MEASUREMENT 251 Take a cuboid with length 4 cm, breadth 2 cm and height 3 cm. l = 4 cm, b = 2 cm, h = 3 cm What is its volume ? For this, at first divide the length, breadth and height equally into 4, 2 and 3 parts respectively as shown in figure. Now, count all the small cubic blocks so formed. There are 24 small cubic blocks. This is the volume of the given cuboid. \ Volume of cuboid (V) = 24 cm3 . = 4 cm × 2 cm × 3 cm = l × b × h Hence, the volume of the cuboid (V) = length (l) × breadth (b) × height (h) We know that the cube has the same length, breadth and height. i.e., l = b = h. \ Volume of cube (V) = l × b × h = l × l × l = l3 l l l l b h 4cm 2cm 3cm 4cm 2cm 3cm 2 24 2 12 2 6 3 cm3 = cm×cm×cm i.e., 24 = 2 × 2 × 2 × 2 × 3 = 4 × 2 × 3 19.7 Volume of Cuboid and Cube Read, Think and Learn
252 The Leading Mathematics - 4 CLASSWORK EXAMPLES Example 1 Write the length, breadth and height of the adjoining cuboid. Also, find its volume. Solution : From the given cuboid, length (l) = 4 units, breadth (b) = 2 units, height (h) = 3 units. Now, we know that The volume of the given cuboid (V) = l × b × h = 4 × 2 × 3 = 24 cu. units. Example 2 What is the volume of a cubical Rubik with length 48 cm ? Solution : Here, The length of the cubical Rubik (l) = 48 cm Now, we know that The volume of the given Rubik (V) = l3 = 483 = 48 × 48 × 48 = 110592 cm3 . EXERCISE 19.7 Your mastery depends on practice. Practice like you play. 1. Write the length, breadth and height of the following cuboids. Also, find their volume. (a) (b) (c) (d)
MEASUREMENT 253 (e) (f) (g) (h) 2. Find the volume of the following cuboids. (a) (b) (c) (d) (e) (f) (g) (h) 3. Find the volume of the following cubes : 4. (a) What is the volume of a box with length 25 cm, breadth 18 cm and height 12 cm ? (b) Find the volume of a wooden block with length 95 cm, breadth 42 cm and height 24 cm. (c) What is the volume of a cubical box with length 14 cm ? (d) Find the volume of a cubical dice with length 32 cm. Measure the length, breadth and height of a rectangular soap in cm, and draw their sketch. Find the volume of the soap by cutting each sides as its measures and using formulae. Prepare a report and present it in the classroom. PROJECT WORK 12 cm 6 cm 8 cm 7 cm 14 cm 9 cm 12 cm 9 cm 6 cm 15 cm 10 cm 6 cm 5 cm 7 cm 12 cm 15 cm
254 The Leading Mathematics - 4 Read, Understand, Think and Do 1. Convert the following unit to the given unit. (a) 3 days (Hour) (b) 5 hour (Secon(d) (c) 420 minute (Hour) (d) 120 hour (Day) 2. The time duration taken by a bus from Kathmandu to Butwal is 7 hour 30 minutes. Ram traveled 2 hr. 45 min. by bus and 1 hr. 30 min. by Scorpio. (a) Calculate the time he spent in traveling. (b) In which vehicle did he travel more by how much? Calculate it. (c) How much time is needed to more traveling? Calculate it. 3. Sabina works 6 hours 30 minutes in the first day and she works 5 hours 30 minutes in the second day in an industry. (a) How much time did she work in the first and the second days? (b) How much time did she work in the first day? 4. Convert the following. (a) 3 km (Meter) (b) 500 meter (centimeter) (c) 420 centimeter (millimeter) (d) 2000 millimeter (Meter) 5. The length of red ribbon is 1 m 55 cm and length of yellow ribbon is 1 m 75 cm. (a) Find the total length of both ribbons. (b) How long is the length of yellow ribbon than red ribbon? MIXED PRACTICE–V
MEASUREMENT 255 6. Rachana bought 8 kg 400 gm tomatoes, 2 kg 130 gm brinjal and 7 kg 300 gm watermelon from the green grocer. (a) Convert 1 kilogram into grams. (b) How many kilograms of fruits and vegetables did she buy? (c) How much more tomatoes did she buy than brinjal?. 7. Convert the following. (a) 3 liter (milliliter) (b) 5000 milliliter (liter) (c) 3 l 500 ml (millimeter) 8. In a square grid, a shape of square is shaded as shown in the figure. A small square has length 5 mm. (a) How many small boxes are there? (b) How many boxes are there in the side of length and breadth? (c) Find the length and breadth in cm. (d) Is the l × l = total shaded boxes? (e) Is the 4 × l = total side length of square? 9. Observe the given rectangular garden having the length and breadth 125 m and 75 m. (a) Write the formula to find the area of the garden. (b) Calculate the area of the garden. (c) Find the length of the wall around the garden.
256 The Leading Mathematics - 4 Attempt all the questions. FM : 20 Time : 40 Min. CONFIDENCE LEVEL TEST V MEASUREMENT 1. Anjana reads 3 hours 34 minutes whereas Jamuna reads 225 minutes. (a) How many days are there in a leap year? [1] (b) How many minutes does Anjana read for? Find it. [1] (c) How many hours and minutes does Jamuna read for? Find it. [1] (d) Who is reading more Anjana or Jamuna and how many minutes? [1] (e) If Anjana gets 20 paisa for 1 minute reading from his father, how much does she get in total? [2] 2. Jacson reached in two days at his home by walking a 56 km 400 m long road. He remembered that "he walked 32 km 500 in the first day". (a) How many millimeters are there in a meter ? [1] (b) Convert 56 km 400 m into meter. [1] (c) How long road did he walk in the second day? Find it. [2] (d) If he walked as long as the second day in the first day, when would he reached at his home? Why ? [2] 3. Arman bought 4 kg 450 gm potatoes, 3 kg 225 gm tomatoes and 2 kg 235 gm cauliflower from a shop. (a) How many grams are there in 3 kg? [1] (b) How many kilograms of vegetables did he buy from the shop? Find it. [2] (c) How much more potatoes than cauliflower did he buy ? Find it. [1]
MEASUREMENT 257 4. Convert the following. (a) 8000 milliliter (liter) [1] (b) 5.2 liter (milliliter) [1] (c) 2 l 456 ml (millimeter) [1] 5. Observe the rectangular open ground alongside and answer the following questions. 150 m 175 m (a) Write the formula to find the area of the rectangular ground. [1] (b) If a man wants to construct a wall around the open ground, how long wall would he construct? [2] (c) What is the area of the ground? Find it. [2]
258 The Leading Mathematics - 4 STATISTICS UNIT VI COMPETENCY Prepare the information of household income and expenditure as well as take and give the necessary information by looking bill. CHAPTERS 20 Bill and Budget 21 Statistics 0 10 Food Clothing Education House Rent Miscellaneous 20 30 40 50 60 70 80 AR Graphics Hetauda, Makawanpur Invoice Date: ..................... Invoice No.: PAN: 607662850 MS: _________________________________________________ _________________________________________________ SN Description Qty. Rate Amount Total Amt. in NRs. In words (NRs.): .................................................................... ................................................................................ E. & O. E. Signature PAN/VAT 9845456078, 9849098999 LEARNING OUTCOMES After completion of this content area, the learner is expected to be able to: take and give information from the bill. calculate the conclusion from a tabulated information. represent the given simple information into bar graph. plot the ordered pair in the graph to indicate any place.
STATISTICS 259 CHAPTER 20 Bill and Budget How many members are there in your family ? What types of goods do you need in your family ? What other things do your parents buy for foodstuff ? What house appliances do your parents buy ? How do your parents manage all these things ? From where do your parents earn money ? What types of jobs do your parents do ? Do you know how much money do your parents earn? Do you know how much money do your parents expense for you? How do you know how much expense is on different goods? Can you see the bill of different goods? WARM-UP Lesson Topics Pages 20.1 Reading Bill 260 20.2 Preparing Bill 263
260 The Leading Mathematics - 4 Observe the price list of vegetables in Kalimati Vegetables and Fruits Centre. Kalimati Vegetables and Fruits Rate Today SN Vegetable Rate/Kg SN Fruit Rate/Kg 1. Tomato (Local) Rs. 35 11. Guava Rs. 125 2. Tomato (Tunnel) Rs. 40 12. Kiwi Rs. 250 3. Potato (Red) Rs. 70 13. Pear Rs. 75 4. Potato (White) Rs. 45 14. Mango Rs. 150 5. Carrot Rs. 130 15. Pineapple Rs. 135 6. Radish Rs. 55 16. Lemon Rs. 45 7. Cauliflower Rs. 70 17. Pomegranate Rs. 300 8. Cabbage Rs. 30 18. Banana Rs. 95 9. Brinjal Rs. 41 19. Orange Rs. 115 10. Green Peas Rs. 190 20. Apple Rs. 190 Discuss the following questions: (a) Which vegetable is the most expensive ? (b) Which vegetable is the cheapest ? (c) Similarly, which fruit is the most expensive ? (d) Which fruit is the cheapest ? (e) Which vegetable has the cost of Rs. 55 ? (f) Which fruit has the cost of Rs. 135 ? (g) Which two vegetables are bought with Rs. 100? (h) Which three fruits are bought with Rs. 500? (i) What is the cost of 1-1 kg of radish, carrot and brinjal ? (j) What is the cost of 1-1 kg of lemon, mango and orange ? (k) What is the cost of 3 kg of potatoes? 20.1 Reading Bill Read, Think and Learn
STATISTICS 261 (l) What is the cost of 5 kg of pear ? (m) What is the cost of 2 kg of tomatoes and 3 kg of cauliflowers ? Study the given bill and discuss. Jyoti bought potatoes, onions, eggs and sugar from City General Store, Gongabu. After paying the amount for the items she purchased, the shopkeeper handed a bill, as the following : S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Maths book 1 pc 425 425 2. Copies 18 pcs 45 810 3. Pencils 6 pcs 12 72 4. Erasers 5 pcs 10 50 Total 1357 City General Store Gongabu, Kathmandu BILL NO. 017. Customer’s Name : Jyoti Thapa Address : Gongabu Date : 2080/10/11 In words NRs. : One thousand three hundred fifty seven. Signature of shopkeeper ........................... Shopkeeper Name of shop Address of customer Name of customer Date of buying Grand total paid by customer This is a paper bill which contains description, name, quantities, date, amount, etc. Generally, the bill has the following titles : Name and address of the shop. Bill number that helps to trace out the details. Name and address of customer. Date that helps to record the sold information. Name of items, their quantities and rates. Sum of price of each item with grand total. Other charges (e.g. tax, service charge etc.) (sometimes) The grand total amount in words. Signature of shopkeeper at last.
262 The Leading Mathematics - 4 1. Observe the price list of a shop and answer the given questions: SN Vegetable Rate/Kg 1. Rice Rs. 112 2. Gram Rs. 65 3. Pea (Green) Rs. 67 4. Pea (Dry) Rs. 60 5. Soybean Rs. 93 6. Red Lentils Rs. 80 7. Peas Rs. 72 8. Split green gram Rs. 80 9. Split Black gram Rs. 85 10. Dry Beans Rs. 96 2. Study the given bill and answer the following questions: S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Potatoes 3 kg 55 165 2. Onions 2 kg 60 120 3. Eggs 1 crate 425 425 4. Sugar 2 kg 85 170 Total 880 City Kinara Store Lagankhel, Kathmandu BILL NO. 032. Customer’s Name : Jamuna Malla Address : Bhotahiti, Kathmandu Date : 2080/08/05 In words NRs. : Eight hundred eighty only. Amar. ........................... Shopkeeper (a) Which is the most expensive item? (b) Which is the cheapest item? (c) Which item has the cost of Rs. 85 ? (d) Which two grams are bought with Rs. 150? (e) What is the cost of 1-1 kg of rice and green pea? (f) What is the cost of 1-1 kg of soybean, red lentils and dry beans? (m) What is the cost of 5 kg of rice and 2 kg of split green gram ? PRACTICE 20.1 Your mastery depends on practice. Practice as you play.
STATISTICS 263 (a) Write the name and address of the shop. (b) What is the bill number? (c) Who sold the items described in the given bill ? (d) Who is the shopkeeper ? (e) In which date are the good sold? (f) Write the name of items, their quantities and rates. (g) What is the amount of 3 kg of potatoes ? (h) What is the amount of each item ? (i) What is the grand total of all goods ? (j) Write the grand total amount in words. Look at the price list for the fruits in the Grocers shop. Prepare a bill for Hari if he bought 2 kg apples, 1 kg oranges and 2 dozen bananas and answer the following questions : (a) How many items did Hari buy ? (b) How much did Hari pay for apples ? (c) How much more money for banana did Hari pay than the oranges? (d) How much did Hari pay in total ? (e) If Hari handed in Rs. 300 to the shopkeeper, what amount did he get back ? GENERAL STORES Apples Rs. 60 per kg Oranges Rs. 45 per kg Bananas Rs. 29 per doz Grapes Rs. 60 per kg 20.2 Preparing Bill Read, Think and Learn
264 The Leading Mathematics - 4 Solution : Again, from the above bill, (a) Hari bought 3 items in all. (b) Hari paid Rs. 120 for apples. (c) Hari paid Rs. 3 more for bananas than the price of oranges. (d) Hari paid Rs. 213 in total. (e) If Hari gave in Rs. 300 to the shopkeeper, he gets Rs. 87 back. PRACTICE 20.2 Your mastery depends on practice. Practice as you play. 1. Read the following bill carefully. Allied Publication Bagbazar, Kathmandu BILL NO. ........... Customer’s Name : Sweta Mishra Address : Samakhusi Date : 2080/10/13 S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1. The Leading Maths - 4 12 435 5220.00 2. The Leading Maths - 5 15 447 6705.00 3. The Leading Maths - 6 18 528 9504.00 Total 21429.00 In words : Twenty one thousand four hundred twenty nine rupees only .................. Salesperson Sharmila 0052 Grocers Shop Kathmandu Costomer’s Name : Hari Pudasaini Address : Samakhusi Date : 2080/10/12 S.No. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Apples 2 60 120 2. Oranges 1 45 45 3. Bananas 2 24 48 Total 213 In words : Two hundred and thirteen rupees only. ...................... Shopkeeper Ashok BILL NO. 0050
STATISTICS 265 Now, answer the following questions: (a) Who sold the books ? (b) Who bought the books ? (c) From where did Sweta bought the books ? (d) What types of books did she buy ? (e) Which book is the most expensive ? (f) What amount did she pay in all ? (g) Who was the salesperson ? (h) What change did she receive if she gave Rs. 32,000 ? 2. Read the bill given below : Koirala Grocers Dharan, Sunsari BILL NO. 1248 Customer’s Name : Ramita Address : Koteshwor Date : 2080/10/15 S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1. Oranges 3 kg 40 120.00 2. Apples 4 kg 60 240.00 3. Grapes 2 kg 80 160.00 4. Bananas 1 kg 30 60.00 Total 580.00 In words : Five hundred eighty rupees only ................................ Salesperson : Sharmila Answer the following questions: (a) Who sold the fruits ? (b) Where is the Grocers’s shop ? (c) Who bought the fruits ?
266 The Leading Mathematics - 4 (d) What items of fruits did Ramita buy ? (e) What was the rate of oranges per kg ? (f) What was the rate for bananas ? (g) How much in all did she pay for the fruits ? (h) How much did Sharmila return when Ramita gave Rs. 1000 ? 3. Look at the Restaurant Menu. Now make the bill for the following orders : (a) Bread with butter 4 pcs Egg toast 2 pcs Tea 6 cups (b) Cream roll 3 pcs Veg. burger 2 pcs Coffee 2 cups Tea 3 cups (c) Egg toast 4 pcs Cream roll 7 pcs Veg. burger 2 pcs Tea 10 cups Coffee 3 cups 4. Look at the following price list. Now prepare the bills for the following purchases. (a) Rice 3 kg Egg 1 Crate Potatoes 4 kg Onion 1 kg (b) Potatoes 2 kg Dal 1 kg Onion 2 kg Egg 2 Crates CHILDREN’S MENU Bread with butter Rs. 35 Bread with jam Rs. 25 Cream roll Rs. 45 Egg toast Rs. 25 Veg. burger Rs. 80 Tea Rs. 20 Coffee Rs. 25 GENERAL STORES Eggs Rs. 310 Dal Rs. 115 Rice Rs. 78 Potatoes Rs. 60 Onion Rs. 45
STATISTICS 267 CHAPTER 21 Bar Graph How many books are there in the above picture ? 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 red apples are there in the above pictures? How many total apples are there ? Which apples are more and by how much ? WARM-UP Lesson Topics Pages 21.1 Bar Graph 268
268 The Leading Mathematics - 4 A traffic police recorded the following means of transport they passed by Thapathali in Saturday morning during 7 AM to 8 AM. Bus School Bus Private Car Government Van Taxi 30 20 15 25 10 In this information of the means of transport, we find the answers of the following questions : (a) Which is the most popular means of transport ? (b) Which means of transportation passed the least ? (c) How many private cars passed by Thapathali ? (d) How many school buses passed by Thapathali ? (e) How many government vans passed by Thapathali ? (f) How many total vehicles passed by Thapathali ? This information may be represented by means of bar graph as the following : 21.1 Bar Graph Read, Think and Learn
STATISTICS 269 0 5 10 15 20 25 30 35 Bus Means of Transport Number of Transportation School Taxi Bus Private Car Govt. Van Y X Here, the horizontal line OX represents the types of means of transport and the vertical line OY represents the number of transportation. Now, answer the following questions : (a) What does the height of the bar represent ? (b) Which bar is more or less wide ? (c) Is the space between the two bars equal ? (d) Which vehicle ran the most during one hour ? (e) How many vehicles are represented by square ? (f) How many vehicles are represented by one square box ? In this way, the bar graph provides the means for comparison at a glance. You can use bar graph in a variety to compare height, weight, achievement of students of a class, temperature, rainfall during the week, profit ratio of items sold in a week etc. and many more depending upon the situation. Therefore, the graph is useful way to display the collected information or data.
270 The Leading Mathematics - 4 EXERCISE 21.1 Your mastery depends on practice. Practice like you play. 1. Table below has the height of the students in a class. Height (cm) 110 cm 115 cm 120 cm 125 cm 130 cm No. of students 5 12 10 6 2 Draw bar graph to represent this data. 2. Table below has the rainfall record for the 7 days of a week in Bhadra. Days Sunday Monday Tuesday Wednesday Thursday Friday Saturday Rainfall (mm) 30 50 70 10 15 0 40 Draw bar graph to represent this information. 3. Classwise student number in Mangaldaya Ma. Vi. Chuwa in the primary section is given below : Class 1 2 3 4 5 No. of students 60 50 40 70 30 Draw bar graph to represent this information. 4. The table below shows certain favourite fruits among 170 students. Fruits Apples Mangoes Oranges Grapes Pineapples No. of students 30 50 60 20 10 Draw bar graph to represent this information. 5. Draw the bar graph for the given data : Height (in inch) 20 25 30 35 40 No. of students 8 6 10 12 6
STATISTICS 271 6. Read the following bar graph about the favourite subjects for the students of class 4. Favourite Subjects of Students 0 5 10 15 20 25 30 35 40 45 50 55 60 English Nepali Maths Science Social Subjects Number of Students Y X Now, answer the following questions : (a) What is the bar graph about ? (b) How many students are represented by one square ? (c) What does the height of the bar represent ? (d) How many subjects were asked in the interview for the favourite subjects ? (e) Which is the most favourite subject ? (f) How many students like English ? (g) How many more students like Nepali than English ? (h) How many students like social studies ? (i) Which is the subject less favoured by the students ? (j) How many students were interviewed ?
272 The Leading Mathematics - 4 1. Ramita bought some fruits and she got a bill as follows. Sharma Grocers Biratnagar, Morang Bill No. 1248 Customer’s Name : Samir Yadav Address : Sangam Tole S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1 Orange 3 kg 40 120.00 2. Apples 4 kg 60 240.00 3. Grapes 2 kg 80 160.00 4. Bananas 3 kg 30 60.00 Total 580.00 In Words : Five hundred eighty rupees only. Salesperson Susula Date : 2080-05-11 (a) Which fruits is the most expensive? (b) How much quantity of apples did she buy? (c) If she gave Rs. 1000 to the shopkeeper. How much amount does she get return? 2. The income and expenditure of a family in a month is given as follows. Income Rs. Expenditure Rs. 1. Father’s Salary 2. Rent from house 3. Income from others 50,000 20,000 10,000 1. Food 2. Electricity 3. Transportation 4. Education 5. Internet 6. Medicine 7. Miscellaneous 10,000 1,500 6,000 8,000 1,300 2,000 7,000 Total 80,000 Total 35,000 MIXED PRACTICE–V Read, Understand, Think and Do
STATISTICS 273 (a) In which items the family expense most? (b) How much amount did the family save in a month? (c) In the festival the family expense Rs. 2,00,000, how much amount did the family save in a year 3. The expenditure of two families’ in a month on dated 2080/03/16 are showing here. Items of expenditure Expenditure (in Rs) Family A Family B Food 12000 15000 Clothing 4000 3500 Clothing 8000 12000 Light & Fuel 1500 1700 Miscellaneous 2300 2700 (a) Which family expense is more in total? (b) What is the total expenditure of family B? Calculate it. 4. The following bar graph shows the average age of men in various countries in a certain year: 10 67 20 30 40 50 60 70 80 90 X Y Countries Average age of people (in year) India Nepal China Pakistan UK USA 62 75 80 62
274 The Leading Mathematics - 4 (a) Which country has high average age of men? (b) Compare the high and low average age of the countries. (c) How many countries have low average age of men than Nepal? 5. The favorite subjects of students are presented below in the bar graph. 5 10 15 20 25 30 35 40 45 55 50 X Y Subjects Number of students English Nepali Maths Science Social (a) Which subjects is more favorite of the maximum number of students? (b) How many students told my favorite subject is English? (c) How many students said my favorite subject is Math than said to English?
STATISTICS 275 Attempt all the questions. FM : 15 Time : 40 Min. CONFIDENCE LEVEL TEST IV STATISTICS 1. Observe the given bill below and answer the following questions. Star Grocers Kapan, Kathmandu Bill No. 1235 Customer’s Name : Kapil Nepal Address : Baluwakhani S.N. Particulars Quantity Rate (Rs.) Price (Rs.) 1 Rice 5 kg 120 600.00 2. Daal 2 kg 155 310.00 3. Soap 6 pcs. 80 480.00 4. Oil 2 ltr. 350 700.00 Total 2670.00 In Words : Two thousand six hundred seventy only. Salesperson Hari c. Date : 2080-06-17 (a) Who is customer? [1] (b) How many items does he/she buy? [1] (c) How much total amount is shown on the bill? [1] (d) She gave Rs. 3000 to the shopkeepers. How much amount does she get return? [2] 2. Observe the budget of a family. Income Rs. Expenditure Rs. 1. Parents Salary 2. Rent from house 3. Income from others 75,000 20,000 10,000 1. Food 2. Electricity 3. Transportation 4. Education 5. Internet 6. Medicine 7. Miscellaneous 20,000 2,500 7,000 10,000 1,500 1,800 9,000 Total 1,05,000 Total 51,800
276 The Leading Mathematics - 4 (a) From which sectors does the family earn money? [1] (b) How much amount is the family expense in a month? [1] (c) How much amount did the family save in a month? [1] 3. Observe the income and expenditure below. Incomes: Salary: Rs. 45000, Shop: Rs. 65500, Rent: Rs. 23000 Expenditure: Food: Rs. 24500, Education: Rs. 35000, Health: Rs. 15500, Clothes: Rs. 12000, Vehicles: Rs. 15000 (a) In which topic does the family expense the most? [1] (b) Prepare a budget from the above information. [2] 4. The following bar graph shows the days of snow fall in Manang. Answer the following equations. 0 2 4 6 8 10 12 14 16 X Y Month Number of days of snow December January February March (a) In which month was snow fall more and how many days? [1] (b) In which month was snow fall least and how many days? [1] (c) How many total days was snow fall at Manang? [2]
ALGEBRA 277 ALGEBRA UNIT VII COMPETENCY Solve the simple problems related to algebraic expression and identify the constant and variable. CHAPTERS 22 Algebraic Expressions LEARNING OUTCOMES After completion of this content area, the learner is expected to be able to: solve the problems of one variable including the operations of addition, subtraction, multiplication and division by inspection or hit and trial method by using notation. identify constant and variable.
278 The Leading Mathematics - 4 CHAPTER 22 Algebraic Expression What is the value of 100 rupees note ? How much money did her mother to the girl ? How much amount does she expense on shopping ? How much amount does she have left? 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 22.1 Algebraic Expressions 279 22.2 Value of Algebraic Expression 277 22.3 Solving Problems Involving Addition and Subtraction 280 22.4 Solving Problems Involving Multiplication and Division 287 22.5 Constant and Variable 289
ALGEBRA 279 Which number comes after 8 in the given number pattern ? How ? 4, 5, 6, 7, 8, ........... Here, 4 = 1 + 3 5 = 2 + 3 6 = 3 + 3 7 = 4 + 3 8 = 5 + 3 Any number = n + 3 If n > 5, we find the numbers after 8. If n = 6, it will be 6 + 3 = 9 If n = 7, it will be 7 + 3 = 10 and so on. Here, n + 3 is an algebraic expression in which n and 3 are its terms. Further, 2x – 1 is also algebraic expression with terms 2x and –1. The product of constant or variable or both constant and variable is called an algebraic terms. For example, 2, a, – 3, x, y, 2x, 2ab, 3xy, etc. are algebraic terms. The sum or difference of algebraic terms is called an algebraic expression. For example, 2 + 0, 3x + 1, x – 1, 3x + y – 4, etc. are algebraic expressions. Oh! n is the natural number, i.e. n = 1, 2, 3, 4, 5, 6, .......... . x + x + x = 3x Oh! It is also algebraic expression. 1. Write down the following statements in algebraic form. (a) The sum of double of x added to 3 = 2x + 3. (b) The sum of a and 2. = ....................... (c) The sum of 5a and 3. = ....................... EXERCISE 22.1 Your mastery depends on practice. Practice like you play. 22.1 Algebraic Expressions
280 The Leading Mathematics - 4 (d) The sum of 7a and 2b. = ....................... (e) The difference of 2a and 3. = ....................... (f) The difference of 5a and 2b. = ....................... (g) The difference of 3a and 2l. = ....................... (h) The product of 3a and b. = ....................... (i) The product of 2a and 3b. = ....................... (j) Three times the sum of 2a and 4. = ....................... (k) Four times the difference of 7x and 5. = ....................... (l) Two times the number a added to 7. = ....................... (m) Five times the number x taken from 12. = ....................... (n) The quotient when a is divided by b. = ....................... (o) The quotient when 3a is divided by the difference of a and b. = ....................... 2. Write the word phrase for the following expressions. (a) 2x + 3 = “the sum of 2x and 3”. (b) a + 2 (c) 3 + b (d) 3x (e) x – 3 (f) 3x – 5 (g) 2x – y (h) 2b – 3a (i) 5a + 2b (j) 9x + y (k) 2mn (l) 3xy (m) xyz 3. If 3x means x + x + x, write down the meaning of the following : (a) 2a (b) 3a (c) 7x (d) 6y (e) 9m (f) 8n (g) –2z (h) –3p (i) 9q
ALGEBRA 281 4. Write the expression for the following length of the segments. (a) (b) (c) (d) 5. Write algebraic expression for each of the following : (a) Sohan had Rs. x. He spent Rs. 9. How much is left with him ? (b) Bimala had Rs. y. Her husband gave her Rs. 25. How much money has Bimal now ? (c) Rohit had 15 marbles. He lost x marbles. How many marbles has Rohit now ? (d) Navaraj had x oranges. He bought 5 more oranges. How many oranges has Navaraj now ? (e) Jigyasha had 12 pencils. She gave x pencils to Sunaina. How many pencils are left with Jigyasha now ? (f) There were x students in a hostel. y students joined the hostel. How many students are now in the hostel ? (g) There are ‘a’ passengers in a bus. ‘b’ passengers are went out at a bus stop and ‘c’ passengers are come in. How many passengers are now in the bus ? x 2 2 a b y y y 3 x y z
282 The Leading Mathematics - 4 Numerical Value of Algebraic Expression Consider the expression x + 7. x + 7 = 10, when x = 3 x + 7 = – 3, when x = – 10 + 7 = 11, when x = 4 If we replace the given value for the variable in the given expression, we get the numerical value of that expression. CLASSWORK EXAMPLES Example 1 Find the value of expression, when x = 4 2x + 1 Solution : Here, 2x + 1 = 2 × 4 + 1 = 8 + 1 = 9 Example 2 Find the value of 2p – 3q when p = 4 and q = 1. Solution : Here, 2p – 3q = 2 × 4 – 3 × 1 = 8 – 3 = 5 EXERCISE 22.2 Your mastery depends on practice. Practice like you play. 1. In each of the following expressions, find the numerical values of the expressions if x = 3. (a) 4x (b) 2x – 1 (c) 7x + 3 (d) 5x + 4 (e) 3 – 7x (f) –2x + 9 (g) 4 – 3x (h) 8 – 2x (i) 5x + 7 3, –10, 4 etc. are the values of x + 7 depending upon the different replacements for x 22.2 Value of Algebraic Expression Read, Think and Learn
ALGEBRA 283 2. (a) If x = 2, find the value of 3x – 2. (b) If l = 4 and b = 5, find the value of l × b (c) If l = 8 and b = 6, find the value of 2(l + b ) (d) If a = 5 find the value of a2 . (e) If b = – 4 find the value of b2 . (f) If p = 7, find the value of 2p – 12. (g) If x = 2, y = 3, find the value of 2x + 3y. (h) If p = 4, q = –2, find the value of 3p – 2q. (i) If m = –2, n = 3, find the value of m – 4n. (j) If x = 5 and y = 4, find the value of 3xy. (k) If m = 2 and n = – 1 find the value of 4mn. 3. Write an expression for the length of each of the following segment and find the actual length if x = 5cm. (a) x 2 (b) x x 3 (c) 3x 4 (d) x x x 5 (e) 2x – 1 3 (f) 3x + 5 2 4. Write an expression for the perimeter of the following shapes and find the actual perimeter if a = 4 cm. (a) (b) (c) A C 3a a 2a B 3a 2a 2a a a
284 The Leading Mathematics - 4 (d) (e) (f) (g) (h) (i) 3a a a 4a 3a 2a 2a 2a a a 2a a a a a 3a 2a 2a 2a 2a 2a 2a 2a 2a 5a 2a a a 2a2a 2a2a 5a 2a 4a 3a 2a a a a a a 22.3 Solving Problems Involving Addition and Subtraction Read, Think and Learn In the adjoining beam balances, 7 and 3 + 4 are equal in the first. What is the value of in the second beam balance, 9 and 4 + are equal ? 7 3 + 4 9 4 + 8 – 5 What must subtract from 8 to be equal 5 in the third beam balance ? These linear equations can be solved by the following rules :
ALGEBRA 285 Let x represent . Then 9 = 4 + or, 9 = 4 + x Since 9 = 4 + 5 \ x = 5 4 + 1 = 5 4 + 2 = 6 4 + 3 = 7 4 + 4 = 8 4 + 5 = 9 Let y represent . Then 8 – y = 5 Since 8 – 3 = 5 \ y = 3. 8 – 1 = 7 8 – 2 = 6 8 – 3 = 5 9 = 4 + or, 9 – 4 = 4 + – 4 [Subtracting on both sides] or, 5 = \ = 5 Next Method Next Method 8 – = 5 or, 8 – + = 5 + y [Adding on both sides] or, 8 = 5 + or, 8 – 5 = 5 + – 5 [Subtracting 5 on both sides] or, 3 = \ = 3 1. Write the suitable number in the box. (a) + 2 = 7 (b) + 13 = 20 (c) + 24 = 30 (d) 27 + = 32 (e) 50 + = 57 (f) + 87 = 94 (g) 27 – = 24 (h) 32 – = 28 (i) 48 – = 42 (j) – 4 = 8 (k) – 18 = 5 (l) – 54 = 49 EXERCISE 22.3 Your mastery depends on practice. Practice like you play.
286 The Leading Mathematics - 4 2. Solve the following problems. (a) x + 4 = 7 (b) x + 5 = 12 (c) x + 38 = 47 (d) 44 + x = 50 (e) 80 + x = 87 (f) 98 + x = 102 3. Find the value of x for equal pair of pencils : (a) x + 3 12 (b) 15 + x 24 (c) x – 15 22 1. If you have more than Rs. 100, you spend Rs. 78 from it to buy some things. How much money do you have now? Solve it by using mathematical open sentence. Prepare a report and present it in your classroom. 2. If your mother gives Rs. 50 to you, you have Rs. 152 now. How much did you have in the beginning ? Solve it by using mathematical open sentence. Prepare a report and present it in your classroom. PROJECT WORK
ALGEBRA 287 4 × 1 = 4 4 × 2 = 8 4 × x 12 4 × 3 = 12 4 × = 12 = ? Since 4 × 3 = 12 \ = 3 4 × 1 = 4 4 × 2 = 8 y ÷ 3 4 4 × 3 = 12 ÷ 3 = 4 = ? Since 12 ÷ 3 = 4 \ = 12 Next Method Next Method 4 × = 12 or, 4 × 4 = 12 4 [dividing 4 on both sides] or, = 3 ÷ 3 = 4 or, 3 = 4 or, 3 × 3 = 4 × 3 or, = 12 1. Write the suitable number in the box. (a) 3 × = 9 (b) 5 × = 30 (c) 7 × = 49 (d) × 2 = 18 (e) × 9 = 45 (f) × 8 = 40 (g) ÷ 4 = 24 (h) ÷ 5 = 35 (i) ÷ 7 = 35 (j) ÷ 4 = 3 (k) ÷ 6 = 36 (l) ÷ 8 = 32 EXERCISE 22.4 Your mastery depends on practice. Practice like you play. 22.4 Solving Problems Involving Multiplication and Division Read, Think and Learn
288 The Leading Mathematics - 4 2. Solve the following problems : (a) 3 × x = 18 (b) 5 × y = 20 (c) 6 × y = 24 (d) a × 4 = 16 (e) b × 7 = 28 (f) c × 8 = 56 (g) x ÷ 5 = 25 (h) x 6 = 3 (i) x 7 = 5 (j) 14 ÷ p = 7 (k) 35 q = 7 (l) 64 r = 8 3. Find the value of x for the equal pair of pencils : (a) 4x 36 (b) x ÷ 3 27 (c) 24 ÷ x 8 4. Write in mathematical sentences and solve for x : (a) x added to 5 the sum is 12. (b) 7 taken out from x the difference is 5. (c) 7 times of x is 63. (d) x is divided by 8, the quotient is 32. (e) x multiplied by 5 the product is 49. (f) 24 is divided by x, the quotient is 8. If you bought 15 eggs from Rs. 240, what is the cost of each egg ? Solve it by using mathematical open sentence. Prepare a report and present in your classroom. PROJECT WORK
ALGEBRA 289 Letters for Numbers What numbers go for a, b, c ...... etc in the number line. 0 1 2 3 7 8 a b c d e Here, a = 4, b = 5, c = 6, d = 9 and e = 10. In algebra, we use letters for numbers. These letters may represent 1 or more numbers. For example : if x + 2 = 5, x = 3 is the only value of x for which x + 2 = 5 is true. If x > 2, then x may take any values which are greater than 2. In this case x = 3, 4, 5 ... The alphabet to represent a fixed number is constant. a = 4, b = 5, c = 6 etc. are constants in the above number line. When x > 2, then x = 2, 3, 4, ... etc. and x is variable, which may take any value greater than 2. The constant has only one value and the variable has more than one value. Observe the monthly calendar of Mansir, 2079 in below and discuss the following questions; a) How many Saturday are there in this month ? If x represents the days of Saturday, is x constant or variable ? b) How many Poornima are there in this month ? If a represents the days of Poornima, is a constant or variable ? 22.5 Constant and Variable Read, Think and Learn
290 The Leading Mathematics - 4 c) How many Astami are there in this month ? If y represents the days of Astami, is y constant or variable ? d) If y represents the holidays of this month, is y constant or variable ? e) If b represents the day of the 1st December, is b constant or variable ? EXERCISE 23.1 Your mastery depends on practice. Practice like you play. 1. What are the values of a, b, c, d, and in the number line given below: 5 6 a 8 9 b c 12 13 d e 2. Observe the number line given below and answer the following questions: 6 7 8 (a) a represents a number that is 3 units before than 6. What is the value of a ? Show the value of a in the number line. (b) b represents a number that is 4 units after than 7. What is the value of b ? Show the value of b in the number line. (c) c represents a number that is 5 units left from 8. What is the value of c ? Show the value of c in the number line. (d) d represents the number between 6 and 8. What is the value of d ? Mark the value of d in the number line. (e) x represents the numbers between 3 and 7. What are the values of x ? Show the values of x in the number line. (f) y represents the numbers less than 5 in the given number line only. What are the values of y ? Show the values of y in the number line. (g) z represents the numbers greater than 7 in the given number line only. What are the values of z ? Show the values of z in the number line. (h) Identify the numbers a, b, c, x, y and z mentioned above as constants or variables.
ALGEBRA 291 3. State whether each of the following is a constant or a variable. (a) a represents the number 5. (b) b represents the counting numbers less than 5. (c) c represents the set of capital city of Nepal. (d) x represents the whole numbers between 18 and 20. (e) y represents the name of the students in the class 3. (f) w represents the days of a week. (g) z represents the number of months in a year. (h) d represents the sum of 5 and 7. (i) t represents the number of trees in the garden. (j) n represents the odd numbers less than 5. (k) k represents the product of 5 and 7. (l) l represents the letters of Devanagari script. 4. Observe the monthly calender of Mansir, 2079 in below and discuss on the following questions. (a) a represents the day of Constitution day and write the day. Is a constant or variable?
292 The Leading Mathematics - 4 (b) x represents the days of Monday and write the days. Is x constant or variable? (c) b represents Poornima. Is b constant or variable? What is the value of b? (d) c represents the day of Fulpati. Is c constant or variable? What is the value of c? (c) y represents the days of Dashain vacation. Is y constant or variable? What are the values of y? (d) z represents the days of holiday. Is z constant or variable? What are the values of z? (e) d represents the World Tourism Day. Is d constant or variable? What is the value of d? (f) e represents the World Peaceful Day. Is e constant or variable? What is the value of e? . Select a monthly calendar including your birth day. Prepare a report about the constant and variable for the occasion of your birth day, different festival days, Sunday, Tuesday and holiday, and present it in the classroom. PROJECT WORK
ALGEBRA 293 1. Two terms are equal in each other. Put the correct number in the box. (a) + 3 = 5 (b) – 3 = 5 (c) × 3 = 15 (d) ÷ 3 = 5 2. Two algebraic terms are given below. x and 3 (a) Distinguish between variable and constant. (b) Can we add x and 3? Justify. (c) If in the box, there is x and x + 3 = 7, what should be the value of x? 3. Some variables and constants are given below. 3, 5, x, y, 0, 7x, 9y (a) Define variable with example. (b) 7x is variable. Do you agree? (c) List out variables and constants. 4. In the given instrument, the weight of one side is given and another side is given with box. (a) Write a variable in box. (b) What will be the condition in terms of x for its balance? Write it. (c) For what value of x, the given instrument is balanced? Write it. 4 × x 12 MIXED PRACTICE–V Read, Understand, Think and Do
294 The Leading Mathematics - 4 5. Red ribbon is x m long and Blue ribbon is 2 m long. (a) Do you know the length of red ribbon exactly? Justify. (b) Which ribbon has the longest length by how much? Do you calculate? (c) For equal length of ribbons, what should be the length of red ribbon? Find it. 6. The age of father is 35 years and age of son is years but sum of their ages is 42 years. (a) Whose age is given in constant terms? (b) Is the condition of the above facts 35 + = 42? (c) Which number should be kept in the ? 7. Two terms are equal in each other. Put the correct number in the box. (a) 2 × + 3 = 9 (b) 5 × – 3 = 7 (c) × 12 = 4 (d) 2 × ÷ 3 = 4
ALGEBRA 295 Attempt all the questions. FM : 20 Time : 40 Min. CONFIDENCE LEVEL TEST V ALGEBRA 1. Observe the line segment AB which is intersected by a point C. (a) What is the length of AB in terms of x and y? [1] (b) If x = 6 cm and y = 7 cm, find the length of AB. [2] (c) Are the both segments AC and BC equal ? why [1] 2. A boy has Rs. 100 and he buys a note copy of Rs. x. (a) How much does he have left? [1] (b) If he has Rs. 45 left after buying note copy, find the cost of the note copy. [1] (c) If he wants to buy two note copies, does he have enough money?[2] 3. Two algebraic terms are as 2x and 5. (a) Write the definition of constant. [1] (b) Can we add 2x and 5? Justify. [1] (c) If 2x + 5 = 13, what should be the value of x? [2] 4. Observe the given figure and answer the following questions. (a) Find the perimeter of the given shape. [2] (b) If its opposite sides AB and CD are equal, what is the value of x? Find it. [2] (c) For what value of y are the sides AD and BC equal ? Find it. [2] 5. In the given beam balance, the weight of one side is given and another side is given with box. (a) For balancing the given beam balance, what will be the condition in terms of x? Write it. [1] (b) For what value of x, the given beam balance is balanced? Write it. [1] 4x – 8 y + 5 2x + 4 2y 4 × x 12
296 The Leading Mathematics - 4 SN Chapter/ Lesson Participation (Attendance + Class work): 4 marks Project and Practical Work 36 marks Terminal Exams: 10 marks (50) Total Marks Percent % Grade Grade point Result Remarks Based on main learning outcomes (20) Records of Project and Practical Work (10) Figure, leveling, Oral (6) First (5) Second (5) Subject: The Leading Mathematics Student's Name : ………………………………………………………. Class : 4 Internal Evaluation Sheet Practical Recorded Form for 50 Marks