Preliminary preparation

Transcription

1 Preliminry preprtion Syllus prerequisite: Preliminry preprtion This topi provies the si knowlege, skills n unerstnings require in Yer. Outomes Simplify n lgeri expression. Estlish n pply inex lws in lgeri form. Multiply n ivie lgeri terms. Fin unknown sies using Pythgors theorem. Clulte the perimeter of simple figures. Clulte the re of omposite shpes. Clulte simple interest. Clulte the future vlue, ompoun interest n present vlue. Crete frequeny tles to orgnise t. Clulte the mein n men. Digitl Resoures for this hpter In the Intertive Textook: Vieos Litery worksheet Quik Quiz Wigets Spresheets Stuy guie Solutions (enle y teher) In the Online Tehing Suite: Tehing Progrm Tests Tehing Notes Review Quiz Knowlege hek In the Intertive Textook you n tke test of prior knowlege require for this hpter, n epening on your sore you my e irete to revision from the previous yers work. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

2 Chpter Preliminry preprtion A A Simplifying lgeri expressions A pronumerl (letter) represents numer. It my stn for n unknown vlue or series of vlues tht hnge. For exmple, in the eqution x 5 = 8, x is pronumerl tht represents vlue. Its vlue n e etermine euse we know 3 5 = 8, so x = 3. Terms tht hve extly the sme pronumerls suh s n 5 re lle like terms. Only like terms n e e n sutrte. It involves ing n sutrting the oeffiients. Aing n sutrting like terms simplifies the lgeri expression. It is often lle olleting the like terms. Aing n sutrting like terms Fin the like terms or the terms tht hve extly the sme pronumerls. Only like terms n e e or sutrte, unlike terms nnot. 3 A or sutrt the oeffiients or numers efore the pronumerl of the like terms. Exmple : Aing n sutrting like terms A Simplify 5x 6x 3x x. Group the like terms together. A n sutrt oeffiients of like terms. 5x 6x 3x x = 5x 3x 6x x = x 5x Aing n sutrting lgeri frtions An lgeri frtion ontins pronumerls. Fin ommon enomintor (preferly the lowest) if enomintors re ifferent. Express eh frtion s n equivlent frtion with the ommon enomintor. 3 Simplify the numertor y ing or sutrting the like terms. Exmple : Aing n sutrting lgeri frtions Write s single frtion. 3 6 The lowest ommon enomintor of 3 n 6 is 6. Write the equivlent frtions y multiplying top n ottom of the first frtion y (3 = 6). 3 Sutrt the like terms in the numertor n simplify. A 3 6 = = = 3 = 6 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

3 A Simplifying lgeri expressions 3 Exerise A LEVEL Simplify y olleting like terms. 6g g 5x x 8 7 5x x e 8 f r 5r g m ( m) h t ( 3) t i 9 v ( 3 v ) Exmple Simplify y olleting like terms. 3w w q 3q 8 5x y x e 6 5 f 3 3 n m n g 3x y 5x 3y h 5 3 i 5z y z 8y 3 Simplify y olleting like terms. 7u u u q q q q x 6x 3x x e p 3p p p f w w 3w w g 3 3 h 5 k k i y 5 3y y Exmple Express eh s single frtion. 5 y 3 x 7 7 w t A or sutrt these lgeri frtions. m m 3y y e f 7n n x 3x 3 3 g 7 h 3 5w w A or sutrt these lgeri frtions. x x 7y y 8 e 5x x f 6 3 m m 0 g h h 0 0 h r r 8 p p 7 A or sutrt these lgeri frtions. k k k 5x x 5x y 3y y LEVEL 8 Copy n like terms where possile to omplete the tle. x 3x x 3x x x 6x x x x Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

4 Chpter Preliminry preprtion A 9 The perimeter of retngle is the istne roun the outsie of the retngle. The retngle elow hs length of x n reth of x. Write n expression for the perimeter of this retngle y olleting like terms. x x 0 Av went shopping. She spent $y on jket, $3 y on pir of shoes, n $y on skirt. If she h $500, write n expression in simplifie form for how mny ollrs she hs left. A pentgon hs four sies of length x y n one sie of length x y. The pentgon is shown elow. Write n expression in simplifie form for the perimeter of this pentgon y olleting like terms. x y x y x y x y x y A or sutrt these lgeri frtions. k k 3 e 3n n 5 6 f r r 3 g 5 0 h 6r 3r 7 0 3y y LEVEL 3 3 A or sutrt these lgeri frtions. m m m 6 x x x 5 9m m m h h h 3 7 A or sutrt these lgeri frtions. 3 y 7 y e y 3 5 x f 3x x 8m m 7 g 3 3 h y y 9 5w 6w 5 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

5 B Inex lws 5 B Inex lws Inex form or inex nottion is use to write expressions in shorter wy, suh s =. When numer or pronumerl is expresse in inex form, the inex or power represents the numer of times the multiplition ours. Tht is, n = ( n ftors) Inex lws shown elow re rules for working with lgeri terms expresse in inex form. Inex lws m n m n = Multiplying terms in inex form (sme se), the inies. m n m n = Diviing terms in inex form (sme se), sutrt the inies. m or m n n = 3 m n mn ( ) = A term in inex form rise to power, multiply the inies. 0 = A term in inex form rise to the power of zero is one. Multiplying n iviing using the inex lws Algeri terms re multiplie n ivie to form single lgeri expression. This requires the lgeri expression to e expresse in expne form. If the lgeri terms ontin frtions, it is esier to first nel ny ommon ftors in the numertor n enomintor. The oeffiients re then multiplie n ivie, n finlly the pronumerls re multiplie n ivie. The inex lws shown ove re often pplie to mke the lultions esier. Multiplying n iviing lgeri terms Write in expne form. If the lgeri term is frtion, nel ny ommon ftors. 3 Multiply n ivie the oeffiients. Multiply n ivie the pronumerls. Use the inex lws if require. 5 Write the oeffiient efore the pronumerls. 6 Write the pronumerls in lphetil orer n express in inex form. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

6 6 Chpter Preliminry preprtion B Exmple 3: Multiplying lgeri terms using the inex lws B Simplify the following. 3 5y y 3 ( 5 ) Write in expne form. Multiply the oeffiients (5 = 0). 3 Multiply the pronumerls using the inex 3 5 lws ( y y = y ). Express nswer using inex nottion. 5 Write in expne form. 6 Multiply the oeffiients ( = ). 7 Multiply the pronumerls using the inex lws ( =, = ). 8 Write the pronumerls in lphetil orer. 9 Express nswer using inex nottion. 5y y = 5 y y = 5 y y = 0y ( ) = = = Exmple : Diviing lgeri terms using inex lws B Simplify the following Write in expne form. Divie the oeffiients (0 5 = ). 3 Divie the pronumerls using the inex 3 9 lws ( = ). Express nswer using inex nottion. 5 Write in expne form. 6 Divie the oeffiients ( 7 = 3). 7 Cnel the pronumerl ( m) in oth the numertor n enomintor ( m m = m ). 8 Write the oeffiient efore the pronumerls. 9 Write the pronumerls in lphetil orer. mn 7m = = 3 = = mn = m m n 7m 7 m 7 3 m m n = 7 m = 3 m n = 3mn 9 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

7 B Inex lws 7 Exmple 5: Applying the inex lws B Simplify the following. 0 x 5 (6 ) pq 3 Write in expne form. 0 0 Reple x with ( = ). 3 Evlute. Write in expne form. 5 Rise eh term to the power of. Use the m n inex lw ( ) = mn to simplify eh lgeri expression. 6 Write the pronumerls in lphetil orer. 0 0 x 5 = x 5 = 5 = 7 (6 pq ) = (6 p q ) = 6 p q 6 = 36pq Exmple 6: Simplifying lgeri terms with frtions B Simplify x xy 6y 0 5 Write frtions in expne form. Determine ny ommon ftors in the numertor n the enomintor. 3 Cnel out the ommon ftors ( is ommon ftor of n 6, is ommon ftor of n 0). Cnel the pronumerl ( y ) in oth the numertor n enomintor s it is 5 3 ommon ftor ( y y = y ). 5 Multiply the numertors together 5 ( x x = x ). 6 Multiply the enomintors together. 7 Express nswer using inex form. 8 Write the oeffiient efore the pronumerls. However it is eptle 5 3 x y to leve nswer s. 5 x 6y 5 xy = x x y 0 6 y 0 = x 3 y = x x y 3 y 5 = = = x x y 3 5 x y x y = x y 5 (or 5 3 x y ) x y Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

8 8 Chpter Preliminry preprtion B Exerise B LEVEL Multiply these lgeri terms. 5w 3 6 p 6f 5 x 8y e 3 m ( 3 n) f 7r 5k 3 g h 6g 3 h 7n n 3 i 30 5 Exmple 3 Simplify eh of the following using the inex lws. 3 6 m m y 3y z 3 z 5 y y e 3 f w 0 w g st 7st h pq ( 5 pq) i 6 5 ( 3 8 ) 3 Fin the prout of these terms. 3 5, n 7 x,3x n ( x) 6y 3, y n 3y mn, mn n mn 3 e ( 7 ), ( ) n f e, n e 3 Exmple Divie these lgeri terms. 6w 8 s ( 3) e ( 0 gh) g f 5xy 5xy 3 g h m m i 3 ( 8 z ) 7z 5 Simplify eh of the following using the inex lws. 7 y y 8 5 x 5 ( 5 x ) m 0m e 30 t ( 0 t ) f ( 5 ) 5 5 g 6 9 h 6q q i e 5 e Simplify. 5 3w w w y y y 6 x 9x 3x ( ) 0 5 e ( 0 ) 0 f 3 7e e e Exmple 7 Simplify these lgeri terms. 7 ( 5 ) 30 g 7 6y 3 ( 8 y ) e 0t 5 n n 6 5 h e e f i 6z z 0 ( x y) 3 ( x ) 3x y 3 6x y Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

9 B Inex lws 9 Exmple 5 Exmple 5 8 Remove the grouping symol n express eh nswer in simplest form. 3 ( x ) 3 ( s ) ( 3 y) 3 3 ( n ) 5 e ( ) f (5 r t) 3 3 g ( ) ( ) 7 3 h ( 6 ef ) ( e f) i (6 ) ( ) 9 Express in simplest inex form. 0 y m e f g 5 x h ( p ) i e f Simplify eh of the following using the inex lws q q u ( u) x x e 3m 7m f 0 ( ) g (5 g ) h ( ) i ( x y) j 5n k ( 9 ) l Copy n multiply the lgeri terms to omplete the tle. x 3 ( 3 x ) xy x 3 5 ( 5 x y ) 7x 5 y 8 Copy n ivie the lgeri terms to omplete the tle. ( ) ( 3 ) 6 Exmple 6 3 Simplify. 8y y 6 5w w x e 3 f 5w w x g 0 3 h m 6mn i 9n 5 e 3 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

10 0 Chpter Preliminry preprtion B LEVEL Simplify eh of the following using the inex lws m 3m 9m x x ( 7 x ) 9 y (5y 3 y ) 3 5 Express 8 x y 6x y 3 in simplest form. 6 The plne shpe elow is retngle with length of 3x n reth of expression in simplest form for the re of this retngle. 5x. Write n 5x 3x 7 Simplify. 3 6x x 3 y 5x 3 3x y g e n m 8n m h 3mn 5n 9 m 7 6 7v v 8 3 f 6e e f 9 f i e e 7 3 LEVEL 3 8 Simplify. 5 3k 3k h 3 3 9m 6mn 3mn 5u 5uv 7 v 3 3( m ) 3 e m 8m ( m ) f 3 7y 5( y ) 0( y ) 6 y 9 Simplify these lgeri expressions x 3yz 3 3yz 6xy 5 mn 6p 3 m p 8mn e 5 8 w ( 6 wu ) 5 5 wu u f 7 6 e e Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

11 C Expning lgeri expressions C Expning lgeri expressions Grouping symols in lgeri expressions inite the orer of opertions. The two most ommonly use grouping symols re prentheses () n rkets []. They re remove y using orer of opertions or the istriutive lw. This is illustrte elow. Using orer of opertions (3 ) = = 8 Using istriutive lw (3 ) = 3 = 6 = 8 To expn n lgeri expression using the istriutive lw, multiply the numers or terms insie the grouping symols y the numer or term outsie the grouping symols. The resulting lgeri expression is simplifie y olleting the like terms. Expning lgeri expressions Multiply the numer or term outsie the grouping symol y the first term insie the grouping symol seon term insie the grouping symol. Simplify n ollet like terms if require. ( ) = = ( ) = = Mke sure you rememer to multiply ll the terms insie the grouping symol y the numer or term outsie the grouping symols. Notie tht in the numeril exmple of the istriutive lw t the top of the pge, (3 ) = 3 is the sme s ( ) =, with =, = 3 n =. Exmple 7: Expning lgeri expressions C Expn 5(y 3). Write in expne form. Multiply the first term insie the prenthesis ( y) y the numer outsie the prenthesis (5). 3 Multiply the seon term insie the prenthesis ( 3) y the numer outsie the prenthesis (5). Write in simplest form. 5(y 3) = 5 (y 3) 5 (y 3) = 5 y 5 ( 3) = 0y 5 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

12 Chpter Preliminry preprtion C Exmple 8: Expning lgeri expressions C Expn ( m 5). Multiply the first term insie the prenthesis ( m) y the numer outsie the prenthesis ( ). Multiply the seon term insie the prenthesis ( 5) y the numer outsie the prenthesis ( ). 3 Write in simplest form. ( m 5) = ( m 5) = m 5 = m 5 Exmple 9: Expning n simplifying lgeri expressions C Remove the grouping symols for (3x ) 3( x ) n simplify if possile. Write the expne form Multiply the first term insie the prenthesis (3 x) y the numer outsie the prenthesis (). 3 Multiply the seon term insie the prenthesis ( ) y the numer outsie the prenthesis (). Repet the first two steps for the seon prenthesis. 5 Simplify y olleting the like terms. (3x ) 3( x ) = (3x ) 3 ( x ) = 3x 3( x ) = 6x 8 3( x ) = 6x 8 3 x 3 = 6x 8 3x 3 = 9x 5 Exmple 0: Expning n simplifying lgeri expressions C Expn y(5y ) yy ( 6) n simplify if possile. Write in expne form Multiply the first term insie the prenthesis (5 y) y the term outsie the prenthesis ( y). 3 Multiply the seon term insie the prenthesis ( ) y the term outsie the prenthesis ( y). Repet the first two steps for the seon prenthesis. 5 Simplify y olleting the like terms. y(5y ) yy ( 6) = y (5y ) y ( y 6) = y 5y y y( y 6) = 5y y y y y 6 = 5y y y 6y = y 7y Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

13 C Expning lgeri expressions 3 Exerise C LEVEL Ryn ws require to remove the grouping symols. This ws his solution. 3(3x ) = 9x Where is the error in Ryn s working? Exmple 7 Expn eh of the following. 3( ) ( ) 7( ) (3x ) e (5x 7) f (9 ) g (5 t) h 6( w) i 5(3 9 ) j 8(5e ) k 5( 9 ) l 7(h 8 g) Exmple 8 3 Expn eh of the following. ( x 3) 3( y 5) ( 8) 7( k ) e 6( w ) f ( x 3) g ( q) h 5(3 r) i 7(8 s) Expn eh of the following. yy ( ) vv ( ) nn ( 0) x(x 3) e e(3e 5) f (6 ) g z(7e 3 f ) h ( 3) i ( e) 5 Remove the grouping symols n simplify if possile. ( g ) g 7( s ) s 3( y 9) y 5x ( x ) e 6z ( z ) f 3q 7( q 5) 6 Remove the grouping symols n simplify if possile. ( x ) x 5 7(3y ) y (5 ) 8 r 7 5( r 3) e n 8 3( n ) f 5q ( q 9) 7 Expn eh of the following. ( y 3) ( ) 5( 3) 3( 3 e ) e ( ) f 6(3 g ) 8 Expn eh of the following. kk ( 3) ( 6) y(5 3 y) x(x ) e ( ) f p(5q 3) r Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

14 Chpter Preliminry preprtion C Exmple 9 9 Simplify the following. ( x ) 5( x ) 3( y ) ( y ) 5( ) 3( ) 8( 3) 5( 3) e 6( s 6) (s ) f 5( h 7) (h 7) g (3x ) ( x ) h 9( z 5) 7( z ) i 5( ) 3( 7) j 5(5g ) ( g ) k 7(u 3) ( u 3) l ( ) 3( 3) LEVEL 0 Tylh ws require to remove the grouping symols n simplify. This ws her solution. 6(3x ) ( x 3) = 8x x 6 = 6x 6 Where is the error in Tylh s working? Exmple 0 Remove the grouping symols n simplify if possile. xx ( 5) xx ( ) ( 3) ( ) yy ( 3) yy ( 8) g(g 3) gg ( 3) e vv ( 7) v(6v ) f (5 ) ( ) g uu ( ) uu ( 9) h nn ( 6) nn ( ) i 3 ( 7) ( 5) j ee ( ) 7 ee ( 9) k 6 kk ( 3) kk ( 3) l t(5 3) t 7( t t) Expn n simplify the lgeri expression ( 3) ( ). LEVEL 3 3 Simplify the following. x (x 3) ( x ) ( ) ( 3) y (5y ) 3( y 7) ( 7) (3 ) e z(3z ) z ( z 5) f e(7 e) e (e 6) g xx ( 7) xx ( ) h ( ) ( ) i v( v ) v( v ) j ( ) ( ) k x ( x y) x( x 3 y) l y ( y z) y( z ) Remove grouping symols. m (5m ) x ( x 3x ) 8 ( 3 ) 5 Expn n simplify the following lgeri expressions ( ) ( 7) 3 3 x (3x ) 3 x ( x 9) ( n r) n (7 n rn ) 6 Simplify the following lgeri expressions y nelling ftors. 3 7 ( ) 6 3( ) 5r 3( r ) ( r ) 0r 6( 3) ( 3) Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

15 D Pythgors theorem 5 D Pythgors theorem Pythgors theorem links the sies of right-ngle tringle. In right-ngle tringle the sie opposite the right ngle is lle the hypotenuse. The hypotenuse is lwys the longest sie. Hypotenuse (longest sie) Pythgors theorem Pythgors theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sies. h (Hypotenuse) = (sie) (other sie) h = Pythgors theorem is use to fin missing sie of right-ngle tringle if two of the sies re given. It n lso e use to prove tht tringle is right ngle. Exmple : Fining the length of the hypotenuse D Fin the length of the hypotenuse, orret to two eiml ples. 5 m h m Write Pythgors theorem. Sustitute the length of the sies. 3 Clulte the vlue of h. Tke the squre root to fin h. 5 Express nswer orret to two eiml ples. h = = 5 9 h = 5 9 h = m 9 m Exmple : Fining the length of shorter sie D Wht is the vlue of x orret to one eiml ple? x mm mm 5 mm Write Pythgors theorem. Sustitute the length of the sies. 3 Mke x the sujet. Tke the squre root to fin x. 5 Express nswer orret to one eiml ple. h = = x 5 x = 5 x = mm Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

16 6 Chpter Preliminry preprtion D Exerise D LEVEL Exmple Fin the length of the hypotenuse, orret to one eiml ple. 5 m 6 m h m 8 m m h m mm h mm 0 mm e f 5 m.5 m h m h mm 0 mm. m 63. m h m 0 mm Exmple Fin the vlue of x, orret to two eiml ples. 5 m 5 m x m m x m m x mm 3 mm 6 mm e.3 m f x m. m mm 3 mm x m.8 m 9.5 m x mm Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

17 D Pythgors theorem 7 LEVEL 3 Clulte the length of the sie mrke with the pronumerl. (Answer to the nerest millimetre.) y mm 6 mm 35 mm mm 30 mm 63 mm x mm 8 mm mm 0 mm e 33 mm f m mm 8 mm 7 mm mm 5 mm mm mm Fin the length of the igonl of retngle with imensions 7.5 metres y 5.0 metres. Give the nswer orret to one eiml ple. 5.0 m 7.5 m 5 Fin the vlue of the pronumerls, orret to two eiml ples. LEVEL 3 90 m 7 m 7 m y m x m x m m 6 m 6 Clulte the length of the sie mrke with pronumerl, orret to one eiml ple. 7 m 5 m 6 m x m m 8 m x m y m m Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

18 8 Chpter Preliminry preprtion E E Perimeter Perimeter is the totl length of the outsie eges of shpe. It is the istne of the ounry. Perimeter formule Nme Shpe Perimeter Tringle P = Qurilterl P = Squre s P = s Retngle l P = ( l ) Cirle r Cirumferene C = π r C = π Exmple 3: Fining the perimeter of retngle E Fin the perimeter of the following retngle. 3 m The shpe is retngle, so use the formul P = ( l ). Sustitute the vlues for l n (l = 8 n = 3). 3 Evlute. Write the nswer in wors. P = ( l ) = (8 3) 8 m = m Perimeter of the retngle is m. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

19 E Perimeter 9 Exmple : Fining the perimeter of tringle E Fin the perimeter of the tringle. Answer orret to one eiml ple. 3.7 m Fin the length of the hypotenuse or h. Write Pythgors theorem. 3 Sustitute the length of the sies. Evlute the vlue of h. 5 A the lengths of sies to fin the perimeter. 6 Express nswer orret to one eiml ple. 7 Write the nswer in wors. h = h = m h = m P = = 3.5 m Perimeter of the tringle is 3.5 m. Exmple 5: Fining the irumferene of irle E Fin the perimeter of irle with rius of 9mm. Answer orret to two eiml ples. The shpe is irle, so use the formul C = πr. Sustitute the vlue for r ( r = 9). 3 Evlute. Write the nswer in wors. 9 mm C = π r = π mm Perimeter of the irle is mm. Exmple 6: Fining the perimeter of semiirle E Fin the perimeter of semiirle with imeter of m. Answer orret to two eiml ples. The shpe is semiirle, so use the formul C = π. Sustitute the vlue for ( = ) to fin the urve istne. 3 Evlute. A the urve istne to the imeter. 5 Evlute. 6 Write the nswer in wors. C = π = π 6.8 m P = 6.8 = 0.8 m Perimeter is 0.8 m. m Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

20 0 Chpter Preliminry preprtion E Perimeter of omposite shpes A omposite shpe is me up of two or more plne shpes. The perimeter of omposite shpe is lulte y ing the lengths tht mke up the ounry of the shpe. Perimeter of omposite shpe Sies with the sme mrkings re of equl length. Unknown sie lengths re etermine y using the given lengths of the other sies. 3 Pythgors theorem is use to fin unknown sie lengths involving right tringle. Lengths tht re prt of irle re foun using C = π r. 5 A the Lengths tht mke up the ounry of the shpe to lulte the perimeter. Exmple 7: Fining the perimeter of omposite shpes E Fin the perimeter of eh of these shpes. m 5 m 6 m 8 m 5 m 5 m Fin the unknown sie lengths using the mesurements given in the question. A the lengths of ll the eges to fin the perimeter. 3 Evlute. Write the nswer in wors. m 6 m 8 = 6 m 6 5 = m 5 m 8 m P = = 38 m Perimeter is 38 m. 5 Use the formul C = π r for the urve length. 6 Sustitute the vlue for r ( r = 5). 7 Evlute. 8 A the urve length to other eges. 9 Evlute. 0 Write the nswer in wors. C = π r = π m P = = 7.85 m Perimeter is 7.85 m. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

21 E Perimeter Exmple 3 Exerise E Fin the perimeter of eh qurilterl. Answer orret to one eiml ple. 7. m 0 m 5. m 3. m LEVEL 9.5 m Fin the perimeter of squre with sie length of.3 m. Answer orret to one eiml ple. Exmple 3 Fin the perimeter of eh right tringle. Answer orret to one eiml ple. 7 mm 8 m 5 m 5 m m 8.5 mm Fin the perimeter of right tringle with se of 0.5 m n height of 5.5 m. Answer orret to two eiml ples. 5 Fin the perimeter of eh irle. Answer orret to one eiml ple. 3 m m mm 6 Fin the irumferene of eh irle. Answer orret to one eiml ple. Rius of m Rius of 9 m Rius of 3 mm Dimeter of 50 mm e Dimeter of m f Dimeter of 6m Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

22 Chpter Preliminry preprtion E LEVEL Exmple 6 7 Fin the perimeter of eh semiirle. Answer orret to one eiml ple. 5 m. m 6 m 8 Fin the perimeter of eh shpe. Answer orret to two eiml ples. 3 mm 5 m 7 m Exmple 7 9 Fin the perimeter of eh omposite shpe. Answer to the nerest whole numer. 6 m 8 m m m 8 m m m m 0 m m 0 m m e f m 3 m 6 m m 5 m m 6 m LEVEL 3 0 An nnulus is shpe, like oughnut or tyre, etween two irles with the sme entre. Fin the perimeter of n nnulus if the inner imeter is 3m n the outer imeter is 6m. Answer orret to the nerest entimetre. 3 m 6 m A retngle ABCD hs length AB = m n with of BC = 6m. Answer orret to two eiml ples. Fin the vlue of x. Clulte the perimeter of qurilterl AECF. A D E x m F B C Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

23 F Are 3 F Are The re of shpe is the mount of surfe enlose y the ounries of the shpe. It is the numer of squres tht fit insie the shpe. When lulting re, the nswer will e in squre units. 00 mm = m m = m m = h m = km To lulte the re of the most ommon shpes, we use formul. These formule re liste elow. Are formule Nme Shpe Are Tringle h A = h Squre s A = s Retngle A = l l Prllelogrm h A = h Trpezium h A = ( h ) Rhomus x y A = xy Cirle r A = πr Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

24 Chpter Preliminry preprtion F Exmple 8: Fining the re of tringle F Fin the re of the tringle. The shpe is tringle, so use the formul A = h. Sustitute the vlues for n h ( = 8. n h = 5.5). 3 Evlute. Write the nswer using the orret units. 5.5 m 8. m A = h = =.75 m The re of the tringle.75 m. Exmple 9: Fining the re of trpezium Fin the re of the qurilterl..9 m 3 m F The shpe is trpezium, so use the formul A = h ( ) Sustitute the vlues for, n h. 3 Evlute. Write the nswer using the orret units. 5. m A = h ( ) = (.9 5.)3 = m The re of the shpe is m. Exmple 0: Fining the re of prllelogrm F Fin the re of the following qurilterl. The shpe is prllelogrm, so use the formul A = h. Sustitute the vlues for n h ( = 6.5 n h = ). 3 Evlute. Write the nswer using the orret units. mm 6.5 mm A = h = 6.5 = 6 mm The re of the shpe is 6 mm. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

25 F Are 5 Exmple : Fining the re of irle F Fin the re of irle with rius of 5 metres. Give your nswer orret to one eiml ple. 5 m The shpe is irle, so use the formul A = πr. Sustitute the vlue for r ( r = 5). 3 Evlute orret to one eiml ple. Write the nswer using the orret units. A = πr = π m The re of the irle is 78.5 m. Are of omposite shpes A omposite shpe is me up of two or more plne shpes. The re of omposite shpe is lulte y ing or sutrting the res of simple shpes. Are of omposite shpes Composite shpes re me up of more thn one simple shpe. Are of omposite shpe n e foun y ing or sutrting the res of simple shpes. Exmple : Fining the re of omposite shpe F Fin the re of the omposite shpe. Answer orret to one eiml ple. 0 m m Divie the shpe into retngle n semiirle. Use the formul A = l for the retngle. 3 Sustitute Evlute. 5 Use the formul A = πr for the semiirle. 6 Sustitute 7 Evlute. 8 A the re of the retngle to the semiirle. 9 Evlute. 0 Write the nswer using the orret units. A = l = 0 = 0 m A = πr = π m A = = 59.3 m The re of the shpe is 59.3 m. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

26 6 Chpter Preliminry preprtion F Exerise F LEVEL Exmple 8 Fin the re of eh tringle. Answer orret to one eiml ple where neessry. 3 m 6 mm 3 m m m 6.5 mm 9 m 7.6 m e 5.5 mm 3 mm f 9.5 m 8.5 m Fin the re of eh shpe. Answer orret to one eiml ple where neessry. Exmple 9 6. m 9 m 6. m. m m Exmple 0 e m f 3.8 m 7 mm 7 m m 0 mm 6.7 m 3 Fin the re of tringle with se of 8.5 m n height of 0.5 m. Answer orret to the nerest squre entimetre. Fin the re of squre with sie length of 05. m. Answer orret to the nerest squre metre. Exmple 5 Fin the re of irle with rius of 7 m. Answer orret to the nerest squre entimetre. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

27 F Are 7 6 Fin the re of eh shpe. Answer orret to one eiml ple where neessry. LEVEL.3 km 9. m 5 km 9. m Digonls re. mm n 6.8 mm.8 m 5.7 m. m 9.8 m e 6 mm.5 mm 8 mm f 7 m 5 m 8 m 7 Jsmine is plnning to uil irulr pon. The rius of the pon is.5 m. Wht is the re of the pon, orret to the nerest squre metre? 8 A 5 m swimming pool inreses in epth from.3 m t the shllow en to.6 m t the eep en. Clulte the re of one sie wll of the pool. Answer orret to the nerest squre metre. 5 m.3 m.6 m 9 Philip wnts to tile retngulr re mesuring.5 m y 3m in his kyr. The tiles he wishes to use re 50 m y 50 m. How mny tiles will he nee? Give your nswer s whole numer. 0 Fin the re of eh omposite shpe. m 30 m 3 m 6 m 8 m 3 m 5 m m 0 m Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

28 8 Chpter Preliminry preprtion F An nnulus onsists of two irles with the sme entre. Fin the re of n nnulus if the inner imeter is 6m n the outer imeter is 0 m. Answer orret to the nerest squre entimetre. 6 m 0 m A metl prllelogrm hs two ientil squres remove from its shpe. The two squres hve sie length of m. Fin the she re. Answer orret to the nerest squre entimetre. 3 A lwn is to e li roun retngulr gren e. Wht is the mount of lwn require? Fin the ost of the new lwn if the require turf osts $0 per squre metre. 5 m 0 m 3 m 9 m 5 m 7 m 0 m LEVEL 3 Exmple A light fitting hs the shpe of hlf of sphere mounte on n upsie-own one. Its ross setion is shown opposite. Wht is the rius of the semiirle? Wht is the height of the tringle? Clulte the re of the ross-setion. Answer orret to one eiml ple. 0 m 6 m m 5 Wht is the re of qurnt if it hs rius of 8mm? Answer orret to two eiml ples. 6 Deking for house onsists of squre n tringle. The squre hs sie length of 8 metres n the tringle is isoseles. Use Pythgors theorem to fin the vlue of x. Clulte the re of the she region. 8 m 8 m x m x m 7 A metl worker ut irles with imeter of m from retngulr sheet of tin m y 8m. Wht is the re of the retngulr sheet? How mny irles n e ut from the retngulr sheet? Wht is the re of the remining metl fter the irles hve een remove from the retngulr sheet? Answer orret to two eiml ples. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

29 G Simple interest 9 G Simple interest Interest is the mount pi for orrowing money or the mount erne for lening money or investing. There re ifferent wys of lulting interest. Simple interest (or flt interest) is fixe perentge of the mount investe or orrowe n is lulte on the originl mount. For exmple, if we invest $00 in nk ount tht pys interest t the rte of 5% per nnum (per yer) we woul reeive $5 eh yer. Tht is, Interest = $00 5 = $5 00 This mount of interest woul e pi eh yer. Simple interest is lwys lulte on the initil mount, or the prinipl. An investment in the ontext of interest is money tht you put into nk, or other finnil institution, in return for interest pyments. It is like lon tht you mke to the nk. Formul for lulting Simple interest I = Prn I Interest (simple or flt) erne for the use of money, pi y orrowers to leners P Prinipl is the initil mount of money orrowe, lent or investe r Rte of simple interest per time perio expresse s eiml, e.g. 5% = 0.05 n Numer of time perios (ys, weeks, months or yers) Exmple 3: Fining simple interest G Clulte the mount of simple interest pi on n investment of $ 000 t 0% simple interest per nnum for 3 yers. Write the simple interest formul. Sustitute P = 000, r = 0.0 n n = 3 into the formul. 3 Evlute. Write the nswer in wors. I = Prn = = $3600 Simple interest is $3600. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

30 30 Chpter Preliminry preprtion G Amount owe or future vlue The interest is e to the prinipl to etermine the mount owe on lon or the future vlue of n investment. formul for Amount owe or future vlue A = P I A Amount owe or future vlue I Interest (simple or flt) erne P Prinipl is the initil quntity of money orrowe, lone or investe Exmple : Clulting the mount owe G Fin the mount owe on lon of $ t 7% per nnum simple interest t the en of two yers n six months. Write the simple interest formul. Sustitute P = , r = 0.07 n n =.5 into the formul. 3 Evlute. Write the mount owe formul. 5 Sustitute P = n I = 8750 into the formul. 6 Evlute. 7 Write the nswer in wors. I = = Prn = $8750 A = P I = = $ Amount owe is $ Exmple 5: Clulting vlue of n investment G Joel plns to mke n investment of $ t 7 %p.. simple interest for yers. Wht is the totl vlue of his investment t the en of yers? Write the simple interest formul. Sustitute P = , r = n n = into the formul. 3 Evlute. Write the mount owe formul. 5 Sustitute P = n I = into the formul. 6 Evlute. 7 Write nswer in wors. I = Prn = $ = $ A = P I = $ $ = $ Totl vlue is $ Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

31 G Simple interest 3 Exerise G LEVEL Exmple 3 Exmple Clulte the mount of simple interest for eh of the following. Prinipl = $5 000, Interest rte = 3%p.., Time perio = 3yers Prinipl = $000, Interest rte = 6 %p.., Time perio = 7yers Prinipl = $00 000, Interestrte = 9 %p.., Timeperio = yers Prinipl = $3600, Interestrte = 9%p.., Time perio = 3 yers e Prinipl = $0 000, Interestrte = 7.5%p.., Time perio = 5 yers Clulte the mount owe for eh of the following. Prinipl = $500, Simple interest rte = 5%p.., Time perio = yers Prinipl = $900, Simple interest rte = 3%p.., Time perio = 7yers Prinipl = $000, Simple interest rte = 8 %p.., Time perio = 3yers Prinipl = $6900, Simple interest rte = 0%p.., Time perio = yers e Prinipl = $0 000, Simple interest rte = 6.75%p.., Time perio = yers 3 The simple interest rte is given s.8% per nnum. Wht is the interest rte per qurter? Wht is the interest rte per month? Wht is the interest rte per six months? Wht is the interest rte per nine months? Clulte the mount of simple interest for eh of the following. Prinipl = $800, Interest rte = %p.., Time perio = month Prinipl = $600, Interest rte = 8%p.., Time perio = 6months Prinipl = $60 000, Interest rte = 9.6%p.., Time perio = 3months Prinipl = $0 000, Interest rte = 6%p.., Time perio = 9months 5 Anrew tkes lon of $ for perio of 6 yers, t simple interest rte of % per nnum. Fin the mount owing t the en of 6 yers. 6 A lon of $800 is tken out t simple interest rte of 5.5% per nnum. How muh interest is owing fter 3 months? Exmple 5 7 A sum of $ ws investe in fixe term ount for yers. Clulte the simple interest erne if the rte of interest is 5.5% per nnum. Fin the vlue of the investment t the en of yers. 8 Joshu investe $00 t 8% per nnum. Wht is the simple interest erne etween 30 Septemer n Jnury? Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

32 3 Chpter Preliminry preprtion G 0GQ LEVEL 9 Sophie eies to uy r for $ She hs sve $7000 for the eposit n tkes out lon over two yers for the lne. The flt rte of interest hrge is % per nnum. Wht is the totl mount of interest to e pi? 0 Domenio hs orrowe $ 000 to uy furniture. He wishes to repy the lon over four yers. Clulte the simple interest on the following rtes of interest. 8% per nnum for the entire perio 9% per nnum fter 6-month interest-free perio 0% per nnum fter -month interest-free perio Crete the spresheet shown. Cell D5 hs formul tht lultes the simple interest. Enter this formul. The formul for ell E5 is =A5 D5. Fill own the ontents of E6 to E using this formul. Iselle uys TV for $00. She pys it off monthly over yers t n interest rte of.5% per nnum flt. How muh per month will she py? 3 Riley wnts to ern $000 yer in interest. How muh must he invest if the simple interest rte is 0%p..? LEVEL 3 Smir invests $6 000 for yers. Wht is the minimum rte of simple interest neee for her to ern $3000? 5 Gurrumul pys k $0 000 on $5 000 lon t flt interest rte of 0%. Wht is the term of the lon? 6 Hrry orrowe $ t flt rte of interest of 8.5% per nnum. This rte ws fixe for yers. He pys k the interest only over this perio. How muh interest is to e pi over the yers? After pying the fixe rte of interest for the first yer, Hrry fins the nk will erese the flt interest rte to 7.5% if he pys hrge of $000. How muh will he sve y hnging to the lower interest rte for the lst yer? Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

33 H Compoun interest 33 H Compoun interest Compoun interest is lulte on the prinipl t the strt, n then t eh time perio, on the prinipl plus interest. It lultes interest on the interest. For exmple, if $00 is investe in svings ount t ompoun interest rte of 0% per nnum: Firstyer Interest = $ = $0 Amount owe = $00 $0 = $0 Seon yer Interest = $0 0.0 = $ Amount owe = $0 $ = $ Thiryer Interest = $ 0.0 = $.0 Amount owe = $ $.0 = $33.0 These lultions show tht the interest erne inrese eh yer. In the first yer it ws $0, the seon yer $ n the thir yer $.0. Compoun interest formul A= P( r) n A Amount (finl lne) or future vlue of the lon P Prinipl is the initil mount of money orrowe or present vlue of the lon r Rte of interest per ompouning time perio expresse s eiml n Numer of ompouning time perios Totl ompoun interest erne or pi The totl ompoun interest is lulte y sutrting the prinipl from the finl lne i.e. the mount of money t the en. Interest erne or owe I = A P A Amount of money or finl lne I Interest (ompoun) erne or pi P Prinipl is the initil mount of money Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

34 3 Chpter Preliminry preprtion H Exmple 6: Fining the ompoun interest H Pige invests $5000 over 5 yers t ompoun interest rte of 6.5%p.. Clulte: the mount of the investment fter 5 yers, orret to the nerest ent the interest erne fter 5 yers, orret to the nerest ent. Write the ompoun interest formul. Sustitute P = 5000, r = n n = 5 into the formul. 3 Evlute. Write nswer in wors. 5 Write the formul. 6 Sustitute A = n P = 5000 into the formul. 7 Evlute. 8 Write in wors. A = P( r) n = 5000( 0.065) = $ Amount of investment interest erne is $ I = A P = = $850.3 Interest erne is $ Exmple 7: Fining the ompoun interest H Jmes orrowe $ for yers t %p.. interest ompouning monthly. Clulte: the mount owe fter yers, orret to the nerest ent the interest owe fter yers, orret to the nerest ent. Write the ompoun interest formul. Clulte the numer of time perios ( yers months) n the interest rte per time perio. 3 Sustitute P = , r = 0. n n = 8 into the formul. Evlute. 5 Write nswer in wors. 6 Write the interest owe formul. 7 Sustitute P = n A = into the formul. 8 Evlute. 9 Write in wors. A = P( r) n n = r = 0. 8 A = $ Amount owe is $ I = A P = $ $ = $ Interest owe is $ ( ) Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

35 H Compoun interest 35 Exerise H LEVEL Exmple 6 Exmple 7 Clulte the mount owe, to the nerest ent, for eh of the following lons. Prinipl = $800, Compoun interest rte = %p.., Time perio = 3yers Prinipl = $9000, Compoun interest rte = 6 %p.., Time perio = yers Prinipl = $ 000, Compoun interest rte = %p.., Time perio = yers Prinipl = $ 000, Compoun interest rte = 5.5%p.., Time perio = yers Clulte the mount of ompoun interest to the nerest ent for eh of the following. Prinipl = $5 000, Interest rte = 7%p.., Time perio = 5yers Prinipl = $ , Interest rte = 0 %p.., Time perio = 3yers Prinipl = $6500, Interest rte = 3%p.., Time perio = Prinipl = $80 000, Interest rte = 8.5%p.., Time perio = yers 3 yers 3 Amy is investing $ Wht sum of money will she reeive if she invests for yers t 8%p.. ompoun interest? Answer to the nerest ent. Use the formul A = P( r) n to lulte the vlue of n investment of $0 000 over perio of yers with n interest rte of 0.8% ompouning monthly. Answer to the nerest ent. 5 Ryn investe $0 000 for 5 yers t %p.. interest ompouning monthly. Wht is the mount of interest erne in the first yer? Answer to the nerest ent. 6 Fin the mount of money in nk ount fter 6 yers if n initil mount of $000 erns 8%p.. ompoun interest, pi qurterly. Answer to the nerest ent. 7 Christopher investe $3 500 over 7 yers t 6.%p.. interest ompouning qurterly. Clulte: the vlue of the investment fter 7 yers to the nerest ent. the ompoun interest erne to the nerest ent. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

36 36 Chpter Preliminry preprtion H 0HQ LEVEL 8 Wht sum of money woul Biley nee to invest to umulte totl of $ t the en of yers t 6%p.. ompoun interest? Answer to the nerest ent. 9 Clulte the mount tht must e investe t 9.3%p.. interest ompouning nnully to hve $ t the en of 3 yers. Answer to the nerest ent. 0 Wht sum of money nees to e investe to umulte to totl of $ in 0 yers t 7.5%p.. ompoun interest? Answer to the nerest ent. Crete the spresheet elow. Cell D5 hs formul tht lultes the ompoun interest. Enter this formul. The formul for ell E5 is = D5 A5. Fill own the ontents of E6 to E using this formul. LEVEL 3 How muh more interest is erne on $0 000 investment if the interest t 6%p.. is ompoune nnully over 6 yers, ompre with the simple interest t 6%p.. erne over the sme time? 3 Hmish hs $ to invest for two yers. Whih is the etter investment n y how muh? Investment Simple interest rte %p.. Investment Compoun interest rte %p.. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

37 I Frequeny tles 37 I Frequeny tles A frequeny tle is listing of the outomes n how often (its frequeny) eh outome ours. The outomes my lso e lle sores. The tlly of the frequeny n the finl ount re liste in seprte olumns. When you rete frequeny tle from set of t, you shoul inlue tlly olumn to reor your ount of how often eh sore ours. The tlly olumn my e omitte in frequeny tle given to you tht hs lrey een rete. A frequeny tle is lso lle frequeny istriution. Sore Tlly Frequeny Lowest sore Highest sore Lowest frequeny Highest frequeny Frequeny tle Sores or outomes re liste in the first olumn in sening orer. Tlly olumn reors the ount of the numer of times the sore ourre (groups of 5s). 3 Frequeny olumn is the totl ount of eh outome. Exmple 8: Construting frequeny tle I The tempertures for 39 ys re shown elow. Construt frequeny tle Drw tle with three olumns n lel them sore, tlly n frequeny. List the tempertures in the sore olumn from the lowest (8) to the highest (8). 3 Reor mrk in the tlly olumn for eh temperture. Count the tlly mrks n write the totl in the frequeny olumn. 5 A the frequeny olumn to fin the totl numer of sores. This shoul mth the totl numer of tempertures (39). Sore Tlly Frequeny Totl 39 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

38 38 Chpter Preliminry preprtion I Exerise I LEVEL Note: inlue tlly olumn, where you re ske to rete frequeny tle in this exerise. Exmple 8 The ges, in yers, of plyers in footll tem re reore in frequeny tle. Copy n omplete the tle. Wht ws the most ommon ge? How mny plyers re in the tem? The numer of times fire engine is lle out on given y ws reore in frequeny tle. Copy n omplete the tle. Wht ws the most ommon numer of lls? On how mny ys ws the fire engine lle out four times? On how mny ys ws the fire engine lle out fewer thn three times? Sore Tlly Frequeny Numer of lls Tlly Frequeny Exmple 8 3 The numer of rothers n sisters for 30 stuents is reore elow. Construt frequeny tle for this t The shoe sizes of 0 seventeen yer ols re reore elow. Construt frequeny tle Exmple 8 5 The ssessment result for 30 stuents is reore elow. Construt frequeny tle for this t Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

39 I Frequeny tles 39 LEVEL 6 A ie ws rolle n the results liste elow Construt frequeny tle. How mny times ws the ie rolle? How mny results re higher thn? Wht ws the most ommon numer rolle? e Do you think the ie is ise? Give reson for your nswer. Exmple 8 7 Dvi reore the following times, in seons, for the 50 m freestyle Construt frequeny tle. How mny times were reore? How mny times were elow 35 seons? Wht ws the most ommon time for the 50 m freestyle? e Wht perentge of times re 37 seons or more? LEVEL 3 8 Count the numer of letters in eh wor of the prgrph elow. A frequeny tle is listing of the outomes n how often (frequeny) eh outome ours. The outomes re often liste uner heing lle sore. The tlly of the frequeny n the finl ount re liste in seprte olumns. Frequeny tles re lso lle frequeny istriution. Crete frequeny tle for the length of wors use in the ove prgrph. Using the frequeny tle, wht is the most frequent wor length in the English lnguge? Comment on the firness of the onlusion me in prt. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

40 0 Chpter Preliminry preprtion J J Men, mein n moe Men The men is mesure of the entre of t istriution. It is lulte y summing ll the sores n iviing this y the numer of sores. For exmple, onsier the sores, 6, 3 n. The men is 6 3 = 3. The men of set of t is wht most people ll the verge. Mein The mein is the mile sore or vlue of t istriution. To fin the mein, list ll the sores in inresing orer n selet the mile one. For exmple, the mein of,, 5, 7, 8 is 5. When there is n even numer of sores, the mein is the verge of the two mile sores. For exmple, the mein of,,, 5, 7, 8 is foun y sorting the six sores n fining the verge of n 5 or.5. Moe The moe is the sore tht ours most often in t istriution. It is the sore with the highest frequeny. The moe is useful for tegoril t tht o not llow you to o numeril lultions, for exmple, if the t ollete is olour. Moes my our t the eginning or en of rnge of vlues. Men Mein Moe Sumofsores Men = Numer of sores x x = n Arrnge ll the sores in inresing orer. Mein is the mile sore. Determine the numer of times eh sore ours. Moe is the sore tht ours the gretest numer of times. Exmple 9: Clulting the men, mein n moe J The tle elow shows the numer of riny ys for the first six months. J F M A M J Fin the men, mein n moe. Write the formul for the men. Sum ll of the sores n ivie y the numer of sores. 3 Evlute. Write the nswer in wors. 5 Write the sores in inresing orer. 6 Count the totl numer of sores. There re 6 sores. 7 Mein is the verge of the 3r (sore 0) n the th sores (sore ). Write the nswer in wors. 8 Moe is the sore tht ours the most. = x x n = = Men is 8, 8, 0,,3,5 Mein = 0 = Mein is. Moe is 8. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

41 J Men, mein n moe Exerise J LEVEL Fin the men of eh t set. 6, 9, 9,5,9,3,7,9,3,0,8,,3, 3, 5, 5, 8,,3, 3, 7, 9,0 7, 5,,3,0,,,9,7 e 6, 5, 6,7,3, 6, 0, 9, f 39,35, 39,, 7,9, g 3, 5,, 9, 3, 3, 3, 9, 3, 8, h 55,,, 8, 3, 3,,,, 7, Fin the men of eh t set. Answer orret to one eiml ple. 3,,5 5, 6, 7, 8, 9 5, 7, 8, 9,3,5 6, 8,,3 e 6, 7,0,, 3,9 f,,,,,, 3, 3, 3, 3 g 9, 9,0,0,0,,,, h 6, 6, 9, 9, 9,0,0,0,0 3 Twenty people mesure their hert rte using hert-rte monitor. The results were 6,68, 6,7, 75,67, 9,80, 77,73, 68,8, 73,7, 60,6, 7,68, 55 n 6. Wht is the sum of these hert rtes? Fin the men hert rte. Answer orret to two eiml ples. Another person with hert rte of 63 is inlue in this t. Wht is the new men? Answer orret to two eiml ples. There is n o numer of sores. Fin the mein of these numers. 3, 9,0 3,, 6, 7, 9,0,5 3, 8, 9, 30, 3, 5, 6, 9, 50 00,00, 00, 0, 60 e,,,,,,, 9, 9 f,5,00,0,0 g 0, 0, 0,,,, 7, 8, 8, 8, 8 h, 7, 9,0,0,,8 5 There is n even numer of sores. Fin the mein of these numers., 5, 6, 8, 8, 9,,8, 0, 0, 0,,, 3,, 5, 9, 0,, 6, 8, 8, 8 e 00,0,30,0 f,,,, 3, 3, 3, 5, 5, 5, 6, 6 g 0,0,,0, 60,6, 70,80 h 3, 5, 8,0,,,8,9 6 The numer of senior itizens entering resturnt in the pst 9 hours ws 8, 7, 6, 7, 9, 3, 0, 6 n 5. Wht is the mein? 7 Fin the moe of eh t set. 3,9,3,,9,3,5,3, 8,,3 5, 8,,3, 3, 7, 9, 5,0 5,7,9,3, 6, 9, 9, 3, 9,0 7,9, 39,, 39,35, Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

42 Chpter Preliminry preprtion J LEVEL 8 Use the stem-n-lef plot opposite to nswer these questions. How mny sores re there? Wht is the lowest sore? Wht is the mein? Remove from the t. Wht is the new mein? Stem Lef Use the ot plot opposite to nswer these questions. Wht is the highest sore? Wht is the lowest sore? 3 Clulte the mein. How mny sores woul nee to e e to mke 0 the mein 3? e Wht is the mein if single sore of 6 is inlue in the t? The tle opposite shows the ges of plyers in the lol footll tem. Wht is the ge of the olest plyer? Wht is the ge of the youngest plyer? Wht is the rnge of ges? Wht is the mein ge? Age Frequeny Nine stuents were surveye on the numer of hours they slept lst night. Their nswers were 0,8,7,7,,7,8,9 n. Fin the men numer of hours slept. Fin the moe numer of hours slept. A tenth stuent ws then surveye n the men hnge to 7.5. Wht ws the numer of hours slept y the tenth stuent? LEVEL 3 The men height of five sketll plyers t the strt of the gme is.9 m. During the gme plyer who is.80 m tll is injure n reple y plyer who is.98 m tll. Wht is the men height of the five plyers now? Answer orret to two eiml ples. 3 A sore ws e to the set of sores: 5,8, 0,, n 6. The new men is equl to 0. Wht sore ws e? Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

43 3Summry Chpter Summry Key ies n hpter summry Simplifying lgeri expressions Fin ommon enomintor if the enomintors re ifferent. Express eh frtion with the ommon enomintor 3 Simplify the numertor y ing or sutrting the like terms. Inex lws m n m n = Multiplying terms in inex form, the inies. m n m n = Diviing terms in inex form, sutrt the inies. m n mn ( ) = Inex form rise to power, multiply the inies. 0 = A term in inex form rise to the power of zero is one. Expning lgeri expressions Pythgors theorem Perimeter Multiply the term outsie the grouping symol y the first term then the seon term insie the grouping symol. Simplify n ollet like terms if require. (Hypotenuse) = (sie) (other sie) Perimeter is the totl length of the outsie eges of shpe. It is the length of the ounry. Are Tringle A = h Squre A = s Retngle A = l Prllelogrm A = h Trpezium A = ( h ) Rhomus A = xy Simple interest I=Prn A = P I Compoun interest A = P( r) n I = A P Frequeny tle Sores re liste in sening orer. Tlly olumn reors the numer of times the sore ourre. 3 Frequeny olumn is ount of eh outome or sore. Men Men = Mein Sumofsores Numer of sores The mein is the mile sore or vlue. Moe The moe is the sore tht ours the most. Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

44 Chpter Preliminry preprtion Review Multiple-hoie questions Simplify 5 3. A 7 B 0 C 3 3 D 3 3 Simplify x 3x. A 7x B 7x C x D x 3 Expn (p 3 q). A 8p q B 8p q C 8p 3q D 8p 3q Wht is the length of the hypotenuse if the two other sies re m n 6 m? A 00 m B 0 m C 8 m D 7.6 m 5 Wht is the perimeter of qurnt with rius of 5mm? A 3.9 mm B 7.9 mm C 3.9 mm D 7.9 mm 6 Wht is the re of tringle with se of 5m n perpeniulr height of 8m? A 3 m B 0 m C 0 m D 80 m 7 Wht is the simple interest on $500 t 8%p.. for yers? A $0 B $60 C $660 D $600 8 Jmes orrows $3000 t 0%p.. interest ompouning nnully. Wht is the mount owe fter yers? (Answer to the nerest ollr.) A $3030 B $3060 C $3600 D $ The frequeny tle shows the results of test. Wht is the most ommon sore? A 3 B C 5 D 6 Sore Tlly Frequeny Wht is the mein of, 0,9,,6, n 3? A B C D 3 Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

45 5Review Chpter Review Short-nswer questions Simplify y olleting like terms x x 6x 6m m m 7m A or sutrt these lgeri frtions. x 5x y y 3 Simplify eh of the following m 3m 36y 3y v 6v n 5 e ( x ) f ( x y ) Remove grouping symols n simplify if require. ( x y ) 6( w 6 m) (6 ) 3 (8 3 ) e 3( y 5) 6( y 6) f ( ) 5 Fin the vlue of x orret to two eiml ples. x m x mm m 8 m mm mm 7 m 7 m x m 6 Fin the perimeter of eh shpe. Answer orret to one eiml ple. 5 m 0 m m m m 7 m 9 m Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press

46 6 Chpter Preliminry preprtion Review 7 Fin the re of eh shpe. Answer orret to one eiml ple. 6 m 3.5 m 9 m 0 m 8. m 5.6 m e mm 7 mm f. m 3 m m 8 Wht is the simple interest on $50 t flt rte of 8%p.. over 3 yers? 9 Riley is investing $0 000 with supernnution fun. How muh will he reeive if the money is investe for yers t the following rules? (Answer to the nerest ollr). 3%p.. ompoun interest 6%p.. ompoun interest 0 Arrnge these sores in orer n fin the mein. 8,, 5,,, 0,,,6 37,6, 3,37, 58,37 Fin the men of eh t set.,3, 9, 9,,6,7,7,,,,3,8,6,,0,8 Extene-response questions Zoe wnts to ern $9000 yer in interest. How muh must she invest if the simple interest rte is %p..? Answer to the nerest ollr. 3 The time (in hours) spent ompleting n ssessment tsk is liste in the tle. Copy n omplete the tle. How mny stuents spent greter thn 3 hours? Wht perentge of stuents spent less thn 9 hours? Clss Frequeny Mthemtis Stnr Yer ISBN Powers 08 Cmrige University Press