3x 5 – 2 = 13 Solution: Here, 3x 5 – 2 = 13 or, 3x 5 – 2 + 2 = 13 + 2 [Adding both sides by 2] or, 3x 5 = 15 or, 3x × 5 5 = 15 × 5 [Multiplying both sides by 5] or, 3x = 15 × 5 or, 3x 3 = 15 × 5 3 [Dividing both sides by 3] or, x = 25 Let's check it : 3 × 25 3 – 2 = 15 – 2 = 13, which is true. PRACTICE 16.2 Your mastery depends on practice. Practice as you play. 1. Solve for x. (a) x – 3 = 0 (b) x + 4 = 0 (c) x + 3 = 6 (d) x – 3 = 2 (e) x + 6 = 2 (f) 3x = 6
Algebra 295 (g) –6x = 18 (h) –4x = 2 (i) x 2 = 4 (j) x 3 = 5 (k) x 4 = 1 2 (l) 3x 4 = 9 2. Solve for x. (a) 3x – 4 = 2x + 1 (b) 5x – 15 = 12 – 4x (c) 5x + 7 = 2x – 14 (d) 13x – 12 = 10x + 27 (e) 7 – 2x = 3x – 8 (f) 19 – 7x = 3x – 11 3. Solve for x: (a) 3x – 4 = 2x + 1 (b) 5x – 15 = 12 – 4x (c) 5x + 7 = 2x – 14 (d) 13x – 12 = 10x + 27 (e) 7 – 2x = 3x – 8 (f) 19 – 7x = 3x – 11 4. Write an equation in x if the two pencils are of equal length and hence find the value of x. (a) x + 7 11 (b) x – 5 12 (c) 2x – 6 14 (d) 3x + 5 14 (e) 3x 4 – 1 11 (f) x + 6 5x 3 + 4
296 The Leading Mathematics - 5 5. Write an equation in x and hence find the value of x. (a) 2x x + 4 (b) x – 12 17 (c) 2x + 4 20 (d) 21 x 4 + 1 (e) 2x + 1 3 7 (f) 9x 4 – 1 2x + 1 (g) 7x + 1 6 6 (h) 13 3x 5 – 2
Algebra 297 16.3 Verbal Problems Related to Equations At the end of this topic, the students will be able to: ¾ solve the verbal problems related to the equations. Learning Objectives Read, Think and Learn CLASSWORK EXAMPLES Example 1: The age of father is double of the age of his son exceed 15 years. If the age of father is 23 years, what is the age of his son? Solution: Let the required age of son be x. Then double of son's age is 2x. Father's age is 23 years. Now, 2x + 15 = 23 or, 2x + 15 – 15 = 23 – 15 or, 2x = 8 or, 2x 2 = 8 2 or, x = 4 Hence, the required age of the son is 4 years. Example 2: If both sticks are of equal length, find the value of x writing an equation in x. Also, find the actual length of the stick if the lengths are measured in cm. Solution: Since both sticks have the same length So, 3x – 7 = 2x + 1 or, 3x – 7 – 2x = 2x + 1 – 2x [∵ Subtracting 2x from both sides] or, x – 7 = 1 or, x – 7 + 7 = 1 + 7 or, x = 8 2x + 1 3x – 7
298 The Leading Mathematics - 5 And, actual length of the stick may be obtained by substituting x = 8 in any expression. 3x – 7 or 2x + 1 Substituting x = 8 in 3x – 7 we get, 3 × 8 – 7 = 24 – 7 = 17 cm Substituting x = 8 in 2x + 1 we get 2 × 8 + 1 = 16 + 1 = 17 cm Hence, the actual length of the stick is 17 cm. PRACTICE 16.3 Your mastery depends on practice. Practice as you play. 1. Observe the following two equal sticks in each picture and answer the given questions. (a) 2x + 4 x + 7 (b) 5x – 4 4x + 16 (c) 3x + 5 x + 21 (d) 15x – 17 10x + 18 i. Find the equation in x. ii. Solve it and find the value of x. iii. Find the actual length of the stick. 2. Write the equation and solve. (a) 6 added to a number gives 13. (b) 5 added to a number gives 17. (c) The difference of a number and 5 is 17.
Algebra 299 (d) The difference of a number and 11 is 9. (e) 6 times a number is 36. (f) 15 times a number is 36. (g) One fifth of a number is 15. (h) 5 times a number added to 10 gives 30. (i) 7 times a number taken from 20 gives 2. (j) 5 times a number taken from 22 gives 2. 3. Write the equations to represent the following verbal problems and solve them: (a) Amar spent Rs. 750 for shoe and he has left Rs. 150 now. How much did he have at first? (b) There are 16 girls in a class. If the total number of students in the class is 38, find the number of boys. (c) If three times of John's money is added with Rs. 15, it becomes 45. How much did he have at first? (d) In a school, the number of girls is two times of the number of boys. If the total number of students of the school is 456, how many boys are there in the school? (e) If two-thirds of a work can be finished in 40 days, in how many days the whole work can be finished? (f) What is the number when one-third of it is 14? (g) A ribbon is 3x m long and another is (x + 2)m long. If the total length of both ribbons is 22 m, what is the length of each ribbon?
300 The Leading Mathematics - 5 Read, Understand, Think and Do 1. An algebraic equation is given below. x + 3 = 5 (a) Write the differences between algebraic term and algebraic expression with example. (b) Identify algebraic term and algebraic expression from the given equation. (c) Find the value of x. 2. Two algebraic terms are given below. x and 3 (a) Define algebraic expression with example. (b) Make an algebraic expression by using the above two terms. (c) If x + 3 = 7, what should be the value of x? (d) x is both algebraic term and algebraic expression. Do you agree? Justify. 3. Two algebraic expressions are given below. 2x + 3y and 5xy – 3xy (a) Define like terms with example. (b) 2x + 3y is an algebraic expression. Why? (c) What is the value of 5xy – 3xy? 4. The algebraic expression is given below. 5x2 – 3y2 – 7x2 + 5xy + 4y2 + x2 – 2ab (a) List out the like and unlike terms. (b) Add and subtract the like terms. (c) Is it possible to add unlike terms? Why? 5. The length and breadth of a rectangle are 2x + 3y and x + 2y with x > y. [Perimeter of rectangle = 2(l + b)] (a) Are the length and breadth of rectangle in like terms? (b) Find the perimeter of the rectangle. (c) If x = 3 cm and y = 2 cm, what will be the actual length and breadth of the rectangle? Find it. MIXED PRACTICE–V