MATHEMATICS AND STATISTICS 1.2


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1 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr Simplifying lgebric epressions In lgebr, vrible is letter such s, used to stnd for number. Algebric terms nd epressions re formed by pplying opertions (such s +,,, ) to vribles.. The number which is double y is y which is written y. The number multiplying y,, is clled the coefficient of y, nd y is clled the product of nd y.. The number which is 5 more thn hlf is written + 5 or + 5. An lgebric epression is mde up of terms dded (or subtrcted) together. Like terms (terms with the sme vribles) cn be simplified by ddition or subtrction.. 7y + y = 0y. 8 = 5. b b = 6 b Eercise A: Algebric epressions. Write down s lgebric epressions: The squre of less thn The squre root of more thn y p is multiple of 6. Wht is the net multiple of 6 fter p? A dog costs $d per month to fee A ct costs $7 less per month to fee Wht is the cost of feeding dog nd ct for month? Simplify your nswer. Mum is yers older thn her son, Mrk. Mrk is 4 yers younger thn his sister, Disy. Dd is two yers older thn Mum. If Mum is m yers old wht is the sum of ll four ges? Simplify your nswer. i. Hn buys kg of pples t $ per kilogrm nd kg of pers t $p per kilogrm. i. How much does this cost ltogether? Ans. p. 4 The product of 7 nd The sum of y nd 4 ii. How much chnge does Hn get if she pys $d for her fruit? The number which is more thn five times the number w iii. Wht restrictions re there on the vlue of d?
2 Achievement Stndrd 907 (Mthemtics nd Sttistics.). Simplify where possible the followin b b + b s t s + t Multiplying nd dividing terms Any lgebric terms cn be multiplied together. Indices re used to simplify repeted multipliction, y y is written y. 4b = b [ 4 =, b = b]. y.6y = yy [dot mens ] = y [using indices] p + 8q + 9p 4q 7f f 0f f 6b + 4b + Division of lgebric terms is best written in frction form. Simplify using =. Note tht if ll fctors in the numertor cncel, then fctor of will remin in the numertor. For emple, 9b = b =. 0y 5 = y [dividing top nd bottom by 5] [simplifying nd cncelling ] 4w 5 w i j. bc bc cb + cb. Find the missing term in ech of the following simplifictions. 5b 4 + = 5b If powers of vribles occur, these cn be written s repeted multiplictions before cncelling s befor Q. Simplify y 4 y A. Using repeted multiplictions: y 4 y = y y y y = y y y y = y [cncelling (twice) nd y y ] Ans. p. 4 5pq qp + r = pq r m + n m + mn = n mn Eercise B: Multiplying nd dividing terms. Write ech of the following in simplest form. 7 b.y + = 5 w 6 b 7d + = d d. b 5c y y y 5.
3 Apply lgebric procedures in solving problems. Simplify ech of the following divisions. 0 6y 5 9y 0 0 8b 9pqr rst pq 4q y 4y b Order of opertions The correct order of opertions must be followed when simplifying epressions. The mnemonics BEMA or BEDMAS (brckets, eponents, multipliction nd division, then ddition nd subtrction) re useful reminder = + 5 = 8 [ before +]. y + y 0 (y + y) = 0 = 5y 0 = y [division line mens brckets] [divide top nd bottom by 5]. Simplify the following (write the powers s repeted multiplictions). b b 4 Eercise C: Order of opertions. Simplify using the correct order of opertions. 4 ( + ) Ans. p. 4 4y 5 8 y y + y 4 ( + 5) (4 ) 9w z 6w 4 z m np 4 4m p. Insert brckets to mke correct sttements. + = 6 4. Insert the missing numertors or denomintors. = 5 b = b = = y + y y y = y p + 4p p p = w w w = w 6 = = p q 4 = p q p + q q q = 0
4 4 Achievement Stndrd 907 (Mthemtics nd Sttistics.) Eponents A whole number eponent (or inde or power) is used to show repeted multipliction of the sme bse number or vribl n =... n fctors For emple, is written 5 which equls. y 6 mens y y y y y y [si fctors of y multiplied] 6 6 is written ( 6) which is 6 Note tht 6 = 6 = 6 0 =, for ny nonzero number. 4 = 4 which is written Simplify the following products y y 4 b.b. 5y y 5 = 5yy yyyyy = 5y 7 Note: =, for ny number For indices with the sme bse: multiply by dding the powers: b = + b divide by subtrcting the powers: 9 4 b b. Simplify the following divisions. p 6 p 8 6 b = b If b >, use b = b = 5 Numbers re multiplied or divided in the usul wy. 4b 9 b =..5 4 = bc b c = 0 b 4 c p p 7 = p 5 6. p 9 p 5 = p 4 4y 6 y = y5 8 4 y 6 4y 8 7p 4p q Ans. p. 4 Eercise D: Multiplying nd dividing with indices. Evlut ( ) ( ) Find the missing terms. b = b y = y
5 Apply lgebric procedures in solving problems 5 Powers nd roots of indices To rise n eponentil term to new power, multiply the powers: ( ) b = b All fctors inside the brcket hve their powers multiplied by the new power: y b d = d y bd. ( 0 ) = 0 w c w cd. Epress in simplest form. y y 4 p 6 t 0. ( y 4 ) 5 = 5 5 y 4 5 [ mens ] = 5 y 0. 4 = = 8 8 To find the squre root of n inde epression, hlve the power: n n = Tke the squre roots of numbers in the usul wy =. 6 6 = 6 = 8 = 6 8 Eercise E: Indices with powers nd roots. Epress in simplest form. ( 5 ) ( 5 ) 6. Use ll your inde rules to simplify the followin ( ) 4 (p 5 ) 6p 4 5 b 4 (b) Ans. p. 4 ( ) 4 (5y ) 0 () 4 ( w 4 ) b () (5 + ) 4 y y ( b ) 4 6 9b 4
6 6 Achievement Stndrd 907 (Mthemtics nd Sttistics.) Bsic lgebric frctions These re frctions with vribles, such s or 54 y Algebric frctions obey the sme rules s numericl frctions. Simplify frctions by cncelling common b 0bc 4b 49 b y 6y b b bc fctors (using = ) nd the lws of indices. For emple: 5y7 y simplifies to 5y 4 Add (or subtrct) lgebric frctions with the sme denomintor, by dding (or subtrcting) numertors nd putting the result over the denomintor. For emple: = 7 If the denomintors re different, use equivlent frctions to epress ech frction with the sme denomintor first. For emple: + = = = = 0 Multiply lgebric frctions, by multiplying numertors together to get the numertor of the nswer, nd multiplying denomintors together to get the denomintor of the nswer. For emple: 4 = Simplify nswers where possibl For emple: b 5b 4 = 5b 8 b = 5b 8 To divide by frction, multiply by the. Write s single frction nd simplify if possibl y y 4 8 y 7 y 7 reciprocl (the reciprocl of the frction b is the frction b ) = 4 6 = 4 = 8 Ans. p. 44 Eercise F: Bsic lgebric frctions 4. Simplify ech of the following frctions. 5 0y 6
7 Apply lgebric procedures in solving problems 7. Multiply the lgebric frctions nd simplify your nswer where possibl 4 5 b b 0 b cd b d y 4 5y 6y 0 5 4y 5y b 4b b 5b 8 4 4y 95 y 4 y 9y 7b c c 4 5. Use the correct order of opertions to simplify the following epressions. 5y 0y 4 5p 4q q 5p + 4 y y y 4. Divide the frctions nd simplify your nswer where possibl b p p p b c d c p 8 p + p 4
8 8 Achievement Stndrd 907 (Mthemtics nd Sttistics.) Epnding brckets Brckets re epnded using the distributive lw: (b + c) = b + c. 5( + 7) = = ( ) =. b ( + 4b) = b b 4b = 6 b b. Epnd nd simplify the following epressions ( ) 8 ( + 4) ( ) There my be more thn two terms in the brcket. 6( + b 5) = 6 6 b 6 5 = 6b + 0 Some epressions involve epnding more thn one set of brckets. Tke cre with negtive signs.. 5( + ) 6( ) = = ( 7) ( + ) = 7 6 = (4 + ) 6 5 6( + ). Epnd nd simplify. ( + 6) + ( 4) Ans. p. 44 Eercise G: Epnding brckets. Epnd the brckets. ( + 8) ( + ) 5( + ) ( + 7) 5(y ) 4(y ) ( + 5) ( + 4) (4 ) 5( ) 4( + ) 6( + 4) 7( + 5) (8 ) y( y + y) ( ) + ( 4)
9 ANSWERS Eercise A: Algebric epressions (pge ). 7 y + 4 5w + ( ) y + p + 6 d 7 4m 58 i. i. + p ii. d p iii. d + p. b s p 6q f f 0b + w 5 i. 4 j. 0. r mn; m ; 8d; d Eercise B: Multiplying nd dividing terms (pge ). 4b 6y 5w b 6 5bc 6y 5. p pq 4st. b mnp 4b b y wz b 4 pq 5 Eercise C: Order of opertions (pge ). 6y 8 4. ( + ) = 6 8 (4 ) + = ( ) = (y + y) y y = y (p + 4p) (p p) = w (w w) = w (6 4) + = (p + q) (q q) = 0 Eercise D: Multiplying nd dividing with indices (pge 4) y 6 b c 54 5 b. p 5 9 b pq 4. 8b 5 y 4y Eercise E: Indices with powers nd roots (pge 5) w 4 b y 6 8y 6. y y 5 5 p t 5
10 44 Answers ANSWERS. 5 p b Eercise F: Bsic lgebric frctions. y (pge 6) 6b b 7 8bc y 7 y y 8 y b 5c 7 4 b 6y q 4 0 c 8 5b 5p 8 y 4y Eercise G: Epnding brckets (pge 8) y 5 y + 8 bc bc d 4 y y + y y Eercise H: Epnding pirs of brckets (pge 9) y 6y y + 0y + 5 9y p + 6p Eercise I: Fctorising using the distributive lw (pge 0). 7( + ) ( + b) y( y) 5( + ) p(p 4) b(c b) ( b + c) (4 + y). ( ) y ( + y ) y 4 ( + y) 4b (b ) p (p p + ) (4 ) 8 ( ) 8 ( + 8 ) Eercise J: Fctorising qudrtics (pge ). ( + 5)( + ) ( + 6)( + 7) ( 8)( + 6) ( + 8)( 5) ( + 4)( 4) (5y + )(5y ) ( 8)( ) ( 7)( + ). ( + 5)( 4) 5( + )( ) ( + 6)( ) 5( 4)( + ) ( 7)( 4) 0( + )( + ) 8( + )( ) ( 8)( 4). ( + )( + ) ( + )( + ) (5 + )( ) (7 )( + ) (4 + )( + ) ( )( + ) ( )( ) ( + 5)( ) 4. ( + 4)( 4) 4( + ) ( 4)( + ) ( 5) cnnot be fctorised
11 INDEX dd (frctions) 6 lgebric epression lgebric frctions BEDMAS BEMA chnging subject of formul coefficient difference of two squres 9 distributive lw 8 divide (frctions) 6 elimintion method eponent 4 eponentil equtions 4 fctorised 0 formul 0 inde 4 indices like terms liner equtions 4 liner inequtions 7 multiply (frctions) 6 opertions perfect squres 9 power (inde), 4 product qudrtic equtions 8 qudrtic epressions reciprocl 6 repeted multipliction simplified frction 6 simultneous equtions subject of the formul substituting 0 substitution method 6 terms (lgebric) vrible
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