3.2 PerimeterWarm Up Test1. What is the perimeter of each shape?[Note : length and breadth of each room is 1 cm.](a) (b)Perimeter = cm. Perimeter = cm.(c) (d)Perimeter = cm. Perimeter = cm.(e) (f)Perimeter = cm. Perimeter = cm.2. Find the perimeter of the given book.3. Fill in the blank.(a) Perimeter of the triangle is ............... of its three sides.(b) Perimeter of the quadrilateral is sum of its ........................ .(c) ................. is total length of any object.18 cm24 cmAcme Mathematics 6 151
A. Perimeter of a rectangleObserve this rectangle. Its length is 10 cm and breadth is 8 cm. Its total length from A to A is given below.A to A = AB + BC + CD + DA= 8 cm + 10 cm + 8 cm + 10 cm= 2 × 8 cm + 2 × 10 cm= 16 cm + 20 cm= 36 cmThus, the total length (boarder) of the rectangle is 36 cm.The total length of the boarder of any plane figure is called its perimeter.Perimeter of a rectangle in terms of length (l) and breadth (b).Consider the given rectangle ABCD.Now, Perimeter of rectangle ABCD.= AB + BC + CD + DA= length + breadth + length + breadth= 2 length + 2 breadth= 2 (length + breadth)= 2 (l + b)Thus, the perimeter of a rectangle = 2 (l + b)B. Perimeter of a squareIn the figure, PQRS is a square. Its sides are PQ, QR, RS, SP and PQ = QR = RS = SP.Let, Length of each side is 'l'.Now, perimeter of square PQRS= PQ + QR + RS + SP= l + l + l + l = 4 lThus, the perimeter of a square = 4 l8 cmA 8 cm BD C10 cm 10 cmD lengthA BClengthbreadth breadth= ===SP QRll152 Acme Mathematics 6
Solved ExampleExample 1 : A rectangular piece of paper has length 16 cm and breadth 12 cm. Find its perimeter.Solution: Here, Length of paper (l) = 16 cmBreadth of paper (b) = 12 cmNow, Perimeter of rectangular piece of paper= 2 (l + b)= 2 (16 cm + 12 cm)= 2 × 28 cm= 56 cmPerimeter of a paper is 56 cm.Example 2 : A rectangular park is 70 m long and 30 m wide. Find the distance covered by a man around it in 10 rounds.Solution: Here, Total distance = 10 × perimeter of the park.Length of park (l) = 70 mBreadth of park (b) = 30 mPerimeter = 2 (l + b)= 2 (70 m + 30 m)= 2 × 100 m= 200 mNow, 10 times perimeter = 10 × 200 m = 2000 m = 2 kmThus, the distance covered by a man is 2 km.Example 3 : The poster is square shape. Its perimeter is 64 cm. Find the length of its side?Solution: Here, perimeter = 64 cmor, 4l = 64 cm (l = length of one side)or, l = 64 cm4or, l = 16 cm Hence, length of each side is 16 cm.12 cm16 cmAdmission openforPG to Class - 10Seats are limited.2083Acme Mathematics 6 153
Classwork1. Find the perimeter of the given figures.(a) (b)Perimeter = ................ cm Perimeter = ................ cm2. Fill in the blanks.(a) Perimeter of the triangle is .......................... of its 3 sides.(b) Perimeter is .................................(c) Perimeter of triangle having each side 'x' cm is .................. cm (d) Perimeter of square having each side 'y' cm is ..............cm. (e) Perimeter of quadrilateral is .............................3. Find the perimeter of the square whose each side are:(a) 4 cm (b) 5.5 cm(c) 6 cm (d) 4.5 cm4. Find the perimeter of the rectangle whose length and breadth are given below.(a) Length = 6 cm and breadth = 4 cm(b) Length = 7.5 cm and breadth = 6.5 cm(c) Length = 8 cm and breadth = 5.5 cmExercise 3.21. Write the perimeter of the given figures:(a)3 cm3 cm 2 cm 2 cmIts perimeter = ................. cm(b)2 cm2 cm2 cm 2 cmIts perimeter = ................. cm4 cm2 cm3 cm3 cm154 Acme Mathematics 6
2. Complete the table:Length (l) Breadth (b) l + b 2(l + b)2 cm 1 cm3 cm 2 cm4 cm 2 cm4 cm 3 cm5 cm 3 cm3. Complete the table:Length (l) Breadth (b) 4 l 4 b4 cm 4 cm3 cm 3 cm5 cm 5 cm10 cm 10 cm12 cm 12 cm4. Calculate the perimeter of the given rectangles:(a) (b) (c)5. Calculate the perimeter of the given squares:(a) (b) (c) 6. Find the perimeter of the given rectangles when:(a) Length = 8 cm and breadth = 6 cm(b) Length = 12.5 cm and breadth = 8.5 cm(c) Length = 20 cm and breadth = 16.5 cm(d) Length = 50 cm and breadth = 45 cm7. Find the perimeter of the given squares when:(a) Side = 4.5 cm (b) Side = 6 cm (c) Side = 15 cm(d) Side = 25 cm (e) Side = 16.5 cm (f) Side = 22.5 cm 5 m8 m6 inch4 inch10 ft.12 ft.2 cm 3.5 cm8 cmAcme Mathematics 6 155
8. Find the perimeter of the rectangular picture.[Hint : measure its sides]9. Find the perimeter of the given chessboard.10. (a) A man has 180 m of fence for a rectangular garden. If his garden is 10 m long and 8 m wide, how many times he can round it ?(b) The figure alongside is a rectangular garden. It is 40 m long and 30 m wide. A man wants to fence it 5 times with wire. How long wire is needed to him ?11. (a) How long wire is needed to fence a square field of 12 m long 4 times ?(b) How long wire is needed to fence a square field of 20 m long 4 times ?(c) Madan has rectangular field of 35 m length and 25 m breadth. If he runs his field every morning 5 times as 'morning walk' how many meters does he run ?12. Calculate the length of the given squares:(a) (b) (c)Fig. Q.No. 8 Fig. Q.No. 910 m8 mPerimeter4 cmPerimeter18 cmPerimeter20 cm156 Acme Mathematics 6
13. Calculate the sum(l + b) in the given rectangles:(a) (b) (c)14. Calculate the perimeter of the following rectangular objects (use base only).(a) (b) (c)(d) (e) (f)15. Calculate the length of sides of the square whose perimeter is given.(a) 48 cm (b) 60 cm(c) 80 cm (d) 100 cm16. Use P = 2(l + b) and complete the table.Length (l) Breadth (b) Perimeter (P)80 cm 60 cm ....................7 cm ......................... 24 cm...................... 8.5 cm 50 cm3x = ? ............. 2x = ? ................... 100 cmPerimeter36 cmlbPerimeter90 cmlbPerimeter50 cmlb18 cm27 cm6.5 cm8.5 cm20 m14 m15 cm25 cm18 cm 12 cm15 cm28 cmAcme Mathematics 6 157
3.3 AreaArea is the space enclosed within a closed figure.Figure A Figure BSpace covered by figure B is more than A. So area of figure B is more than area of figure A.A. Area of Irregular objectAreas of Irregular figures (objects) are measured by the square grid by counting the number of squares that are inside the figure.There may be some half squares or a part of some square that are the part of the area. So we can only find approximate area of irregular figure.During calculation of area: We should count half and more half squares as one square. We should leave out all the squares which are less than half in the figure.Thus, Area of irregular shape = Number of full square + number of half or more than half square.Let us find the area of the above figure. Number of full square = 8Number of half or more than half squares = 7Therefore area = 8 + 7 = 15 square units.B Area of Regular objectsArea of several regular figures(objects) can be calculated by using formula.(a) Area of rectangleABCD is a rectangle. Its length is 7 cm and breadth is 5 cm. There are 35 square rooms.ADBC7 cm5 cm158 Acme Mathematics 6
So, Area of rectangle ABCD is 35 square cm.Also, 35 = 7 × 5 = length × breadth.Hence,Area of rectangle = length (l) × breadth (b)A = length (l) × breadth (b),l = Ab, b = Al(b) Area of a square:Area of square PQRS is 4 cm × 4 cm = 16 cm2If length of side of a square = l, then Area of square = Length × LengthA = l× l A = l2Solved ExampleExample 1 : Find the area of a rectangular field whose length and breadth are 20 m and 14 m respectively.Solution: Here, Length (l) = 20 mBreadth (b) = 14 mNow, Area of rectangle = length (l) × breadth (b)= 20 m × 14 m= 280 m2Hence, area of rectangular field is 280 m2.Example 2 : Find the area of square garden whose each side is 40 m long.Solution: Here, Length of side (l) = 40 mNow, Area of square = l2= l × l= 40 m × 40 m= 1600 m2Hence, area of the square garden is 1600 m2.l = lengthb = breadthPS R4 cm Q 4 cm20 m14 m40 m40 mAcme Mathematics 6 159
Example 3 : The area of a rectangular garden is 150 m2. Its length is 15 m. Find its breadth.Solution: Here, Area of rectangular garden (A) = 150 m2Length of rectangular garden (l) = 15 mBreadth of rectangular garden = AreaLength= 150 m215 m= 10 m.Hence, breadth of rectangular garden is 10 m.Classwork1. Count the square room and find the area of the given shapes: [1 room = 1 cm2](a)....... cm2(b)....... cm2(c)....... cm2(d)....... cm22. Join the lines and find the area of the given shapes:(a)5 cm4 cm(b)2 cm6 cm15 m150 m2 b160 Acme Mathematics 6
(c)4 cm4 cm(d)10 cm3 cm3. Fill in the blanks:Length (l) Breadth (b) l × b Area 12 cm 3 cm ....... cm28 cm 8 cm ....... cm210 cm 7 cm ....... cm29 cm 2 cm ....... cm2Exercise 3.31. Find the approximate area of the following irregular shapes.1cm2ECB DA2. Calculate the area of the following figure:(a)3 cm2 cm(b)18 cm18 cm(c)15 cm20 cmAcme Mathematics 6 161
3. Calculate the area of the following rectangles by using the formula:(a) Length (l) = 7 cm (b) Length (l) = 12 cmBreadth (b) = 5 cm Breadth (b) = 9 cm(c) Length (l) = 5 cm (d) Length (l) = 3 kmBreadth (b) = 2 cm Breadth (b) = 1 km4. Find the area of the following rectangular figures.(a) (b) (c)(d) (e) (f)5. Calculate the area of square by using the formula:(a) Length (l) = 9 cm (b) Length (l) = 13 cm(c) Length (l) = 4 cm (d) Length (l) =15 cm6. (a) The length and breadth of a table tennis board is 2 m and 1.5 m. Find the area.(b) The length and breadth of the football ground is 90 m and 60 m. Find the area.3 cm2.5 cm6.5 cm8.5 cm90 cm140 cm17.5 cm27 cm 8 cm18 cm35 m30 m162 Acme Mathematics 6
7. Calculate the area of the following square shapes. (use base only)(a) (b) (c)8. Find the area of the following figures.(a) (b) (c)9. (a) A rectangle is 35 cm long and 20.5 cm broad. Calculate its area.(b) A rectangular room is 3.5 m long and 3 m broad calculate its area.10. My study room is 4 m long and its area is 12 m2.(a) Find its breadth.(b) Find its perimeter11. The perimeter of a square ground is 40 m.(a) Find its length.(b) Find its area.12. Fill in the blanks.Length Breadth Area Perimeter5.5 cm .......................... 13.75 cm2 ................................................... 16.3 cm 299.92 cm2 ............................39.3 m 32.1 m ......................... ............................20 m .......................... 200 m2 ............................15 cm15 cm5 cm5 cm20.5 cm20.5 cm2 cm2 cm2 cm3 cmBA1.5 cm 2 cm 1.5 cm4 cm 2 cmA CB3 cm 5 cm10 cm 2 cmC ABAcme Mathematics 6 163
16413. (a) Length of rectangle is double of its breadth, if its area is 21 m2, find its length and breadth.(b) The area of a square is 49 m2. A rectangle has the same perimeter as the square. If the length of the rectangle is 8 m, find its breadth.(c) A square and a rectangle have the same area of 64 cm2.(i) What is the length of the sides of the square?(ii) If the rectangle has a length which is 4 cm more than the side of the square, find the length, the breadth and perimeter of the rectangle.(d) A rectangle and a square have same area. The area of square is 25 cm2. The length of sides of the square is half of length of side of the rectangle. Calculate;(i) Length and breadth of the rectangle.(ii) Perimeter of the rectangle.14. Find the area of the shaded part in the following figures.(a) (b)(c) (d) 1 cm4 cm6 cm4 cm 5.5 cm8 cm2 cm4 cm4 cm6 cm10 cm10 cm6 cm 2 cm 6 cm2 cm 4 cm 4 cm164 Acme Mathematics 6
15. The given map is of Kushal's house.3 m 3 mGuest room Kitchen roomBed roomBath roomBed roomPassageBaranda2.5 m1 m 3 m4 m4 m4 m4 m1.5 m 3 m 3 m12(a) Calculate the total area of the Bedrooms.(b) How big is kitchen room ?(c) Which is big, 'Baranda' or 'Bathroom' ? (d) Can you find the area of the passage ?16. Take ‘a card board size’ graph paper and draw different shapes on the graph and find the area of different shapes by counting the square rooms. Now discuss with your friends about the shape you draw and their area.17. A rectangular room is 6 m long and 4 m broad.(a) Write the formula to find the area of a rectangle.(b) Find the area of the floor of the room(c) Find the area of the carpet required to cover its floor.(d) Find the cost of the carpet at Rs. 150 per sq. m.Acme Mathematics 6 165
3.4 VolumeLook at the adjoining can carefully.If we want to fill up can (a) with milk we have to know how much space it can hold. This amount of space occupied by milk is called the volume of the can. We can easily state on looking that can (a) occupies less space than (b) or it has smaller volume.The measure of the amount of space a solid occupies is called the volume of the solid.(a) Volume of the cuboidIt is unit cube2 cm3 cm2 cm1 cm 1 cm 1 cmIt is unit cube. Its volume is 1 cm3Here is a cuboid. Whose;Length (l) = 3 cm. Breadth (b) = 2 cm Height (h) = 2 cm It has 12 unit cubes.So, its volume = 12 cm3.Also, We have = 3 cm × 2 cm × 2 cm = 12 cm3Hence, Volume of cuboid = length × breadth × height or, V = l × b × h(b) Volume of cubeIn case of cube length, breadth and height are equal. So, it is cuboid with all sides equal.Now, Volume of cube = length × breadth × heightV = l × l × lV = l3Figure (a)100 liters 200 litersFigure (b)l = vb × h , b = vl × h and h = vl × b It is cube rootl = V3166 Acme Mathematics 6
Solved ExampleExample 1 : Calculate the volume of the given chalk box.Solution: Here, Length of chalk box (l) = 6 cmBreadth of chalk box (b) = 5 cmHeight of chalk box (h) = 4 cmIt is cuboid.So, Volume = l × b × h= 6 cm × 5 cm × 4 cm = 120 cm3The volume of the chalk box is 120 cm3.Example 2 : If the volume of the die is 64 cm3, find the length of its each side.Solution: Die is a cube.Volume of cube (V) = 64 cm3Let, Length of cube (l) = x cmNow, Length of cube (l) = V3or, x = 64 3or, x = 4 × 4 × 4 3or, x = 4The length of its each side is 4 cm.Classwork1. Fill in the blanks : (Count the unit cube.)(a) (b)V = .................. cm3 V = .................. cm3(c) (d)V = .................. cm3 V = .................. cm36 cm5 cmSun-Rise 4 cmchalkVolume 64 cm3Acme Mathematics 6 167
(e) (f)V = .................. cm3 V = .................. cm32. Fill in the blanks:Length (l) Breadth (b) Height (h) l × b × h Volume4 cm 3 cm 2 cm7 cm 5 cm 1 cm10 cm 6 cm 4 cm4 cm 4 cm 4 cm5 cm 5 cm 3 cm3. Calculate the volume of the cubes: (use, V = l × l × l)(a) Length (l) = 4 cm (b) Length (l) = 5 cm(c) Length (l) = 9 cm (d) Length (l) = 11 cm4. Calculate the volume of the cuboids: (use, V = l × b × h)(a) l = 10 cm, b = 2 cm and h = 1 cm(b) l = 8 cm, b = 6 cm and h = 6 cm(c) l = 15 cm, b = 2 cm and h = 3 cm(d) l = 20 cm, b = 3 cm and h = 3 cmExercise 3.41. Write the volume of the given cube in the blanks.(a)1 cm1 cm1 cm(b)1.5 cm 1.5 cm 1.5 cm(c)2 cm 2 cm2 cmV = ............... cm3 V = ............... cm3 V = ............... cm3168 Acme Mathematics 6
2. Write the volume of the given cuboids in the blanks.(a)5 cm1 cm2 cm(b)2 cm6 cm1 cm(c)4 cm2 cm 1 cmV = ............... cm3 V = ............... cm3 V = ............... cm33. Find the volume of the cuboids with the following measure.(a) Length = 100 cm, breadth = 8 cm and height = 2 cm(b) Length = 5 cm, breadth = 4.5 cm and height = 2.5 cm(c) Length = 15.5 cm, breadth = 10.5 cm and height = 7 cm4. Find the volume of the cube with the following measure.(a) Length = 5 cm(b) Length = 6.5 cm(c) Length = 4.3 cm5. Find the length of each side of the cube whose volume is given.(a) 125 cm3 (b) 64 cm3 (c) 8 cm3 (d) 21636. Calculate the measure of the unknown sides in each of the following figure.(a)b6 cm1 cmV = 12 cm3 (b)l 2 cm2 cmV = 16 cm3(c)1 cmV = 10 cm31 cmh(d)3 cmV = 15 cm3b2.5 cm(e)2 cmV = 8 cm3aa(f)bV = 32 cm3b8 cm7. Our water tank is 9 m long and 2 m high. If its volume Acme Mathematics 6 169
is 108 m3, find its breadth.8. A match box is 3 cm broad and 1 cm high. If the volume of the match box is 12 cm3, find the length of the match box.9. A wooden block's volume is 24.75 m3. If its length and breadth are 10 m and 5.5 m. Find its height.10. A cubical tank has volume 1728 m3. Find the length of side of the tank.11. A cuboid has volume 1000 cm3. If length is double of its breadth and height is 5 cm, calculate the length and breadth of the cuboid.12. Measure the length and breadth of the objects given below. Board Floor of the class room Baranda Play groundLength breadth Now, solve the following questions.(a) Calculate the area in square cm.(b) Calculate the area in square m.(c) Calculate the area in square inch.(d) Calculate the area in square feet.13. Measure the length, breadth and height of the objects given below and complete the table. Classroom Math book Chalk box Your bed roomLength breadth heightNow, solve the following questions.(a) Calculate the volume in cubic cm.(b) Calculate the area in square inch.170 Acme Mathematics 6
Project Work1. Objective : To find the area of irregular objects. 2. Materials required :Graph paper, A-4 size paper, Glue, A pair of scissors3. Activities: Make 10 cm × 10 cm or 20 cm × 20 cm or 5 cm × 5 cm or any size of graph paper. Draw any five figures. Cut the shape you make. Paste on A-4 size paper. Count the square room and find the area. [Note: Remember not to count squares less than half room]4. An example is done for you:Its area is 30 Sq. cm.Acme Mathematics 6 171
Project Work1. Objective: To construct different solid shapes.2. Materials required : 10 small dice, Card board, pair of scissors, pencil, scale, colours, gum, etc.3. Activity: Arrange 10 small dice so as to get different shapes.4. An example is given below:172 Acme Mathematics 6
Project Work1. Objective: To construct a cube2. Materials required : card board pair of scissors, pencil, scale, colours, gum, etc.3. Activity: Here is a net of cube. Use it to make cube.Each side is 5 cm.Acme Mathematics 6 173
1. Fill in the blanks.(a) Perimeter of the triangle is ………… of all sides.(b) The perimeter of given square ABC is ………….cm.(c) The perimeter of given rectangle ABCD is…………cm.(d) Area of shaded part is …………………cm2.2. Write the area of the given figures:(a)3 cm3 cm3 cm 3 cm(b)6 cm2 cmArea = ...................... cm2 Area = ........................ cm23. Study the given cuboid.(a) Calculate the volume of the cuboid.(b) Calculate the total area of bigger faces.(c) Calculate the total area of smaller faces.(d) How big is bigger face than smaller face?A 2 cm BD C2 cm8 cm1 cmA BD C6 cm 5 cm2 cmMixed Exercise174 Acme Mathematics 6
4. Study the given figure. In this figure a rectangle and a square is given.(a) Calculate the breadth of the rectangle.(b) Calculate the area of square .(c) Find the difference between area of rectangle and square.(d) How long wire is needed to fence the square shape once?5. A rectangular object has its length two times of its breadth. Its height is 10 cm and volume is 2000 cm3.(a) Find its length and breadth.(b) Are breadth and height equal?(c) Write the relation between length and height.(d) What will happen if length is decreased by 10 cm?6. Binita has a square land, and Dinesh has rectangular land, but the area of both lands are equal (a) If the area of Binita's land is 6400 m2, find the land's length.(b) If the length of Dinesh's land is double of Binita's land, then find the length of Dinesh's land.(c) What will be the width of Dinesh's land?(d) Who needs to spend more on fencing if both of them have to fence a wire for a round the land once.7. A square and a rectangle have the same area of 64 m2.(a) Find the length of the square.(b) If the length of the square is half of the length of the rectangle, then find the breadth of the rectangle.(c) Find the perimeter of the rectangle.8. (a) Find the area of the shaded part of the given figure.(b) Our water tank is 9 m long, 3 m broad and 5 m high, find its volume.9. (a) Write the formula to find the perimeter of rectangle.(b) If rectangular field has length 20 m and breadth 10 m, then what is the perimeter of that field.15 ft3 ft3 ft150 ft2A BD C12 cm3 cm 8 cm5 cmAcme Mathematics 6 175
(c) If the breadth of a rectangular field is equal to the length of square field, then find the perimeter of the square field.10. (a) If your height is 4 ft 4 inch, find your height into inch .(b) A rectangle has its sides 84 m and 60 m.(i) Find its area and perimeter(ii) Find the length of wire required to fence it thrice.(c) Find the length of square having area 144 cm2.11. (a) Fill in the blanks..(i) 1 foot = ............ inch. (ii) 1 meter = ......... cm(b) Write the formula of perimeter of rectangle.(c) If length is 10 cm, breadth is 8 cm and height is 5 cm of a cuboid then,(i) Write the formula of volume of cuboid.(ii) Find the volume of cuboid. 12. Ritika has square land and Parbati has rectangular land. The area of both land is equal to 256 m2.(a) If the length of Parbati's land is 32 m, find the breadth of this land.(b) How long wire is needed to fence Ritika's land 4 times.(c) The land has a water tank having 3 m length, 2 m width and 1 m depth. Find the volume of the water tank.13. (a) Find the length of square garden whose perimeter is 120 ft.(b) Find the area of the garden.(c) Write down your own height in foot and inches and convert it into inches.14. (a) A square field is 20 m long.(i) Find its perimeter.(ii) Find the length of wire required to fence it twice.(b) A rectangle field is 35 cm long and 18 cm board. Find its perimeter.(c) Find the are of given figure.(1 box = 1 square unit)176 Acme Mathematics 6
EvaluationTime: 50 minutes Full Marks: 211. Pari's land is square and Nirmal's land is regtangular in shape . (a) Write the formula to find area of Nirmal's land. [1](b) Find the area of Nirmal's land. [1](c) If the length of Paris's land is half of the length of Nirmal's land, Find the perimeter of Pari's land. [2](d) How much fence wire is required to fence around Pari's land two times? [1](e) Convert 4ft into cm. [1]2. Look at the given figure and answer the following questions. (a) Write the name of the given figure. [1](b) Find the area of the base of the given figure. [2] (c) Find the volume of the given figure. [2]3. Look at the given figure and answer the following questions. (a) Write the name of the given figure. [1](b) Find the area of the base of the given figure. [2] (c) Find the volume of the given figure. [2]4. (a) Write 8m and 95 cm into centimeter. [1](b) Find the perimeter of given figure. [2](c) Find the area of the shaded part of the given figure. [2]3.2cm4.8cmAB CD8 cm8 cm8 cm5 cm8 cm3 cm4 cm10 cm3 cm 7 cmAcme Mathematics 6 177
4UNIT AlgebraWarm Up Test1. Solve the following:(a) What is the coefficient in 4x2 ?(b) If 'x' represents 10 only, is 'x' variable ? (c) Choose the 'Binomials: a, a + 3, x – 40 + xy.(d) Express in algebraic form: Three times the difference of 7 and y.(e) If x = 10 and y = 20, find 5(x + y).(f) Write the sum of 5 m and 7 m.(g) Write formula to find the perimeter of rectangle if 'x', 'y' are its length and breadth.(h) Express in the verbal form a(x ÷ y)2. Find the value of the following expressions when x = –5 and y = 4.(a) x + y (b) 3x – y (c) 2x + 5y (d) 4(x + y)(e) 14(x – y) (f) 20 – xy (g) 20 – xy (h) xy + 103. Add the following using column method.(a) 4a2 + 3a + 2 and 10a2 + 7a + 7 (b) – 10x2 + 9x + 1 and 12x2 – 9x + 8(c) x2 + y2 + xy, 3x2 + 2y2 – 3xy and 4x2 + 3y2 + 7xy4. Subtract the following using column method.(a) 2x + 5 from 6x + 10 (b) a + 3 from 2a – 10(c) a2 + a from 5a2 + 3a (d) 2a3 – a from 10a3 + 10a5. Multiply and remove the brackets(a) 2(2x + 1) (b) 5(3x + 4) (c) y(y – 3) (d) 3p(p + 1)(e) 2x(3x + 7) (f) 2(a + b + c) (g) 3(a2 + 6b + 4c) (h) x2(x2 +x + 7)6. Find the quotient.(a) (8x2 – 4x) ÷ 2x (b) (5x2 – 10x3) ÷ 5x2(c) (ab2 + 2ab) ÷ ab (d) (ab2 + 2ab4) ÷ ab27. Solve and Check the following equations.(a) 3x + x = 16 (b) 4x + 2x = 18 (c) 3m – m = 14(d) 4m – 3m = 10 (e) 6x = 20 + 2x (f) 7x = 15 + 2x(g) 10y – 8 = 2y (h) 12y – 2 = 10y (i) 4x + 6 = 2x + 10178 Acme Mathematics 6
4.1 IndicesA. IntroductionConsider the number 16. We know that, 16 = 2 × 2 × 2 × 2 Therefore, 16 = 2424 is the index (exponential) form of 16. Similarly, 49 = 72, 125 = 53In 24, 2 is called the base. 4 is called the index. Thus,24 means '2 is multiplied by itself 4 times'. 24 = 2 × 2 × 2 × 2 = 16 21 means 2 multiplied by itself once. Therefore, 21 = 2 Any number having index power 1 gives the number itself.Solved ExampleExample 1 : Expand (a) 34 (b) (– 2)3 (c) 234 (d) xy3Solution: (a) 34 means, 3 is multiplied to itself 4 times. Therefore, 34 = 3 × 3 × 3 × 3 = 9 × 9 = 81 Similarly, (b) (– 2)3 = (– 2)×(– 2)×(– 2) = (+ 4) × (– 2) = – 8 (c) 234 = 23 × 23 × 23 × 23 = 2 × 2 × 2 × 23 × 3 × 3 × 3 = 1681(d) xy3 = xy × xy × xy = x × x × xy × y × y = x3y3Example 2 : Express 225 in the index form. Solution: 225 = 5 × 5 × 3 × 3 = 15 × 15= 152We use index, power of exponent for same meaning.5 2255 453 93Acme Mathematics 6 179
Example 3 : Write x × x × x × x × x × y × y in the index form. Solution: x × x × x × x × x × y × y = x5 × y2 = x5y2Example 4 : Simplify: 233 × 322Solution: 233 × 322= 23 × 23 × 23 × 32 × 32= 2 × 2 × 23 × 3 × 3 × 3 × 32 × 2= 23Classwork1. Write the base and index (power) in the blanks. (a) 56 base = …….. and index = ………. (b) a3 base = …….. and index = ………. (c) (– 4)5 base = …….. and index = ………. (d) – (3)7 base = …….. and index = ………. (e) y2 base = …….. and index = ………. 2. Expand the following. (a) 73 (b) (– 2)4 (c) 130 (d) (– 1)3(e) 1004 (f) (– 9)4 (g) – 123(h) (xy)43. Write the following in the base and power form. (a) 4 × 4 × 4 × 4 × 4 (b) x × x × x × x (c) (– 3) × (– 3) × (– 3) × (– 3) × (– 3) (d) 1 × 1 × 1 × 1 × 1 × 1 (e) 89 (f) 2401625(g) – 271000 (h) 9– 64Exercise 4.11. Express as power of number or letter.(a) x × x × x × x (b) a × a × a × b × b(c) x × x × x × a × a × a × a × a (d) 2x × 2x × 2x(e) 5a × 5a × 5a × 5a (e) 5x × x × 3x × 2x × x180 Acme Mathematics 6
(g) 4m × 2m × m × m (h) 3n × 6n × 2n × 2n(i) a2 × a × a3 × a2 (j) b2 × b3 × a3 × a4(k) 2y × 2y3 × 2y2 × 2z × 2z2 (l) 8a × 4x × 2ax2. Express as continued product.(a) z4 (b) b3 × a2 (c) l3(d) x2 × y2 × z2 (e) 23 × x3 × 6x4 (f) l2 × b2 × h3(g) 3a2 × 2b2 × 5a3 (h) 7x × 2y × z3 × x × y (i) (2m)4 × (3n)33. If a = 2, b = 3 and c = 5, express given expression as number.(a) 4a2 × b (b) b3 × c3 (c) 4a2 × b2 × c2(d) 6c2 × b3 × a (e) a5 × b2 × 3c (f) c2 × b3 + a44. Compare the following numbers. (a) 23 and 32 (b) 52 and 33 (c) 27 and 34(d) 102 and 53 (e) 43 and 34 (f) 28 and 825. Simplify: (a) 36 + 10 (b) 42 ÷ 24 (c) 33 ÷ 32(d) (–5) × (16)2 ÷ 45 (e) (– 5)4 + 32 (f) 43 – 24 + 7 (g) (– 1)23 + (7)2 (h) – 124 × 23 (i) x7 ÷ x2 + 4x56. A rectangle has length 'x' unit and breadth 'y' unit.(a) Calculate the area of rectangle.(b) If length and breadth are equal, express the area in terms of x.(c) If x = 9 cm, calculate the area of square.7. 'x', 'y' and 'z' unit are the length, breadth and height of a cuboid respectively.(a) Calculate its volume.(b) If x = y = z, express volume in terms of x, or y or z.(c) If x = 12 cm, calculate the real volume of the cube.xyAcme Mathematics 6 181
4.2 Algebraic ExpressionsA. RevisionFour times x is written as 4x. Seven is added to 10y is written as 7 + 10y.Here, 4x and 7+ 10y etc are called algebraic expressions.Mathematical statements formed by the signs + (plus), – (minus), × (multiply), and ÷ (divide) are called algebraic expressions.The components of an algebraic expression are called its terms. For example, in 7 + 10y, 7 and 10y are called terms.According to the number of terms in the algebraic expression we can have the following expressions.(a) Monomial algebraic expressionThe algebraic expression having only one term is called the monomial expression. 4x, 2s, 12xy, 9x2, 12xyz etc are the examples of the monomial expressions. (b) Binomial algebraic expressionThe algebraic expressions having only two terms is called the binomial expression. 6 + x, a + 2x , a2 + b2 etc are the examples of the binomial expressions. (c) Trinomial algebraic expressionThe algebraic expressions having only three terms is called trinomial algebraic expression. 4x + 5y – 3, a2 – 2ab + b2, x2 + xy – 2y2 etc are the examples of the trinomial expressions.B. Constant and VariableThe numbers 1, 2, 3, 4, .......etc are called constants. In the algebra, we also use the letters A, B, C,......X, Y, Z or a, b, c, .........x, y, z instead of numbers.(i) If 'x' represents the number of provinces in Nepal, then the value of 'x' is 7. It is a single value. Hence it is a constant.(ii) If 'x' represents the even numbers between 10 and 15 the value of x = {12, 14}. It has more than one (single) value. Hence 'x' is a variable.Thus, any symbol (letter) having fixed value is known as constant. Any symbol (letter) having more than one value is known as variable.182 Acme Mathematics 6
C. Coefficient, Base and PowerLet, x2 + x2 + x2 + x2 is an algebraic expression. We have, x2 + x2 + x2 + x2 = 4x2Also, 4 × x2 = 4x2So, In the expression 4x2'x' indicate the base of x2.4 indicate that x2 is repeated 4 times, which is called the coefficient of x2. 2 indicate that 'x' is multiplied 2 times which is called the power of 'x'.x2 is added 4 times.x is multiplied2 times. 4x2Power of xBase of x2Coefficient of x2Hence, an algebraic term consist of:(i) Base (The variable)(ii) Coefficient (repetition of the variable)(iii) Power (multiplication times of the variable) Consider the next term 'x'. It can be written as 1x1. Where, its coefficient is 1. Its power is 1.Again, consider the terms 100x and bx.In 100x, 100 is called numerical coefficient of 'x'. In bx, 'b' is called the literal coefficient of 'x'.Classwork1. Write 'constant' or 'variable' in the blank.(a) '77' represents the number of districts in Nepal. 77 is a .....................(b) 'x' represents the counting numbers less than 5. 'x' is a .....................(c) 'y' represents the whole numbers between 5 and 7. 'y' is a .....................(d) The values of 'z' are 3, 6, 9 and 12. 'z' is a .....................Acme Mathematics 6 183
2. Match the following:(a) x – y + z (i) polynomial(b) 2x + 4y (ii) binomial (c) ab (iii) monomial(d) x2 + xy + y2 – 7 (iv) trinomialExercise 4.21. Write down the base, coefficient and power of the following algebraic expressions:(a) 2x2 (b) 10y (c) z3 (d) – 12b (e) 19x3(f) y8 (g) 9x– 2 (h) 7b– 4 (i) – 19 x– 5 (j) 50x102. Classify as monomial, binomial or trinomial:(a) 3x + 8 (b) 9 – x + 2x3 (c) z (d) x2 + 2xy + y2(e) a + b (f) 100 (g) a2 + ab + b2 (h) 4x – 7y + 10(i) a4 + b4 + x (j) 44x – 10y (k) 99x3 (l) 6p + q – r3. Using the given algebraic terms, make 3 monomials, 3 binomials and 3 trinomials.2m, 5x2, – y, – 2q, + 3r, + cd, c, – d4. Express the following algebraically.(a) The sum of 'x' and 4. (b) 'x' is multiplied by 10.(c) The difference of 'a' and 'b' (d) 4 is added to 'y'.(e) One-fifth of a number (f) 10 more than a number(g) The product of 'a' and 'b'. (h) A number increased by 16.(i) 7 times a number 'z'. (j) One less than three times a number 'r'.(k) A number minus 5. (l) The product of 'x' and 'y' divided by 'z'.5. Study the given figure and express the following statements algebraically.(a) Perimeter of the triangle BCD.(b) Perimeter of the triangle ABD. (c) Area of the rectangle ABCD.AD 4x5x 3xBC184 Acme Mathematics 6
6. Express the following algebraically. I am Krishna, I am 'x' years old now. (a) How old was I, 5 years ago ?(b) How old will I be, 10 years after ? (c) My son is 30 years younger than me, how old is he now ?7. Express algebraically. (a) Think a number. Double it. Subtract 10 from the double of the number.(b) Madhav has 'x' rupees now. He got Rs. 10x from his father and Rs. 'y' from his mother. He spent Rs. 4x and Rs. y respectively.8. Look at the cube given alongside and express the following statements algebraically.(a) Perimeter of the square ABCD.(b) Perimeter of the square ABFE.(c) Area of the square ABCD.(d) Area of the square BCGF.D. Value of an Algebraic Expression Consider the algebraic expression 10x +5.Here 'x' is unknown quantity, where 10 and 5 are known quantity. The expression 10x + 5 can have any numerical value.For example,(i) If x = 2,10x + 5 = 10 × 2 +5= 20 + 5 = 25(ii) If x = – 4,10x + 5 = 10 × (– 4) + 5= – 40 + 5 = – 3525 and – 35 are called the numerical value of an algebraic expression 10x + 5.Hence, when we replace the variable (or variables) of an algebraic expression by a number and simplify it, the expression changes to a number, which is called the value of an algebraic expression.B 2x2xCF GD AE H2xAcme Mathematics 6 185
Solved ExampleExample 1 : Find the value of the expression 6y + 3z7x when x = 1, y = 2 and z= 3.Solution: Here, 6y + 3z7x= 6 × 2 + 3 × 37 × 1 [putting x = 1, y = 2 and z = 3]= 12 + 97= 217 = 3Thus, the required value of 6y + 3z7x is 3, when x = 1, y = 2 and z = 3.Example 2 : If a = – 2, b = 1 and c = 2, calculate the value of the expression 12a + 7b – 10c.Solution: Here, 12a + 7b – 10c12 × (– 2) + 7 × 1 – 10 × 2 [putting a= – 2, b = 1 and c = 2]= – 24 + 7 – 20= – 44 + 7 = – 37Thus, the value of 12a + 7b – 10c when a = –2, b = 1 and c = 2 is – 37.Example 3 : If x = 2 and y = 3, find the value of x2 + y2. Solution: Here, x2 + y2= x × x + y × y= 2 × 2 + 3 × 3 [putting x = 2, y = 3]= 4 + 9 = 13Thus, the value of x2 + y2 when x = 2 and y = 3 is 13.Example 4 : If x = 2y and z = 5y, express 3x + 5z in term of y. Find the numerical value of the expression if y = 10.Solution: Here, x = 2y and z = 5ySo, 3x + 5z = 3 × 2y + 5 × 5y= 6y + 25y = 31 yWhen y = 10= 31y = 31 × 10 = 310 Thus, value of the expression is 310.186 Acme Mathematics 6
Example 5 : In the figure ΔABC is given, where(i) Side AB is 2y more than AC and (ii) Side BC is (y + 9) more than AC. If AC is (x + 2) cm, express the perimeter of ∆ABC in algebraic form and calculate the actual perimeter of the triangle when x = 7 cm and y = 1 cm.Solution: Here, AC = (x + 2) cmNow, AB = 2y more than AC = (x + 2) + 2y = x + 2 + 2yBC = (y + 9) more than AC = (x + 2) + (y + 9) = x + 2 + y + 9 = x + y + 11Perimeter of ∆ABC = AB + BC + CA= (x + 2 + 2y) + (x + y + 11) + (x + 2)= x + 2 + 2y + x + y + 11 + x + 2 = 3x + 3y + 15Thus, perimeter of the Δ ABC is (3x + 3y + 15) cm.When x = 7 cm and y = 1 cmPerimeter = 3x + 3y +15= (3 × 7 + 3 × 1 + 15) cm = (21 + 3 + 15) cm = 39 cmHence, actual perimeter is 39 cmClasswork1. Match the following:(a) The sum of x and 70 (i) 10a(b) x divided by 7 (ii) x + 70(c) The product of 10 and a (iii) z – 4(d) 4 taken away from z (iv) x72. Find the value of the following expressions when x = 5,(a) x + 10 (b) 2x + 15 (c) 3x – 8(d) 12 – x (e) 10 – 2x (f) 25 – 4xAB CAcme Mathematics 6 187
188Exercise 4.31. Find the numerical value of the following expression when x = 2, y = 3 and z = 4(a) x + 5 (b) y + 10 (c) z + 12(d) 30 – 13x (e) 14 – 2y (f) 100 – 25z(g) xy + 1 (h) yz – 10 (i) zx + 22. Find the numerical value of the following expressions when x = – 1, y = – 2, z = – 3, a = 2 and b = 3,(a) x2 (b) y2 (c) z2 (d) a2 (e) b2 (f) x33. If x = y = z and x = 10, find the value of the following expressions.(a) 2x + 7y (b) 12y + 4z (c) x + y + z(d) 3x – 6y + 18z (e) 8y – 10x + 23z (f) 100z – 23x – 25y4. If x = 4 and y = 2, find the value of the following expressions.(a) 3x + 4y (b) 8y – 2x (c) 10(x + 4y) (d) 12(5y – 3x)(e) x2 + y2 (f) x2 – y2 (g) 5xy (h) 20y2x(i) 3x + 2yy + 6 (j) 12y – 3x x + 25. If x = 1, y = 2 and z = 3 find the value of each of the following expressions.(a) x2 – y2 (b) z2 – x2 (c) (x + y)2 – z2(d) (z – y)2 + x2 (e) x2 + y2 + z2 (f) (x + y)3 – z3(g) (x + y + z)3 – 3xyz (h) x + y + z xyz (i) (xy – yz – zx)2 (z + y – x)26. Find the perimeter of the given triangle algebraically. Calculate actual perimeter if x = 10 cm and y = 20 cm.7. If x = 9 cm, find the length of the each line segment.(a)(b)(x + 7) cm15 cm(2y + 1) cmx x 2xA BC D2x x 5x188 Acme Mathematics 6
189(c)(d)(e)(f)8. Study the given figure and find the value of the following expressions: [Where, l = Length, b = breadth and h = height].(a) 2(l × b) (b) 2(b × h)(c) 2(l × h) (d) 4(l + b + h)(e) lbh(l + b + h) (f) 2(lb + bh+ lh)9. (a) x and 3x are two numbers, if x = 20, find the numbers.(b) 6y and 9y are two numbers. if y = 5, find the numbers.(c) x2 , 5x and 3x – 7 are three numbers, if x = 4, find the numbers. (d) (x + 1)2 , (6x +3) and (50 – 2x) are three numbers, if x = 19, find the numbers.10. If x = 2.5 cm, find the length of the following sides.(a) AB(b) BC(c) AC11. If R = {3, 4, 5} and x ∈ R , find the length of the following sides.(a) AB(b) BC(c) CD(d) ADE F4x 3x 4xG H2x x 10xI Jx + 1 x + 3 xK Lx + 1 x – 2 2x10 cm 6 cm5 cmA 3x cm6x cm4x cmBCA BD 3x cm(6x – 5) cm(x + 6)cm(3x – 2) cmCAcme Mathematics 6 189
E. Like and unlike terms3x, x, 5x are like terms. Similarly 4a2, 2a2, a2 are also like terms. But 10x and 9x2 are unlike terms, 3x, x and 5x have the same base, 'x' and same power '1'. 10x and 9x2 have same base 'x' but different powers '1' and '2'.Similarly, 4a2, 2a2, a2 have the same base 'a' and same power 2.Hence, the algebraic terms having same base and the same power are called like terms. Otherwise, they are unlike terms.10x and 9x2 have same base 'x' but different powers, hence they are unlike.Similarly, 100x and 2a are also unlike terms.F. Addition and subtraction of algebraic termsMeaning of x3, 4xy and 3x2Meaning of x3 It is cube of x. x3 is volume of cube whose each side is 'x' units.Meaning of 4xy xy is the product of x and y It represents the area of the rectangle whose length is 'x' units and breadth is 'y' units or vice-versa. 4xy represents the sum of 4 rectangles whose area is xy square units.For example,xyxyxy + xy + xy + xy = 4xyHere, xy + xy + xy + xy = 4xyMeaning of 3x2 x2 is the product of 'x' and 'x'. It represents the area of the square whose each side is 'x' units. 3x2 represents the sum of three 'x2'.xxxxx2 x190 Acme Mathematics 6
x2 x2 x2 + + = 3x2Here, x2 + x2 + x2 = 3x2Hence, Only like terms of the algebraic terms can be added. Similarly, Only like terms of the algebraic terms can be subtracted.Examples(i) 4x + 6x (ii) a2 + 3a2 (iii) x3+ 10x3 (iv) 9x4 – 2x4= 10x = 4a2 = 11x3 = 7x4From the above examples, in the case of addition and subtraction only coefficient is changed but the power is not changed (same).Solved ExampleExample 1 : Add: 4x , 7x and 10x.Solution: Here, 4x + 7x + 10x= (4 + 7+ 10) x= 21xExample 2 : Subtract: 6xy2 from 8xy2.Solution: Here, 8xy2 – 6xy2= (8 – 6)xy2= 2xy2Example 3 : Add: 7xy, 2xy and 9xySolution: Here, 7xy + 2xy + 9xy= (7 + 2 + 9) xy= 18xyExample 4 : Simplify: 9xy + 7xy – 3xySolution: Here, 9xy + 7xy – 3xy= (9 + 7 – 3) xy= (16 – 3)xy = 13xyAcme Mathematics 6 191
Classwork1. Match the following:(a) 3x + 2x 7y(b) 10y – 3y 6a(c) 5a + a 2x + y(d) 2x + y 5x(e) y + y + y 12x(f) 4x + 4x + 4x 3y2. Add the following like terms.(a) 5a and 2a (b) 12c and 5c (c) 5x and 8x(d) 9y + 3y (e) 5a + 10a (f) 21r + 11r(g) 3x + 5x (h) 16y + 4y (i) 6q + 13q(j) 18a + 12a (k) 3b + 12b (l) 9m + 3m(m) 3a + 1a + 6a (n) 5x + 6x + 7x (o) 6ac + 5ac + 8ac3. Subtract the following:(a) 5m – 2m (b) 12y – 4y (c) 12m – 7m(d) 7x – 3x (e) 10b – 5b (f) 20a – 12a(g) 7y – 2y (h) 13y – 7y (i) 21y – 2y(j) 9xy – 7xy (k) 8ab – 4ab (l) 11pq – 2pqExercise 4.41. Identify the like and unlike terms with reason.(a) 6x and x (b) 2y and 8y (c) 4a and 7b(d) 7p and 11p (e) 2a2 and a2 (f) – pq and 7pq2. Add the following terms:(a) 3x + 2x + 6x (b) 3y + 9y + 6y (c) 4z + 4z + 3z(d) 10 a2 + 5a2 + 20a2 (e) 9xy + 5xy + xy (f) 7ab + 2ab + 2ab(g) 2abc + abc + 25abc (h) 8a2bc + 7a2bc + 6a2bc (i) 3ax3 + 2ax3 + 6ax33. Subtract the following terms:(a) 13x – 6x (b) 6d – 2d (c) 5c – c (d) 15y2 – 11y2(e) 16h – 9h (f) 9c – 6c (g) 11z2 – 5z2 (h) 3s – 2s(i) 26x – 13x (j) 21y – 5y (k) 15z – 8z (l) 3x3 – 2x3192 Acme Mathematics 6
4. Fill in the blanks:(a) The sum of 2x and 3x is ………………………..(b) The sum of x and y is ………………………..(c) The difference of 7a and a is ………………………..(d) The difference of x2 and – 2x is ………………………..5. Complete the operation and \"Go to home\".Start10x 3x 2x 3+ – ×6. Add the following terms:(a) 3d + 5d + 2d (b) 8mn + 3mn + mn (c) 13wt + 2wt + wt(d) 15z2 + 11z2 + 4z2 (e) a + 2a + 4a + 5a (f) 7b + 5b + 2b + 3b (g) 5x + 3x + 2x + 6x (h) 17y + 11y + 12y + y (i) 8p + 2p + 3p+ 13p(j) 2r + 3r + 4r + 5r + r7. Simplify:(a) 4x + 8 + 5x – 3 (b) 6x + 1 – x + 3(c) 5x – 3 + 2x + 7 (d) 4x – 3 + 2x + 10 +x(e) 5x + 5 – 3x – 2 (f) 4x – 6 – 2x + 1(g) 10x + 5 – 9x – 10 (h) 4a + 6b + 3 + 9a – 3b(i) 8m – 3n + 1 + 6n + 2m (j) 6p – 4 + 5q – 3p – 14 –7q(k) a – 2b – 7 +a + 2b + 8 (l) 6x – 5y + 3z – x + y + z(m) 12a – 3 + 2b – 6 – 8a + 3b (n) 3x + 2y + 5z – 2x – y + 2z8. Calculate the total length.(a)(b)(c)A x cm B 5x cm CA z ft B 2z ft C 3z ft DA y inch B (y + 2) inch C (3 – y) inch DAcme Mathematics 6 193
F. Addition and subtraction of algebraic expressionsTwo or more algebraic expressions can be added or subtracted by grouping the like terms, either horizontally or vertically. The examples gives the clear idea about it.Solved ExampleExample 1 : Add: 6x + 7y and 3x + 2ySolution: Method -I: Horizontal arrangement6x + 7y + 3x + 2y = 6x + 3x + 7y + 2y [like terms are collected]= (6 + 3) x + (7 + 2)y= 9x + 9yThus, the sum is 9x + 9y. Method-II: Vertical arrangementThus, the sum is 9x + 9y.Example 2 : Add 5xy + 9ab + 7bc and 8xy – 7ab – 9bcSolution: Addition by vertical arrangement:Thus, the sum is 13xy +2ab – 2bc.Addition by horizontal arrangement:5xy + 9ab + 7bc + 8xy – 7ab – 9bc= 5xy + 8xy + 9ab – 7ab + 7bc – 9bc [Collecting the like terms]= 13xy + 2ab – 2bcThus, the sum is 13xy + 2ab – 2bc. 6x + 7y+ 3x + 2y 9x + 9y 5xy + 9ab + 7bc+ 8xy – 7ab –9bc 13xy + 2ab – 2bc 194 Acme Mathematics 6
Example 3 : Subtract: 3x – 6y + 8z from 8x + 8y – 4zSolution: Subtraction by vertical arrangement 8x + 8y – 4z 3x – 6y + 8z(–) (+) (–) ...... 5x + 14y – 12z Thus, the difference is 5x + 14y – 12z.Subtraction by horizontal arrangement(8x + 8y – 4z) – (3x – 6y + 8z)= 8x + 8y – 4z – 3x + 6y – 8z= 8x – 3x + 8y + 6y – 4z – 8z= (8 – 3) x + (8 + 6) y – (4 + 8)z= 5x + 14y – 12zThus, the difference is 5x + 14y – 12z.Classwork1. Try to answer orally.(a) The sum of x + y and 2x + 3y is .........................(b) The sum of 3x + 2y and –3x + 2y is ..........................(c) The sum of 4 x + 3y and 2x – 5y is..........................(d) The sum of 4a – 3b and a + b is ..........................2. Add the following algebraic expressions:(a) 2x + 3y and 4x + 4y (b) 3x + 7x and 5x + 10y(c) 8a + b and 7a + 7b (d) 7m + 2n and 2m + 3n(e) 3p + 12q and p – 10q (f) 10a – 3b and 12a + 12b(g) 13m – 2n and 2m – 15n (h) 14y + 10z and 12y – 5z3. Subtract the following algebraic expressions:(a) 2x + 3y from 4x + 5y (b) x + 2y from 3x + 7y(c) 10y + 12z from 20y + 20z (d) 2m + 2n from 7m – 3n(e) 9a + 2b from 10a – 7b (f) 13p – 14q from 20p – 2q(g) 4m – 2n from 5m – 2n (h) 2x + 3y from 2x + 3ySign should be changed while subtractingAcme Mathematics 6 195
Exercise 4.51. Complete the addition route A and B.Addx – 3x + yABSubtractx + 3Addx + yAdd4 + 2xAddy + 10Subtract3x – 10Subtract3x – 10Add5x – 62. Add the following terms:(a) 3x + 4x + 2y + 8y (b) 5p + 3p + 6q + 2q(c) 3xy + 2yz + 2yz + 3xy (d) 12ab + 10bc + 24ab + 9bc(e) 9 + 2b + 4a + 5b + 9a (f) 12 + 15b + 2a + 3b + 22a(g) 5x + 3x + 2x + 6y (h) 17y + 11y + 12y + 2x(i) 5x2 + 6xy + 4xy + 5x2 + xy (j) 12ac + 15a2 + 10ac + 15a2 + 9ac (k) 3xy2 + 4xy2 + 5x2y + 6x2y (l) 7x2y2 + 8x2y2 + 9xy + 10xy3. Add the following using column method:(a) 4a2 + 3a + 2 and 10a2 + 7a + 7(b) – 10x2 + 9x + 1 and 12x2 – 9x + 8(c) x2 + y2 + xy, 3x2 + 2y2 – 3xy and 4x2 + 3y2 + 7xy(d) 3x2 – 2y2 + xy, 4x2 – 3y2 + 4xy and – 4x2 – 3y2 – 8xy(e) p2 + q2 + 10, 3p2 –3q2 + 12 and 9p2 + 9q2 + 9(f) x3 + x2 + x + 5, 3x3 + 4x2 + 5x + 6 and 2x3 – 3x2 – 2x – 1(g) 3a3 + 2a2 – 4a – 10, a3 – 2a2 + 4a + 11 and a3 – 3a2 + 11a + 124. Subtract:(a) 12x + 13y from 14x + 17y (b) 2x + 3y from 8x + 9y(c) 6x – 14y from 9x + 15y (d) –5x + 4y from 14x – 5y(e) – 10x – 12y from 8x + 13y (f) – x – y from – 2x – 7y196 Acme Mathematics 6
5. Subtract the following using column method:(a) 2x + 5 from 6x + 10(b) a + 3 from 2a – 10(c) a2 + a from 5a2 + 3a(d) 2a3 – a from 10a3 + 10a(e) 12ab – 7bc from 9ab + 6bc(f) 14mn + 8ny from 10mn + 3ny(g) 5x + 17y + 2z from 6x + 18y + 3z(h) x – 4y + z from 4x – 7y + z(i) 4x2 – 5x + 3 from – 8x2 –9x + 8(j) – 2x2 + 8y – 2z from – 3x2 – 4y – 3z(k) 3xy –2yz + 4xz from 10xy + 12yz + 14xz(l) 4x2 – 3x – 2y from 5x2 –4x + 7y6. Subtract:(a) 2x + 2y + 2z from 5x + 6y + 7z(b) 7a + 9b + 2c from 9a + 12b – 4c(c) 3x + 7y – 9z from 5x – 14y –13z(d) 4x – 6y – 8z from – 2x – 3y – 4z (e) – xy – 5yz – 7zx from 9xy + 3yz – 8zx(f) 6mn – 7mp – 8pn from – 6mn + 7mp – 2pn7. Collect the like terms together and simplify.(a) 5x2 + 5x + 2x2 – 3x (b) 9x2 – 2xy + 2x2 + 4xy + 6x2(c) 4ab2 + 5xy + ab2 + xy – 8ab2 (d) 8x2 – 3x + 8 + 7x2 – 2x – 12 (e) 5y2 + 4xy + 4 – y2 – 3xy – 1 (f) 2xy2 + 3xz – 7 – 3xy2 + 9xz – 2 (g) 3ax2 – 2bx + 3c – ax2 – 2bx – 5c (h) 6zx2 + 7px – 8q – 3zx2 + 5px – 10q8. Simplify:(a) 2(7y + 5k) + 4(5k + 2y) (b) 5(6k – 3w) + 2(8w + 3k)(c) 5(2w + y) – 3(y – 3w) (d) 8(2n – 6) – 5(4n + 20)Acme Mathematics 6 197
(e) 3x2 + 4x + 6 – (x2 – 3x – 3) (f) 5x2 – (3x + 2 – 3x2) + 2x – 2(g) 2x2 – (2x + 3 – x2) – 2x – 5 (h) 6x2 – 7x + (8 – 3x2 + 5x) – 109. (a) What should be added to x + y to make it 2x + 3y ?(b) What should be added to a – c to make the sum 5a – 9c ? (c) If the sum of two expressions is 2a2 – 3b2 and the first expression is a2 + b2, what is the second expression ?(d) What should be added to 4ab – 3ac so that the sum will be 3ab + 4ac ?(e) What should be subtracted from 9x + 4y to get the remainder x + y ? (f) What should be subtracted from 12ab – bc to make the remainder 18ab + bc ? 10. If a = x + 7y, b = 4x – 2y + 2 and c = – 5x + 6y – 9, calculate the following:(a) a + b + c (b) 2a + b + 4c (c) 3a – b + 2c(d) 5a + 4b – 6c (e) – a – b + 2c (f) – 3a + 4b – 6c11. Find the perimeter (sum of all sides) of each of the following shapes.(a)B 3x + 8y Cx + 12y2x + 7A (b) B x + y + 3 CAx + 4y + 13x + 5y + 10(c)CB DA4y – 135x + 2y + 86x + y10 – x – y(d)A a – 3b BCE 6a + 8b D12 + b20 – a 2a+8b+12(e)ABE DC5xy7xyxy – 92xy+8x+72xy + 4(f)A BCE 22b D3b+720–a–2b4a–8b+44a12. Calculate the perimeter of the given figure (Question 11) when x = 2 cm, y = 4 cm, a = 6 cm and b = 1 cm.198 Acme Mathematics 6
EvaluationTime: 60 minutes Full Marks: 251. (a) Write in expand form of x3y2. [1] (b) State whether the given mathematical statement is true of false. 3x is the sum of x and 3. [1] (c) Write any one example of binomial. [1]2. (a) Find whether the given terms are like terms or unlike terms. [2](i) 2x and 5x (ii) 3a and 7b(b) Sum of 4y and 3y is ............. [1](c) Haris has y pens. He gave 3 pens to his friend Uma. How many pens does Haris have now ? [1]3. (a) Write the algebraic expression for 5 times the difference of x and y. [1] (a) Subtract (2x + 3y + 4z) from (6x + 7y + 8z) [2] (b) Find the total length of given line segment. [2]4. Fill in the blanks:(a) The value of x is 8 is represent as ..................................... [2](b) 8xyz and 5a2b are...........terms. [1] (c) Write 3×3×3×3×3×3 in indices form. [1](d) Find the sum of 5ab and 7ab. [1]5. (a) Fill in the blanks The value of 'x is greater than 5' is written as ......................., [1](b) Define an algebraic expression. [1](c) Simplify: (a2 – 2ab + b2) + (a2 + 2ab –b2) [2] 6. (a) Write the value of x°. [1](b) Write the 5×5×5×5×5×5×5×5 as an exponent. [1](c) Find the value of 3°. [1](d) Simplify :4a2 + 3ab – 2a2 – 2ab [1]xcmA B C D2xcm 5cmAcme Mathematics 6 199
G. Multiplication of algebraic termsLet us take a rectangle whose length is 4 cm and breadth is 2 cm then,Its area = 4 cm × 2 cm = 8 cm2. If length is replaced by 4x and breadth is replaced 2x then, Area of rectangle = 4x × 2x = 8x2 square unit.Oh! 4 × 2 = 8 Coefficients 4 and 2 are multiplied4 cm2 cmx x x xxxSq. is short form of squareYes, x × x = x2power of the base (x) is added, 1 + 1 = 21 cm2 1 cm2 1 cm2 1 cm21 cm2 1 cm2 1 cm2 1 cm2x2x2x2x2x2x2x2x2Solved ExampleExample 1 : Multiply: 3x and 7xSolution: Here, 3x × 7x= 3 × x × 7 × x= 3 × 7 × x× x= 21 × x2 = 21x2Example 2: Multiply: 9x and 5ySolution: Here, 9x × 5y= 9 × x × 5 × y= 9 × 5 × x × y= 45xyExample 3 : Multiply: 6x2 and – 4xSolution: Here, 6x2 × (– 4x)= 6 × x2 × (– 4) × x= 6 × (– 4) × x2 × x = – 24x3 While multiplying a monomial, we first multiply their coefficients. While multiplying ,we add the powers of the same base. If the bases are different, we simply write different bases in the form of product.How we multiply the terms. 200 Acme Mathematics 6