H. Multiplication of Binomial and MonomialActivityDiscuss the area of rectangle whose length is (x + y) cm and breadth z cmxz cm2 z cmx cm ABFEDCy cmyz cm2Here, Area of rectangle ABCD = length × breadth= (x + y) cm × z cm = (x + y)z cm2 .......... (i)Area of rectangle ABCD = Area of rectangle ABEF + Area of rectangle ECDF= xz cm2 + yz cm2= (xz + yz) cm2 ................ (ii)From (i) and (ii) (x + y)z = xz + yzConsider the product of 6 and 7.(i) Product of 6 and 7 = 6 × 7 = 42(ii) Product of 6 and 7 = 6 × 7= 6 × ( 4 + 3)= 6 × 4 + 6 × 3= 24 + 18 = 42Here, result of (i) and (ii) are same. In algebra, the multiplication of binomial by monomial is similar as example (ii)Solved ExampleExample 1 : Multiply x and (2x + 3y)Solution: Here, product = x × (2x + 3y)= x × 2x + x × 3y= 2x2 + 3xyThus the product is 2x2 + 3xyIn general we write the English alphabet in their order.2x x 22x 3y3xyAcme Mathematics 6 201
Example 2 : Multiply (6x2 – 3x +2) and 4xSolution: Here, product = (6x2 – 3x + 2) × 4x= 6x2 × 4x – 3x × 4x + 2 × 4x= 24x2 × x – 12x × x + 8x= 24x3 – 12x2 + 8xExample 3 : If a = x2 + 3x and b = 12x, find ab.Solution: Here, a = x2 + 3x and b = 12x Now, ab = a × b= (x2 + 3x) × 12x= x2 × 12x + 3x × 12x= 12x3 + 36x2Hence, ab = 12x3 + 36x2Classwork1. Multiply:(a) 2 × 4x (b) 3x × 6 (c) 4a × (– 8) (d) x × 2x(e) 3x × 10x (f) x × 9x2 (g) 2b3 × b2 (h) x2 × x3(i) y3 × y8 (j) 2p2 × 7p (k) 6x × 3x2 (l) 2xy × x(m) xy × 5y (n) 10x × xy (o) ab × ab × ab2. Complete the table:(a) (b)3. Fill in the blanks:(a) 4 × 5 ……… (b) 16 × x ……… (c) x × x ……...(d) 4x × 7x ……… (e) a × a2 …… (f) 2a × 5a2 ……...(g) p × p × p ……… (h) p × 2p × 3p ...… (i) 4p × 5p × 6p ….....(j) p2 × p2 × p ……… (k) q2 × 4q2 × 6q …… (l) 2z2 × 4z2 × 5z …...(m) 5p × 3q ……… (n) 3x × 8y ……… (n) z × 0 ……I remembered! powers are added, herex2 × x = x × x × x = x3and x × x = x2× x x2 x3 yx xy2x4x× a 2a 4b2 5z xy4a2b3zxy202 Acme Mathematics 6
Exercise 4.61. Multiply:(a) x × x × x (b) 2x × 3x × 4x(c) x2 × x3 × x2 (d) (– 2x) × 3x2 × 4x2(e) 2y × (– 2y2) × (– 3y3) (f) xy2 × x2y × x3y3(g) 4 × x2y2 × x3 × y3 × 102. Multiply:(a) x × (x + 6) (b) x × (2x + 7) (c) y × (x + 4)(d) 3x × (x + 6) (e) 7x × (10 – x) (f) 8x × (8 – x)(g) 5x × (2x – 4) (h) 8x × (4x + 1) (i) 6x × (2x + 9)(j) 4x × (4x2 – x) (k) 2x × (7x2 + x) (l) 3x × (7x2 – x)3. Choose the correct term and fill in the blanks.(a) 3a × (.......) = 6a2 (b) 2xy × (.......) = 10x2y(c) – 10x × (.......) = – 20x2y (d) 12ab × (.......) = – 36a2b2(e) – x2y2 × (.......) = + 20x2y2 (f) a2b2c2 × (.......) = 4a5b5c54. Find the product of the following:(a) xy × xy × xy (b) ab × ab × 3ab(c) mn × 4mn × 5mn (d) a2b × a2b × 3a2b(e) 2x2y × 4x2y × 3x2y (f) 4x2y2 × 5x2y2 × 6x2y2(g) 10ab × 4bc × 3ca (h) 6x2y 10 × 5y2z × 2xyz 35. Multiply and simplify:(a) (x × x) + (2x × 3x) (b) (4x × x) + (7x × 3x)(c) (x × 6x) + (4x × 3x) (d) (3x × y) – (8x × 3y)(e) (x2 × 2y) – (x2 × y) (f) (3x2 × 2y2) – (3y2 × 2x2)(g) xy × 9x + 2xy × 5x (h) ax × 3ax + 2ax × 7ax(i) pq × 5q + 5pq × 6q (j) x(x + y – 6)(k) 7x(x + 4y – 10) (l) 5p(7p – 3q +1)(m) 2x2(x + a + 7) (n) 7x(x2 + 9x + 8)Acme Mathematics 6 203
6. Simplify:(a) x(x + y) + y(x + y) (b) x(x – y) +y(x – y)(c) x(x – y) – y(x – y) (d) 2x(x + 1) + 4(x + 1)(e) 4x( x + 3y) + 2y(x + 3y) (f) 8x(3x – y) + 3x(9x – y)(g) 7x(x – 2y) – 3y(2x + y) (h) 2x(3x + 2y) + 4y(3x + 2y)7. The area of rectangle = length × breadth. Find the area of the following rectangles whose length and breadth are given.(a)5x cm3x cm(b)x cm(x+5) cm(c)(x + y) cm10y cm(d)(10 – x) cmxy cm(e)(5 + y) cm7xy cm(f) (6xy + 4) m(xy + 5) m(g) If x = 2 cm, y = 3 cm. Calculate actual area of rectangles in (a – f)8. If a = x+ 7y, b = 4x and c = – 5x, calculate the following:(a) a × b (b) 2a × 4c (c) 3a × (b + 2c)(d) (a + 4b) × 6c (e) 5a × (b + 2c) (f) 3a × 4b × (– 6c)9. (a) If x = 4a2b2, y = 2a3b, z = 6a3b3 then find xyz .(b) Which are the cofficient, base and power of the expression in xyz ?(c) Compare xy and z, when a = 1 and b = 2.10. Observe the given solid object.(a) How many faces are there in the object?(b) Calculate the volume of given object.(c) If x = 2 cm, then find the length of given object. 4x2x 3x204 Acme Mathematics 6
J. Division of the Algebraic TermsActivityThe area of rectangle is 8x2 cm2 and its breadth 2x cm. How long is its length? Discuss.Here, area of rectangle ABCD (A) = 8x2 cm2breadth of rectangle (b) = 2x cm.length of rectangle (l) = ?Now, Area of rectangle (A) = length × breadthor, 8x2 cm2 = l × 2x cmor,8x2 cm22x cm = lor, 4x cm = lHence, length of rectangle is 4x cm.Here, the process of finding the length i.e. 8x22x is the division. Consider two numbers 10 and 5.Then, 10 ÷ 5 = 105 = 2Where, '2' is the quotient, 10 is the dividend and 5 is the divisorIf we replace 10 by 10x and 5 by 5x then, division of 10x by 5x is denoted by 10x ÷ 5x.Now, 10x by 5x = 10x5x = 2Where, 10x is called the dividend.5x is called the divisor. 2 is called the quotient.So, DividendDivisor = QuotientNow, study the given solved examples carefully.Solved ExampleExample 1 : Divide 20x2 by 4x.Solution: Here, 20x2 ÷ 4x= 20x24x = 5 × 4 × x × x4 × x = 5 × x1 = 5x8x2 cm2 2x cmDACBAcme Mathematics 6 205
Example 2 : Find the quotient 18x4×z2– 3x2zSolution: Here,18x4×z2– 3x2z = 6 × 3 × x2 × x2 × z × z– 3 × x2 × z= 6 × x2 × z– 1 = – 6x2z Hence, the quotient is – 6x2z.K. Division of binomial by monomialActivityThe area of rectangle is (2x2 + 6x) cm2 and its breadth 2x cm. How long is its length? Discuss.Here, area of rectangle ABCD (A) = (2x2 + 6x) cm2breadth of rectangle (b) = 2x cm.length of rectangle (l) = ?Now, Area of rectangle (A) = length × breadthor, (2x2 + 6x) cm2 = l × 2x cmor, (2x2 + 6x) cm22x cm = lor,2x22x + 6x2x cm = lor, (x + 3) cm = lHence, length of rectangle is (x + 3) cm.Here, the process of finding the length i.e. 8x2 + 6x2x is the division of binomial by monomial.Oh! In division power of the variable is decreased.(2x2 + 6x) cm2 2x cmDACB206 Acme Mathematics 6
Study the given example carefully.Example 1 : Divide (9x3 – 6x2) by 3x.Solution: Method IHere, (9x3 – 6x2) ÷ 3x= 9x3 – 6x23x= 9x33x – 6x23x= 3 × 3 × x2 × x3 × x – 3 × 2 × x × x3 × x= 3x2 – 2xHence, when (9x3 – 6x2) is divided by 3x, the quotient is (3x2 – 2x).Classwork1. Think and fill in the blanks.(a) b ÷ b = …….. (b) 4x2 ÷ 2x = ……..(c) a3 ÷ a2 = ….......….. (d) 4x2 ÷ 2 = …….. (e) 8x4 ÷ 4x = …….. (f) 5b7 ÷ b2 = ................(g) 8xy ÷ xy = …….. (h) – 10x2y ÷ x2y =……..(i) 20x7y7 ÷ 5x3y4 = ..…..2. Divide the following and find the quotient.(a) 4xx (b) 6a22 (c) 10a3a (d) – 3x2x2(e) – 12x2y3xy (f) – 25xy3– 5xy (g) 12a4b4– 3a2b3 (h) – 50x7y6– 10x6y7Exercise 4.71. Divide:(a) a4 by a (b) x3 by x2 (c) 6a by 2 (d) 8x by 4x(e) 24x by 12x (f) 15 by 5y (g) 20x2 by 10x (h) 32x3 by 8x2(i) 16p by 32p (j) 2y2 by 8y (k) – 15z by 3(l) (–15a2) by (–3a) (m) 4p3 by 8p2 (n) 20x6 by 4x4 (o) 21y3 by (– 7y2)2. Find the quotient:(a) x2y ÷ xy (b) 3x2y2 ÷ xy2 (c) 4a2b ÷ 2b(d) 3xy ÷ 2x (e) 4x2k ÷ xk (f) 6a3b ÷ 2abMethod IIHere, (9x3 – 6x2) ÷ 3x3x ) 9x3 – 6x2( 3x2 – 2x9x3– 6x2– 6x2 0–+Acme Mathematics 6 207
(g) 12ab3 ÷ (–3ab2) (h) 9x3y2 ÷ x2y2 (i) 10a3b3 ÷ 5a2b2(j) 6xy2 ÷ (–4xy) (k) (– 20ax2) ÷ (– 2ax2) (l) (– 2x6y6) ÷ 2x2y23. Simplify:(a) x3x2 + x2x2 (b) 4x42x + 3xx (c) 5x2y2xy + 2xyxy (d) 16xy8x – 2xy2y(e) 6xy23y – 4xy2x (f) 12ab26ab – 10a2b5ab (g) 8x3a3xa2 – 9x4a4x2a2 (h) 10p3r2pr – 20r3p5rp(i) 30a7b76a2b2 – 50ab10ab4. Find the quotient(a) (x2 + 5x) ÷ x (b) (6y2 – xy) ÷ y(c) (8x2 + 6x) ÷ 2x (d) (5x2 + 15x ) ÷ 5x(e) (9a2 –15ab) ÷ 3a (f) (a2b + ab2) ÷ ab(g) (x2y – xy2) ÷ xy (h) (3a2b – 2ab2) ÷ ab(i) (4a2b + 6ab2) ÷2ab (j) (2a3b + 3ab2) ÷ ab(k) (2a2 + 4ab) ÷ 2a (l) (8a2b + 4b) ÷ 4b5. Divide:(a) (3ab2 + 2ab4) ÷ ab2 (b) (4x2y2 – 5x2y3) ÷ x2y2(c) (20a3b3 + 10a2b2) ÷ 5a2b2 (d) (6x4y4 – 12x6y6) ÷ 6y4x4(e) (8ax5y5 – 16bx4y4) ÷ (– 4x2y2) (f) (– 10a3b3 + 50a2b3) by (– 5a2b3)(g) (4xyz3 + 36xyz2 ) by 4xyz26. If x = 4a2 b2, y = 2ab and z = 6a3 b3 find the following.(a) xy (b) zy (c) zx (d) yzx7. If the product of x and 4ab is 24a3b3, find the value of x.8. If the product of y and – a3b34 is 10a4b4, find the value of y.9. (a) Find the sum of 6a2b2 and 10a2b2 and divide it by 4ab.(b) Fill in the blanks.(i) 8a2b2 ÷ .................... = 2ab (ii) ............ ÷ 3ab = 4ab(c) If A = 20x4y6, B = 8x2y and C = 4xy. Find, (i) AB (ii) BC (iii) A ÷ C (iv) A + BC (v) B – AC208 Acme Mathematics 6
10. Choose the quotient, Divisor and dividend from the given algebraic expressions:(a) 30a2x6 (b) 5ax3 (c) 6ax3 (d) 5a2x211. Study the given rectangles and find the measure of remaining sides.(a) (b)(c) (d)12. Calculate the measure of the unknown sides in each of the following figure.(a)b6 cm1 cmV = 12 cm3(b)l 2 ft2 ftV = 16 ft3(c)1 mV = 10 m31 mh(d)3 ftV = 15 ft3b2.5 ft(e)2 mV = 8 m3aa(f)bV = 32 cm3b8 cm13. The area of rectangle is 40x2y2 square meter.Now, guess the following.(a) How long is BC?(b) How long is CD?Area = 4xy cm2xy cmDACBArea = 30a2b2 cm26ab2 cmDACBArea = 100x2y3z44y2z2DACBArea = (36x3y3 + 18x2y2) cm29x2y2DACBArea = 40x2y2DACBAcme Mathematics 6 209
Mixed Exercise1. State whether the given mathematical statement is true or false: (a) 4x2y and 7x2yz2 are like terms. .............(b) The value of x is below 5, then x is .................(c) The first box contains 6 packets of chocolate; the second box contains 8 packets of chocolate. If each packet contains y number of chocolate, find the total number of chocolate in both boxes.2. (a) State whether the given mathematical statement is true or false.→ 2a2b and 4ab2 are like terms.(b) A rectangular handkerchief has an area of 24x2y2 square centimeter and breadth is 4xy centimeter then what is its length? Find it.(c) If x = 2 and y = 3, what is the length and area of the handkerchief ? Find it.3. (a) Find whether the given terms are like terms or unlike terms.(i) 2x and 5x (ii) 3a and 7b(b) Sum of 4y and 3y is .................... .(c) Hari has y pens. he gave 3 pens to his friend Ram. How many pens does Hari have now?4. (a) Find the sum of : 3x2 + 4x + 5 and 5x2 + 6x + 7(b) Subtract : 2x + 3y + 4z from 6x + 7y + 8z(c) Find the total length of line segment AD.5. (a) Identify whether the given symbols are constant or variable.i. 'P' is the sum of 2 and 3 ii. 'Z' represent the whole numbers less than 7 (b) Define binomial with examples.(c) Define like terms with examples.6. (a) Identify the like and unlike terms.2a, 3a, 2ab, 4a2Like Terms Unlike TermsA Bx 2x x + 4C D210 Acme Mathematics 6
(b) Write numerical and literal coefficient of p in – 5pq2.(c) Write down in product form 7p3q4.7. Study the given figure.(a) If x = 100, then what is the real length of football ground?(b) If x = 100, then what is the real breadth of football ground?(c) Calculate the perimeter of the football ground?8. (a) What is the value of x0 ?(b) Write the 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 as an exponent.(c) Find the value of 36.9. (a) What is the volume of a solid object having length (l) breadth( b) and height (h)?(b) If length, breadth and height of that solid object becomes 'b' unit, then find it's volume.(c) Write in the form of indices : 5 × 5 × 5 × 510. The expression 2x + 3y is given.(a) Classify the expression.(b) Write coefficient of y(c) If x = 3 and y = 4 find the numerical value of 2x + 3y.11. (a) Write the coefficient of 10x2.(b) Classify as monomial, binomial or trinomial. 4x, 3 – 5x, 4x2 + 3x + 2, x(c) Add: 2x, 5x + 3 and 6x – 212. (a) Multiply: 5a2 and 15a2(b) Subtract: 5x – 3y from 7x + 9y(c) Divide: (12x3 + 9x2) by 3x213. (a) Multiply: x and (4x – y)(b) Find the value of the expression 7x + 2y + 10z where x = 2, y = 3 and z = 5.(c) Divide: (20a4b2 + 10a3b3 + 5a2b2) ÷ 5a2b214. (a) Write the perimeter algebraically:(b) If x = 3 and y = – 2 find the value of x2 – y2 + 7.(x + 10) m(x – 51) mB CAx yx + 3Acme Mathematics 6 211
(c) Multiply: (4x – y) and (x + 3y) (d) Simplify: 2x(x – 2y – 3) + 2y(x – 3y +7)15. The area of square shaped garden is 36x4y4 cm2.(a) How long is AB?(b) How long is BC?16. (a) What is the index of 27?(b) Express 72 in words.(c) Express 81 as power of 3.17. (a) Express in the expontial form : (5x) × (5x) × (5x) × (5x) × (5x) × (5x)(b) Find the quotient. : (15x5 – 5x3) by 5x218. (a) Write 54 in the multiplication form.(b) Express the sum of 7 and y in algebraic expression.(c) If x = 1, y = 2 and z = 3 then find (x + y – z)19. (a) Write 72 in exponential form by prime factorization.(b) Add : 5xy, 3xy and xy.(c) Express algebraically: \"7 is subtracted from 5y\".20. In the figure, ABCD is a rectangular field.(a) Find the perimeter of the field.(b) Find the area of the field.(c) Find the area of the field ABC.(d) How much length must be reduced to make it a square field? (e) What is the area of square field?36x4y4 cm2DACB4x mD (5x + 3) mACB212 Acme Mathematics 6
EvaluationTime: 48 minutes Full Marks: 201. (a) Multiply : (3y × 4y × 5y)( x × 7x × 6x) [2] (b) Find the volume of given solid object. [2] (c) Simplify: 2a(4a + 3b) + a(3a – 4b) [2]2. (a) Divide: 21x3y2 - 56x2y3 [2] 7xy (b)Multiply: 2x(3y + 4z) [2] 3. (a) Calculate the length in given figure. [1] (b) If x= 4cm and y = 6cm then calculate the length and breadth of that rectangle. [1]4. (a) If value of x =2, y = 3 and z = 5 then, find the value of x2 + 2xyz +z2 [2](b) Find the value of 2p + q3 if P = 2 and q = 1. [2] (c) If x = 2 and y = 3, find the value of x2 + 2xy + y2. [2] 5. Find the length of AC in the adjoining figure if x = 5cm and y = 2cm. [2] Area:24x2y3 square meter 3xy meter? meter(3x+y)A B C(2x-3y)4z2z3aAcme Mathematics 6 213
4.3 Linear Equation and InequalityA. Mathematical statementsActivityStudy and discuss the following statements:The sum of 4 and 5 is 9.The difference of 8 and 6 is 4.The product of x and 5 is 20.Which of the above sentences are:(a) A true sentence?(b) A false sentence?(c) Neither true nor false sentence? The sum of 4 and 5 is 9. It is written as 4 + 5 = 9. It is true.Hence, it is true sentence. The difference of 8 and 6 is 4. It is written as 8 – 6 = 4. It is false.Hence, it is false sentence. The product of x and 5 is 20. It is written as x × 5 = 20. We can not say whether it is true of false. We don't know the value of x.When we say,(i) The sum of 21 and 3, it is a expression.(ii) The sum of 21 and 3 equals 24.It can be written 21 + 3 = 24. It is arithmetic equation. (iii) The sum of x and 3 equals 24.It can be written as x + 3 = 24, It is algebraic equation. 214 Acme Mathematics 6
(iv) y is greater than 10. It is written as y>10.It is inequality (not equation), The above statements are called mathematical statements. Now, consider the following statements:(i) x = 21 (ii) y < 10(iii) 21 > 3 (iv) 21 is exactly divisible by 4.What we can say about the above statements ?Statements (iii), 21 > 3, is trueStatements (iv), 21 is exactly divisible by 4, is false. But we cannot say about statement (i) and statement (ii) either true or false.Such statements are called an open statement.Open statements are: (i) Equation, like x + 3 = 24 (ii) Inequalities, like y > 10Classwork1. Write True, False or Open sentence in the blanks:(a) The sum of 3 and 5 is 8. .....................(b) The difference between 10 and 4 is 6. .....................(c) The product of 4 and 7 is 25. .....................(d) The product of 9 and 4 is 36. .....................(e) Triangle has 4 sides. .....................(f) Quadrilateral has 'y' sides. .....................(g) The sum of 'm' and 3 is 20. .....................(h) Y divided by 'a' gives 6. .....................(i) Half of 30 is x. .....................(j) 20 is divisible by 4. .....................2. Make the open mathematical sentence:(a) x added to 7 is 14 ................(b) y subtracted from 10 is 6. ................(c) the product of 5 and y is 30. ................(d) x increased by 2 is equal to 10. ................(e) y divided by x is equal to 5. ................x + 7 = 14Acme Mathematics 6 215
Exercise 4.81. Write 'True' statement, or 'False' statement in the blanks.(a) The sum of 2 and 7 is 9 ................................(b) The difference of 7 and 11 is – 2. ................................(c) The product of x and 10 is 110. ................................(d) The quotient of 200 ÷ y is 10. ................................(e) All acute angles are less than obtuse angles. ................................(f) 3 is a factor of 20. ................................(g) – 2 is a multiple of x. ................................(h) x + 3 = 7, where x = 0. ................................(i) 2 + 3 > 1 + 3 ................................(j) 6 + 2 < 7 – 1 ................................2. Choose any number in the blank and make true sentence:(a) The sum of 10 and ................ is 18.(b) 4 is double of ................(c) The difference of ................ and 7 is 3.(d) ................ × ................ = 50(e) 80 ÷ ................ = 2(f) ................ is divisible by 5.(g) The number between 100 and 102 is ................3. Choose the correct value for the symbol x so that the statement is 'True'.(a) x is less than 5, the value of x is ................................(b) x is factor of 10, the value of x is ................................(c) x is multiple of 2, the value of x is ................................(d) The sum of x and – 2 is 3, the value of x is ................................(e) (x + 3) < 4 – 3, the value of x is ................................(f) x + 3 ≥ 10, the value of x is ................................(g) When 50 is divided by x the quotient is 5, the value of x is ......................(h) The product of 1019 and x is 20, the value of x is ......................216 Acme Mathematics 6
B. Linear equationLet us consider the following statements.(i) The sum of x and 4 is 10.(ii) 6 more than x is 20.(iii) 4 times x is 48.(iv) x divided by 7 gives 3.We can write the above statements (i) to (iv) respectively as follows.x + 4 = 10 .................... (i)x + 6 = 20 .................... (ii)4x = 48 .................... (iii)x7 = 3 .................... (iv)We observe that the symbol '=' appears in each of the statements.Hence, all these are equations. All equations are linear equations. Variable with power 1 is called a linear equation.A statement in which equality (=) sign and a variable present is called an equation.Equation has two sides:Left hand side (LHS) and Right hand side (RHS)In equation (i), x + 4 is LHS and 10 is RHS.C. Solution of an equation by Trial and Error methodLet us consider a linear equation x + 10 = 15Here, Its LHS is x + 10 and RHS is 15Let us put some values of x till the LHS becomes equal to the RHS.When,x = 0, LHS = 0 + 10 = 10x =1, LHS = 1 + 10 = 11x = – 3 LHS = – 3 + 10 = 7x = – 10, LHS = – 10 + 10= 0x = 5 LHS = 5 + 10 = 15.We observe that the LHS of x + 10 = 15 become equal to the RHS, only when 5 is substituted for x. For all other values of x the LHS is not equal to the RHS. x + 4 10Acme Mathematics 6 217
Therefore, x = 5 is the solution of the equation. x + 10 = 15Solution is also called the root of the equation.Oh! This method is called Trial and Error Method.CLASS WORK1. Guess the value of x:(a) x + 2 = 8, x = 6 (b) x + 3 = 11, x = ..............(c) x + 5 = 6, x = .............. (d) x + 6 = 6, x = ..............(e) x + 1 = 7, x = .............. (f) x + 10 = 12, x = ..............(g) x + 8 = 10, x = .............. (h) x + 2 = 4, x = ..............2. Find the value of y:(a) y – 2 = 5 (b) y – 3 = 27 – 2 = 5 – 3 = 2y = 7 y = ...........(c) y – 10 = 2 (d) y – 1 = 7– 10 = 2 – 1 = 7y = ........... y = ...........(e) y – 8 = 10 (f) y – 4 = 5– 8 = 10 – 4 = 5y = ........... y = ...........3. Fill in the box and blanks:(a) 2y = 10 → 2 × y = 10 → 2 × 5 = 10 , y = 5(b) 3y = 3 → 3 × y = 3 → 3 × = 3 , y = .........(c) 10y = 20 → 10 × y = 20 → 10 × = 20 , y = .........(d) 5y = 30 → 5 × y = 30 → 5 × = 30 , y = .........(e) 4y = 16 → 4 × y = 16 → 4 × = 16 , y = .........218 Acme Mathematics 6
Exercise 4.91. By the trial and error method guess the value of the variable so that LHS becomes equal to RHS.(a) x + 3 = 7 (b) y + 4 = 7 (c) x – 4 = 10(d) y – 10 = 4 (e) 2x = 10 (f) 3z = 15(g) x3 = 2 (h) y 10 = 120 (i) 2x = 7 – x2. Choose the value of x from the set of numbers {1, 2, 3, ..........., 10} and fill in the blanks.(a) In x + 2 = 7, the value of x is ....…………….(b) In x – 1 = 3, the value of x is ....…………….(c) In 2x = 10, the value of x is ....…………….(d) In x3 = 3, the value of x is ....…………….(e) In 4x5 = 4, the value of x is ....…………….3. By trial and error method, guess the number (value of x) in each of the following.(a) What number should be added to 2 to get 7 ?(b) What number should be added to 7 to get 20 ?(c) What number should be subtracted from 20 to get 12 ?(d) What number should be subtracted from 10 to get 0 ? (e) What number should be multiplied by 3 to get 21 ?(f) What number should be multiplied by 4 to get 20 ? (g) What number should be divided by 3 to get 2 ?(h) What number should be divided by 5 to get 3 ?4. For what value of x the following statements are true ?(a) x + 7 = 10 (b) x + 3 = 5 (c) x – 4 = 8(d) x – 10 = 0 (e) 2x = 6 (f) 4x = 12(g) x3 = 1 (h) x10 = 2 (i) 30x = 10Acme Mathematics 6 219
D. Solving an equationAn equation may be compared with a balance used for weighing. Its sides are two pans and equality (=) tells us that the two pans are in equilibrium.In the case of an equation the equality sign (=) will not be changed in the following rules.Rule 1 We can add the same number to both sides,e.g., if x – 3 = 16, thenx – 3 + 3 = 16 + 3 → 3 is adding on both sides.or, x = 19Rule 2 We can subtract the same number from the both sides,e.g., if x + 3 = 16, thenx + 3 – 3 = 16 – 3 → 3 is subtracting on both sidesor, x = 13Rule 3 We can multiply both sides of the equation by the same non-zero number,eg., if x5 = 2 thenor,x5 × 5 = 2 × 5 → multiplying both sides by 5or, x = 10Rule 4 We can divide both sides of the equation by the same non-zero number,e.g., If 5x = 20,then 5x ÷ 5 = 20 ÷ 5 → dividing both sides by 5or,5x5 = 205 or, x = 4Now, we shall solve some linear equations using the above four rules 1, 2, 3 and 4.Solved ExampleExample 1 : Solve: x – 7 = 2Solution: Here, x – 7 = 2or, x – 7 + 7 = 2 + 7 → using rule 1(adding both sides by 7) or, x = 9LHS RHSOh! Remember non-zero numberWe use ‘or’ while solving equation220 Acme Mathematics 6
Example 2 : Solve: x + 5 = 9 Solution: Here, x + 5 = 9or, x + 5 – 5 = 9 – 5 → using rule 2 (subtracting 5 from both sides)or, x = 4Example 3 : Solve: x 3 = 2Solution: Here, x3 = 2 or,x3 × 3 = 2 × 3 → using rule 3(multiplying both sides by 3)or, x = 6Example 4 : Solve: 7x = 14Solution: Here, 7x = 14or,7x7 = 147 → using rule 4(dividing both sides by 7)or, x = 2Example 5 : Solve and check: 2(x + 1) = 14Solution: Here, 2(x + 1) = 14or, 2x + 2 = 14or, 2x + 2 – 2 = 14 – 2 → using rule 2or, 2x = 12or,2x2 = 122 → using rule 4or, x = 6Check: Putting x = 6 in the equation 2(x + 1) = 14 2(6 + 1) = 14or, 2 × 7 = 14or, 14 = 14Here, LHS = RHSHence x = 6 is true.So, value of x is 6.Classwork1. Fill in the blanks:(a) 3 + ............. = 5 (b) ............. – 2 = 7(c) ............. × 2 = 10 (d) 4 × ............. = 20(e) 10 – ............. = 8 (f) 8 = 4I know, the equation of the type x + 10 = 7 is called the linear equation.Acme Mathematics 6 221
2. Fill in the blanks (Guess the value):(a) If x + 2 = 3 then, x = .....................(b) If x + 7 = 13 then, x = .....................(c) If a – 2 = 9 then, a = .....................(d) If a – 1 = 0 then, a = .....................(e) If 2 y = 4 then, y = .....................(f) If 6y = 24 then, y = .....................(g) If p3 = 1 then p = .....................(h) If p3 = 2 then, p = .....................Exercise 4.101. Use the rule 1 and write the value of x in the blank.(a) If x – 4 = 8, then the value of x is ....…………….(b) If x – 3 = 6, then the value of x is ....…………….(c) If x – 5 = 9, then the value of x is ....…………….2. Use the rule 2 and write the value of y in the blank.(a) If y + 6 = 10, then the value of y is ....…………….(b) If y + 4 = 8, then the value of y is ....…………….(c) If y + 2 = 50, then the value of y is ....…………….3. Use the rule 3 and write the value of z in the blank.(a) If z5 = 1, then the value of z is ....…………….(b) If z10 = 2, then value of z is ....…………….(c) If z3 = 3, then value of z is ....…………….4. Use the rule 4 and write the value of 'a' in the blank.(a) If 4a = 12, then value of 'a' is ....…………….(b) If 10a = 50, then value of 'a' is ....…………….(c) If 6a = 156, then the value of 'a' is ....…………….222 Acme Mathematics 6
5. Solve and check:(a) x + 2 = 10 (b) x + 1 = 6 (c) 4x + 1 = 17(d) 5x – 1 = 19 (e) 6x –1 = 35 (f) 7x – 2 = 40(g) 3x + 2 = 8 (h) 5x + 7 = 17 (i) 2x + 11 = 17(j) x5 = 1 (k) x3 = 25 (l) x10 = 110(m) 8x + 2 = 3x + 12 (n) 5x + 4 = 2x +136. Solve for x.(a) 6x – 1 = 2x + 3 (b) 5x – 10 = 2x + 20 (c) 9x – 9 = 3x + 9(d) 20x – 9 = 14x +3 (e) 11x + 2 = 9x + 4 (f) 8x + 3 = 6x + 13(g) 8x + 1 = 2x + 7 (h) 7x + 5 = 5x + 15 (i) x2 + 2 = 4(j) x4 + 3 = 5 (k) x3 + 4 = 7 (l) x2 – 5 = 3(m) x3 – 6 = 4 (n) x2 – 7 = 17. Solve for m(a) 12 + (2m + 3) = 21 (b) 15 – (3m – 4) = – 2(c) (3m – 5) + 2 = 18 (d) (4m + 3) – 9 = 14(e) 3m – 5 = m + (2 + m) (f) 7m + 10 = 2m + 5(g) 3(m + 6) – 8 = 2(m + 2) (h) 4(m – 2) + 1 = 3(m + 1)(i) 2 (m + 5) + 8 = 7m + 8 (j) 10 (m – 1) + 8 = 9m – 1(k) 4(m + 3) – 8 = 2(m + 2) (l) 5(2 – m) + 10 = 3(1 – 2m)(m) 4(m + 3) = 5 2 (m – 2) (n) (m + 4)2 = 25 (m – 1)(o) 4(m + 3)2 – 3 = 1 (p) 2 + (m + 4)4 = 4Acme Mathematics 6 223
E. Practical application of linear equationNow, we will consider some practical problems. These problems are stated in words. The words can be mathematically formulated as linear equations in one variable. The following steps can be followed to solve the word problems.Step 1: Read the word problems carefully and note down what is given and what is required.Step 2: Denote the unknown by x or any other letter. Step 3: Transform the problems to the mathematical statement. Step 4: Change to the equation.Step 5: Solve for x (or other variables).Solved ExampleExample 1 : The sum of x and 4 is 10. Find x.Solution: Here, two numbers are x and 4Now, the sum of x and 4 is (x + 4) According to question,x + 4 = 10or, x + 4 – 4 = 10 – 4or, x = 6The value of x is 6Example 2 : The sum of two numbers is 30. If one of them is 10, find the other number.Solution: Let, the other number be x, then x + 10 = 30or, x + 10 – 10 = 30 –10or, x = 20Therefore, the other number is 20.Example 3 : The length of a rectangle is 6 m more than its breadth. Find the length and breadth of the rectangle, if its perimeter is 120 m.Solution: Here, we need to find the length and breadth of the rectangle.Length is given in terms of breadth.Let, Breadth of the rectangle (b) = x m thenLength of the rectangle (l) = (x + 6) m ADBCx cm(x + 6) cm224 Acme Mathematics 6
Now, Perimeter of the rectangle = 120 m 2(l + b) = 120or, 2{ (x + 6) + x } = 120or, 2(x + 6 + x) = 120or, 2(2x + 6) = 120or, 4x + 12 = 120The breadth of the rectangle = 27 m and The length of the rectangle = 27 m + 6 m = 33 m.Example 4 : If 30% of the number is 300, find the number.Solution: Let, the number be x.According to question 30% of x = 30030100 × x = 30or, 30 × x = 300 × 100or, x = 300 × 10030or, x = 1000∴ The number is 1000Example 5 : If the sum of three consecutive natural numbers is 30. Find the numbers.Solution: Let, the first natural number = x, then;The second natural number = (x + 1) The third natural number = (x + 2) According to question,x + (x + 1) + (x + 2) = 30or, x + x + 1 + x + 2 = 30 or 3x + 3 = 30or, 3x = 30 – 3 or, x = 273or, x = 9Thus, the first natural number = 9 the second natural number = 9 + 1 = 10the third natural number = 9 + 2 = 11 Thus, three consecutive natural numbers are 9, 10 and 11.or, 4x = 120 – 12or, 4x = 108or, x = 1084 or, x = 27Acme Mathematics 6 225
Classwork1. Look at the balance and solve for x.(a)x x x 1 1 11 1 111 1(b)x xx x x 1 11 1 111 1 1(c) (d)(e) (f)(g) (h)2. Fill in the blanks:(a) If the sum of x and 2 is 10, the value of x is …………………(b) If the difference of x and 4 is 5, the value of x is …………………(c) If the product of ‘a’ and 6 is 24, the value of ‘a’ is …………………(d) If the quotient of x and 3 is 6, the value of x is …………………Exercise 4.111. Make equation and calculate the value of of x from the given figures.(a)5 cmx cmArea 40 cm2(b)4 cm(x + 10) cmArea 56 cm2(c)4 cm4x cmArea 16 cm2(d) (x – 2) cm(x + 2) cmPerimeter(5x – 18) cm4x 8 x + 9 8x 1 1 11 1 11111x xxx x 11 1 1 x1 1 1 xx x x1 11 xx x x 1 1 11 1 1 xx226 Acme Mathematics 6
2. Write the following statement in mathematical sentence.(a) The sum of x and 3 is 12. (b) The sum of 2x and 7 is 22.(c) The difference of x and 6 is 14. (d) The difference of x and 8 is 16. (e) The product of x and 5 is 15. (f) The product of x and 37 is 10. (g) The quotient of x divided by 9 is 9.(h) The quotient of m divided by 3 is 20.3. Solve the following:(a) The sum of two numbers is 30. If one number is 12, find the other number.(b) The sum of two numbers is 50. If one number is 27, find the other number.(c) The difference between two numbers is 2. If the smaller number is 10, find the greater number.(d) The difference between two numbers is 8. If smaller the number is 20, find the greater number.(e) The product of two numbers is 42. If one of the numbers is 7; find the other number.(f) The product of two numbers is 60. If one of the numbers is 20, find the other number.4. Solve the following problems:(a) Mukunda has Rs. x. If its 10% is Rs. 500, find the sum of money Mukunda has.(b) If 35% of y is 70. Find the value of y.(c) If 13 of students of a school is girls and there are 1500 students, find the number of girls.(d) The cost price of a radio is Rs. y. If 10% of cost price is Rs. 100, find the cost price. (e) The figure shows the information about the book. Find its selling price (SP) of the book.5. Solve the following:(a) The difference of two numbers is 10. If one of them is 7, find the other number.(b) Astha is 5 years younger than Monika. If Monika is 16 years old, find the age of Astha.Market price Rs. 250Discount 20%SP Rs. xAcme Mathematics 6 227
(c) The sum of ages of father and his son is 54 years. If the son is 8 years old, find the ages of the father.(d) In a class of 16 children , there are three times as many boys as there are girls. Find the numbers of girls.6. Solve the following problems:(a) The length of first stick is 2x meter and length of second stick is (3x + 2) meter. If the total length of both sticks is 27 meters, find length of each stick.(b) A triangle has its three sides 3x cm, (x + 3) cm and 4x cm. If its perimeter is 27 cm. Find the actual length of each sides.(c) Calculate the length of side AB of ∆ABC.Perimeter25 cm3x cm(x + 8)cm(2x – 1)cmB CA(d) Calculate the length of side DC of the rectangle ABCD.(2x + 3)cm A BD C(x + 1)cmPerimeter56 cm7. From the table, choose any one item from each column to form an equation and solve it.(2x + 1)(x –8)(x + 7)x4+– ÷×72410=1859228 Acme Mathematics 6
F. TrichotomyTrichotomy symbolsStudy the following noticesWANTED !Security Guard Age more than 35 years Ex-army or Expolice will preferredWANTED !Teacher Bsc. or morecan apply. At least 2 years experienced.WEIGHT LIMIT OF THE BRIDGETON ONLY10These notices can be written as,(a) Speed = 20 km(b) Capacity of bridge < 10 tone(c) Age of security guard > 35 years(d) Qualification > B.scThese representations contain the following symbols:= is the symbol for 'is equal to'< is the symbol for 'is less than'> is the symbol for 'is more than'≥ is the symbol for 'is more than or equal to' < is the symbol for 'is less than or equal to'These symbols; '=' (equal to), '<' (less than), and '>' (more than) are called trichotomy symbols.We can compare the quantities using these symbols.For examples9 > 3 or, 3 < 920 = 20 or, x ≤ 7(– 3) > (– 4) or, (– 4) < (– 3)If a and b are any two whole numbers then there are following possible relations:(i) a > b (ii) a < b (iii) a = b Such properties are called trichotomy property.Acme Mathematics 6 229
Negation of Trichotomy symbols7 is greater than 5, Using the sign of trichotomy, we write it as 7 > 5.The negation of 7 > 5 is written as 7 5 (It is read as 7 is not greater than 5).The symbol is the negation of the symbol >. The symbol is the negation of the symbol <. The symbol ≠ is the negation of the symbol =.Trichotomy RulesStudy the following rules of trichotomy.Let us consider any two numbers 12 and 15 then we have the relation15 > 12or, 12 < 15Rule 1: Rule for additionWe have 15 > 12 or, 12 < 15Adding 3 on both sides,15 + 3 > 12 + 3 or, 12 + 3 < 15 + 318 > 15 or, 15 < 18Both are true for addition.Thus when equal number is added to both sides of trichotomy symbol, the symbol remains the same.Generalized rule for addition:If a > b and c is any integer, then a + c > b + c. If a < b and c is any integer, then a + c < b + c.Rule 2: Rule for subtractionWe have, 15 > 12 or, 12 < 15 Subtracting 3 on both sides,15 – 3 > 12 – 3 or, 12 – 3 < 15 – 3∴ 12 > 9 or, 9 < 12Both are true for subtraction.Thus, when equal number is subtracted from both sides of trichotomy symbol, the symbol remains the same.230 Acme Mathematics 6
Generalized rule for subtraction:If a > b and c is any integer, then a – c > b – c.If a < b and c is any integer, then a – c < b – c.Rule 3: Rule for multiplicationWe have 15 > 12 or, 12 < 15Multiplying both sides by 3 then;15 × 3 > 12 × 3 or, 12 × 3 < 15 × 345 > 36 or, 36 < 45Both are true for multiplication by positive number.Again, we have15 > 12 or, 12 < 15But if we multiplied by (– 3) then,15 × (– 3) > 12 × (– 3) or, 12 × (– 3) < 15 × (– 3)– 45 > – 36 or, – 36 < – 45These are not true.To be true there must be: – 45 < – 36 or, – 36 > – 45Thus, (i) When both sides of trichotomy symbols are multiplied by equal positive numberthen the symbol remains the same.Generalized rule of multiplication: If a > b and c is positive number, then ac > bc. If a < b and c is positive number, then ac < bc.(ii) When both sides of trichotomy symbols are multiplied by equal negative numberthen symbol changes as follows:'<' changes to '>' and '>' changes to '<'If a > b and c is negative number, then ac < bc.If a < b and c is negative number, then ac > bc.Acme Mathematics 6 231
Rile 4: Rule for DivisionWe have 15 > 12 or, 12 < 15Dividing both sides by 3 then:15 ÷ 3 > 12 ÷ 3 or, 12 ÷ 3 > 15 ÷ 35 > 4 or, 4 < 5Both are true for division by positive numbers.Again, we have 15 > 12 or, 12 < 15But if we divide by ( – 3) then;15 ÷ (– 3) > 12 ÷ (– 3) or, 12 ÷ (– 3) < 15 ÷ ( – 3)– 5 > – 4 or, – 4 < – 5Here, – 5 > – 4 and – 4 < – 5 are not true.To be true there must be:– 5 < – 4 and – 4 > – 5Thus,(i) When both sides of trichotomy symbols are divided by equal positive number then the symbol remains the same.Generalised rule for division:If a > b and c is positive number, then ac > bc. If a < b and c is positive number, then ac < bc.(ii) When both sides of trichotomy symbols are divided by equal negative numberthen symbol changes as follows:'<' changes to '>' and '>' changes to '<' If a > b and c is negative number, then ac < bc. If a < b and c is negative number, then ac > bc.232 Acme Mathematics 6
Classwork1. Fill in the blanks 'True' or 'False'.(a) 3 × 3 > 10 ............ (b) 13 > 5 × 2 ............(c) (4 – 2) > 1 × 3 ...….… (d) (6 – 0) < 1 ............(e) 6 < 1 × 0 ............ (f) 11 > 4 × 3 ...……(g) (1 × 2) < (8 × 4) ............ (h) 4 × 5 = 5 × 4 ............(i) 30 ÷ 5 > 10 ...…… (j) 2 < 9 ÷ 3 ............(k) (8 + 8) < 13 ............ (l) 13 > (7 + 4) ...……(m) 1 < 2 ............ (n) 8 < 10 ............(o) 0 > (– 4) ...……Exercise 4.121. Insert the appropriate trichotomy symbol.(a) 2 + 1 .............. 3 + 1 (b) 6 + 2 .............. 4 + 2(c) 1 + 7 .............. 1 + 9 (d) 6 – 2 .............. 4 – 2(e) 1 – 2 .............. 1 – 9 (f) 9 – 8 .............. 8 – 9(g) 1 × 3 .............. 2 × 3 (h) 1 × 9 .............. 1 × 4(i) 0 × 5 .............. 0 × 6 (j) 2 × (– 8) .............. 1 × 7(k) – 11 × 2 .............. – 13 × 2 (l) – 18 × (– 3) .............. – 5 × (– 3)(m) 15 ÷ 3 .............. 6 ÷ 3 (n) 15 ÷ (– 3) .............. 6 ÷ (– 3)(o) – 18 ÷ (– 3).............. 12 ÷ (– 3)2. Write the following statements using their negations.(a) 5 > 7 (b) 6 = 6 (c) – 2 < 6(d) x = 6 (e) x + 3 > 4 (f) y –10 < 20 (g) x – 10 = 20 (h) x < – 3 (i) x = – 10(j) – 3 < – 2 (k) 16 > 112 (l) 42 = 843. Rewrite the following statements using trichotomy symbols.(a) 9 is greater than 1 (b) 8 is less than 11(c) 10 is not greater than 11 (d) 2 is not less than 0(e) x is greater than 7 (f) x is not greater than 6 (g) x is less than – 3 (h) x is not less than –3(i) x is equal to 21 (j) x is not equal to 21Acme Mathematics 6 233
G. Inequalities and its graphIntroductionLook at the following notices:WANTED !Security Guard Age more than 35 yearsWANTED !Teacher Age 25 years or lessSpeed Limit40 km/hr(i) In the first notice the age of guard may be 35 years or 36 years or 37 years or so on.(ii) In the second notice the speed may be 39 km per hour, 38 km per hour, 37 km per hours or so on.(iii) In the third notice the age of teacher may be 25 years or 24 years or 23 years or so on.The above information can be written using mathematical symbols as: If the age of guard is x years then, x > 35If the speed is x km per hour then, x ≤ 40If the age of teacher is x years then, x < 25= is the symbol for 'is equal to'≥ is the symbol for 'is more than or equal to'≤ is the symbol for 'is less than or equal to'The expressions including the symbols < or > are called the inequality. Inequality can be shown on a number line.Now study the following number lines carefully.Let, x = 35, x > 35, x < 35, x > 35 or x < 3529 31 33 35x = 3530 32 34 36 37 38 39 40Here the value of x is only 35.Indicates the number 35 is included29 31 33 35x > 3530 32 34 36 37 38 39 40Here the value of x is more than 35.Or, x = {36, 37,38………………….}Indicates the number 35 is not included29 31 33 35x < 3530 32 34 36 37 38 39 40Here the value of x is less than 35.Or, x = {34, 33, 32………………….}Indicates the number 35 is not included234 Acme Mathematics 6
29 31 33 35x > 3530 32 34 36 37 38 39 40Here the value of x is 35 and more.Or, x = {35, 36, 37, 38…………….}Indicates the number 35 is also included29 31 33 35x < 3530 32 34 36 37 38 39 40Here the value of x is 35 or less.Or, x = {35, 34, 33, 32…………….}Indicates the number 35 is also includedExercise 4.131. Study the following number lines. Choose the variable as 'x' and write down the inequality.(a)0 1 2 3 4 5 6(b)–2 –1 0 1 2 3 4(c)0 1 2 3 4 5 6(d)0 1 2 3 4 5 6(e)–3 –2 –1 0 1 2 3(f)0 1 2 3 4 5 6(g)–3 –2 –1 0 1 2 3 4(h)0 1 2 3 4 5 6 7 82. Show the given inequalities in a number line.(a) x > 0 (b) x < – 4 (c) x > 6 (d) x < 4(e) x < – 2 (f) x < 9 (g) x ≥ 12 (h) x ≥ – 10(i) x ≥ 7 (j) x ≤ 11 (k) x ≤ – 12 (l) x ≤ 133. Express the following statements as inequalities.(a) x is greater than 7 (b) x is less than 0(c) x is greater than or equal to 10 (d) x is less than or equal to 10(e) 2x + 1 is greater than 0 (f) 2x – 7 is less than – 8(g) x is greater than 1 and less than 6(h) x is less than 10 and greater than 5(i) 2x is less than or equal to 10 and greater than or equal to – 8Acme Mathematics 6 235
1. Objective : To virtualize of algebraic terms2. Materials required: Card board paper, Colours, Scissors, Gum, Scale, etc.3. Making Expressions:+ + +x x1 1 2x + 2x + + – xx x – x + x = xx2xxx3xxx3. Activity: Make different strips and represent the following expressions: 4x + 5, x – 3, 4x2, 2x3.If white strip indicates 'x' then black indicates the negative of 'x'.It is negative of xIt is negative x x – x of 11 11 – 1Project Work236 Acme Mathematics 6
1. (a) Express algebraically: 2x + 7 is less than 10(b) Solve: x + 2 = 8(c) Read the number line and write the inequality.–3 –2 –1 0 1 2 32. (a) For what value of x, x + 6 = 10 is true?(b) Solve: 2x = 6(c) Draw a number line to represent x > 4.3. (a) Solve: 6x + 5 = 35(b) What number should be added to 2 to get 12?(c) Solve and check: 4x + 12 = 2x + 24. (a) Draw a number line: x ≤ 10(b) Solve: 3x – 53 – x = 35(c) What should be subtracted from 20 to get 8?5. (a) Rajan has Rs. 18 less than Samin. If both of them have altogether Rs. 52. How many rupees each of them have? Find it.(b) Rewrite by using the symbols '>', '=' or '<'.The value of 'a' is greater than 8. ...........................(c) The sum of two number is 30. If one is 18, find the other number.6. (a) Use the symbol >, < or = .When 5 is subtracted from y, it is greater than 10.(b) Solve by making equation: The sum of x and 7 is 15.(c) Nandani has Rs. 12 more than Samjhana. If both of them have altogether Rs. 96. How many rupees each of them have? Find it.7. The present age of Anamol is x years.Mixed ExerciseAcme Mathematics 6 237
(a) How old was he 2 years ago?(b) How old will he be after 2 years?(c) If his father is four times older than him, how old is his father?8. (a) Illustrate the inequality on number line.3 > x > – 1, x ∈ z.(b) Using the trichotomy rule, state wether the given statement is 'True' or 'False'.62 > 42(c) Convert the following statement into equation. A number 'x' is 5 more than 10.9. (a) State whether the given mathematical statement is 'Open' or 'Closed' : x + 5 < 9(b) State whether the given mathematical statement is 'True' or False'.The number 3 is a solution of the equation : y + 2 = 5.(c) Convert the following equation to statement.5y + 7 = 22.10. (a) For what value of 'x' the given open mathematical statement becomes true? The product of 'x' and 8 is 32.(b) Make equation and find the value of x.6 m 3x m24 m(c) Write 'True' or False'. 8 < – 1211. State 'True' or 'False'.The given equation 34 m = 9 can be written as the following statements.(a) Three-fourth of a number 'm' is 9.(b) A number 'm' multiplied by 3 and divided by 4 gives 9.(c) 34 multiplied by a number 'm' is 9.(d) None of all.238 Acme Mathematics 6
EcaluationTime: 62 minutes Full Marks: 261. (a) Complete the given table. [1]x 1 3 5x+5 7 9 11(b) Put appropriate symbol > or < . [1](i) 8 ....... 9 (ii) –5.... –7 [1] 2. (a) Find the value of x when 2x + 1 = 6 [2](b) Solve: 3x + 4 = 10 [2] (c) Solve the equation : x + 12 = 32 [2] 3. (a) Bikash has Rs. 17 less than Bidur. If both of them have altogether Rs. 23. How many rupees each of them have? Find it. [2](b) Write the ststement ''x is less than 7\" with an approprite sign. [1]4. (a) 5 is added to four times of a number, the sum is 33.what is the number ? [2](b) Write \"True\" or :False\". 1 < –7. [1]5. (a) Nandani has 12 times more money than Samjhana. If both of them have altogether Rs. 91. How many rupees each of them have ? Find it. [2] (b) Write the statement \"x–8 is smaller than or equal to 20\" with an approprite sign of tricotomy. [1]6. The age of Kushal is 20 years less than age of Manju. The sum of their ages is 32 years. Find their ages. [2] 7. 4 is added to 5 times a number,the sum is 34. Find the number. [2] 8. Make the equation from the condition given below and find the value of x. [2]+[2](a) (b) 3x ft 2x ft x ft78ft3x 2x 555Acme Mathematics 6 239
5UNIT GeometryWarm Up Test1. Measure the following angles.(a) (b) (c)2. Construct the following angles.(a) 40° (b) 100° (c) 200°3. Classify the angle according to their size.(a) (b) (c)4. Find the value of 'x', in the following angles, when ∠AOB = 90°.(a) (b)5. Classify the following triangles according to their sides.(a) (b) (c).................................. .................................. .................................B CAQ RPY ZXQ RP110°Y ZX90°Y ZX50°O BCxA OC BxA3 cm3 cmB CA3 cm4 cm4 cmQ 5 cm RPYX Z5 cm 3 cm4 cm240 Acme Mathematics 6
6. Name the sides, vertices and angles for the following polygons.(a) (b)7. Measure the angles of the given triangles and classify the triangles.(a) (b) (c)8. Fill in the blanks.(a) A ray has ............... end point. (b) A line has ………………………….(c) A line segment has …………………….. length.(d) 2° is ………………….. angle.(e) 90° is ………………… angle.9. Solve the following.(a) If x, y and z are the angles of a triangle and x = 45° find the value of (y + z).(b) If a, b and c are the angles of triangle, a = 45° and b = 60°, find the value of c.(c) If x°, 2x° and 3x° are the angles of a triangle find the size of all angles.10. Solve the following.(a) x, y, a and b are four angles of a quadrilateral. If a = 50° and b = 150°, find(x + y).(b) a, b, c and d are the angles of a quadrilateral if a = 100°, b = 70°, c = 125°, find thesize of angle d.(c) (x + 10)°, (x + 20)°, (x + 100)° and (x + 26)° are the angles of a quadrilateral. Findthe size of each angle.B CD ABCDE AC DBN OMX YAIt is ray.Acme Mathematics 6 241
5.1 Line and AnglesA. Intersecting and Parallel LinesLook at the given figures carefully.The figure above shows,(a) a road (b) a door (c) a scale (d) a tableSee how both the edges of the alongside figure run. They will never meet to each other. Distance between them is always equal. The edges of straight road, door, scale and table are parallel.Two lines that do not intersect (meet) each other at all are called parallel lines. AB and CD are parallel lines. The symbol // is used to denote the parallel lines.Now, look at the following figures.(a) (b) (c)You can see several lines are intersecting each other. These lines are called intersecting lines.Two lines that intersect (meet) each other at a point are called intersecting lines. ACBDCAOBDCG EDF RQBACUST P242 Acme Mathematics 6The sign || indicates the lines are parallel.
B. Perpendicular lines Two intersecting lines are said to be perpendicular lines if they intersect at right angle (90°). In the figures all intersecting lines are at right angles. Hence they are perpendicular to each other. The symbol '⊥' is used to denote the perpendicular.PMN Q BO CAD AB⊥CD is read as AB is perpendicular to CD.S QP R O In the above figures AB ⊥ CD, PQ ⊥ RS and MN ⊥ PQC. Construction of perpendicular bisector of a line segment using compass Draw a line segment of length 6 cm. AB is 6 cm. At A, put a compass (center) and draw an arc taking radius more than half of the length AB. (fig. i) Repeat the same process from B also. The arcs cut each other at X and Y. (fig. ii) Join X and Y. (fig. iii)A O BYXA 3 cm 3 cmYXA 6 cm BX6 cm BThis line XY divides the line AB into two equal parts as AO = OB = 3 cm.XY is called the bisector of the line AB. ∠XOB is 90°. So, XY ⊥ AB. Hence, XY is called perpendicular bisector of the line AB.A 6 cm B(i)(ii) (iii)Acme Mathematics 6 243
D. Construction of parallel and perpendicular lines (a) Construction of parallel lines using set-square. In the figure alongside AB is the given line and P the given point. Place the one edge of your sets quare along AB as shown. Slide one of the set-square along the point P until the edge touches the point P. Draw a line along the edge of the set-square in this position. This line CD is parallel to line AB.1 2 3 4 5 6 7 8 9 1010 9 8 7 6 5 4 3 2 1AP BAP B1 2 3 4 5 6 7 8 9 1010 9 8 7 6 5 4 3 2 1AP BACPBD244 Acme Mathematics 6
(b) Construction of perpendicular line using set-squares Let, AB is a given line and P the given point. Place the scale along AB. Fix the right angle of the setsquare along the edge of the scale and draw a line PQ. Line PQ is perpendicular to AB. (PQ ⊥ AB). (c) Construction of perpendicular line using compasses Let, AB is a given line and P the given point. Put compass at P and draw an arc below such that arc cut the line at C and D. Repeat the process at C and D same as P. Join P and R such that it cut AB at Q.Now, PQ is perpendicular to AB. (PQ ⊥ AB). APBAPQ B1 2 3 4 5 6 7 8 9 10APQ BA BPA C D BRPQA C D BPAcme Mathematics 6 245
Classwork1. Name the following angles in two ways:(a) MO N(b)A CB (c)R QP(d)OBA(e)M NO (f) EO D2. Measure the following angles and fill in the table given below:(a)P QR (b)C DB (c)C DE(d)NMO(e)S TU (f)B CA(g)QPR(h)F G H(i) AB C246 Acme Mathematics 6
No. Name of the angle Vertex Measurement(a)(b)(c)(d)(e)(f)(g)(h)(i)Exercise 5.11. Name the lines whether they are parallel, perpendicular or intersecting.(a) (b) (c)(d) (e) (f)2. List the parallel lines.(a) (b) (c)D CA BQ RS TPQ DB EAcme Mathematics 6 247
3. Fill in the blanks.(a) AF is parallel to ...…………….(b) AB is parallel to ........................ (c) DE is parallel to …………........(d) EF is parallel to……………….(e) BC is perpendicular to ………. (f) ED is perpendicular to ……….. (g) AF is perpendicular to ………..(h) CD is perpendicular to ……….4. What kind of lines are presented in each of the following figures ?(a) (b)(c) (d)BA FCE D248 Acme Mathematics 6
5. List the parallel and perpendicular lines from the given figures. (a) (b)6. Draw the line through the given points and parallel to given lines (use scale and set-square).(a) (b) (c)(d) (e) (f)7. Draw the perpendicular line through the given points to given lines (use scale and set-square)(a) (b) (c)(d) (e) (f)8. Draw the following line segments. Mark a point 'R' above the line segment and draw perpendicular from the point 'R' to the line using compass and scale.(a) AB = 4 cm (b) BC= 5 cm (c) CD = 6 cm(d) DE = 6.5 cm (e) XY = 7 cmAB CDACDBE FH GAP QBX YCS TPA BQC DRE FXA BYC DZE FWG H P BAR B BAcme Mathematics 6 249
9. Draw the following line segments. Mark a point 'R' above the line segment and draw parallel line through 'R' to the given line using protractor and scale.(a) AB = 5 cm (b) XY= 6 cm (c) PQ = 5.5 cm(d) BE = 4 cm (e) MN = 4.5 cm10. Study the given figure carefully and answer the following questions.(a) Measure BO and OD. Is BO = OD ?(b) Measure ∠AOD and ∠AOB. Is AC⊥BD(c) Is AC, the perpendicular bisector of BD ?11. Construct the following line segments and draw their perpendicular bisector.(a) AB = 4 cm (b) BC = 6 cm (c) XY = 5 cm(d) PQ = 7 cm (e) DE = 8 cm (f) AD = 8.6 cmE. Type of AnglesThere are many types of angle. Types of angle are based on their size or pairs.(a) Acute angleAn angle whose measure is more than 0° and less than 90° is called an acute angle. In the figure ∠AOB is a acute angle as it is 48°.(b) Right angleAn angle whose measure is exactly 90° is called right angle. In the figure ∠AOB is a right angle as it is 90°.(c) Obtuse angle An angle whose measure is more than 90°but less than 180° is called obtuse angle.In the figure ∠AOB is a obtuse angle as it is 105°.AB DCOO48°B A 90°O B A A 105°O B 250 Acme Mathematics 6