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1 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin with the id of spectcle lens Let s Lern to epress n lgebric frction in its simplest form perform the four opertions (,,, ) on lgebric frctions solve frctionl equtions tht cn be reduced to qundrtic equtions 4 solve everydy problems involving lgebric frctions 5 find the vlue of n unknown quntity in given formul 6 chnge the subject of formul A humn eye uses lens to form the imge of n object on the retin. A shortsighted person hs to use concve lens to correct his or her vision. We pply the lens formul u v to find the required focl f length f of lens to correct shortsightedness, where u nd v denote the object distnce nd imge distnce from the lens respectively.
2 5. Simplifying Simple Algebric Frctions We know tht number of the form p, where p nd q re integers, nd q 0, q is clled frction. For emple,, 7 5, nd 9 6 re frctions. An lgebric epression of the form P, where P nd Q re epressions Q involving ddition, subtrction, nd/or multipliction nd Q 0, is clled n lgebric frction. For emple, b c, m n, 4 pq 5rs, nd re lgebric frctions. ( ) ( ) y z It must be emphsized tht the vlue of the denomintor of n lgebric frction cnnot be zero. This is becuse division by zero is undefined. Therefore, in the bove emples of lgebric frctions, c 0, n, or, nd y z. Therefore, in this chpter, we will tke the vlue of ll denomintors to be nonzero. MATH WEB You cn see more emples of nd prctice with lgebric frctions t the website regentsprep.org/regents/ mth/algebra/av5/indeav5. htm. If R 0, we hve R R, nd hence, P Q R R P Q R R P Q nd P R Q R P Q R R P Q. We see tht the vlue of n lgebric frction is unchnged when both its numertor nd denomintor re multiplied or divided by the sme nonzero number or epression. Discuss Is b c for c 0? b c Emple Solution Epress ech of the following in its simplest form. b 5b 8 6 8t b 5b ( b )( 4 ) ( b)( 5b ) 4 5b Fctorize the numertor nd the denomintor. b b 8 6 ( )( 4 ) 8t ( )( 9t) 4 9t RECALL The simplest form of frction (lso clled lowest term) is one where the numertor nd denomintor hve no fctors in common, other thn. Chpter 5 simple lgebric frctions 05
3 Try It! Simplify ech of the following lgebric frctions. 8y 0y 5 5mn 5m 5mn Emple Simplify ech of the following lgebric frctions. 6 y b by 0 8 y 5 b 4by 5 9 Solution 6 y b by 0 8 y 5 b 4by ( y) b( y) ( 5 4 y) b( 5 4y) ( b )( y ) ( b)( 5 4y) y 5 4y Fctorize by grouping terms. 5 9 ( 5)( ) ( )( ) ( 5)( ) ( ) ( ) 5 Using b ( b)( b), 9 ( )( ). ( ) b b REMARKS In generl, y ( y). Try It! Simplify ech of the following. p 5 py q 0qy p py 4 q qy
4 5. Addition nd Subtrction of Algebric Frctions In Grde 7, we lerned to simplify lgebric epressions such s, where the denomintor of ech lgebric frction is n integer. 4 5 Recll tht to dd these lgebric frctions, we cn find the LCM of the denomintors so tht both frctions hve common denomintor s shown below. 5 ( ) 4( ) LCM of 4 nd 5 is Similrly, when we dd or subtrct lgebric frctions with denomintors tht involve vribles, we cn find the LCM of the denomintors first. The LCM of two lgebric epressions is n epression tht is the common multiple of the given epressions with the lest number of fctors. RECALL The lest common multiple (LCM) of two or more numbers is the smllest number tht is multiple of the given numbers. Consider the epressions 8 b nd b. 8 b b b b their LCM b the smllest product tht contins every fctor 4 b of the given epressions For ( ) ( ), ( ) ( ), nd ( )( )( ), their LCM is 6( ) ( )( ). Emple 5 Epress ech of the following s single frction in its simplest form. 7 8n n y y 4 Solution LCM of 8n nd n is 8n. 7 8n n 7 n 8n 8 n 7 n 8 8n 8 LCM of y nd 4 is y. y y 4 4 y y y 4 y y Chpter 5 simple lgebric frctions
5 WORK. A group of people hired bot for $,00 nd ech person pid n equl mount for it. Five people did not turn up, so ech person on the trip hd to py $8 more. Find the number of people in the group who went on the trip.. The numertor of positive frction is less thn its denomintor. When both the numertor nd the denomintor re incresed by, the new frction is greter thn the originl one by. Find the 8 originl frction. BRAIN WORKS. A store owner hs 80 copies of computer gme. Bsed on pst eperience, if the price is set t $60 per copy, ll of them will be sold. For ech $5 increse in price, n dditionl 4 copies of the gme will be unsold. If the price is set t $70 per copy, (i) find the number of copies tht will be sold, (ii) find the totl sles mount. If the totl sles mount is $5,00, find the possible price per copy of the gme. (c) Wht should the price per copy be in order to get the mimum totl sles mount? [Assume tht the price per copy of the gme is multiple of 5 in nd (c).] 5.5 More bout Formuls A Finding the Vlue of n Unknown Quntity in Formul In Grde 7, we lerned tht formul is n eqution tht reltes two or more quntities. Here re two emples of formuls.. The perimeter P cm of rectngle of dimensions cm by y cm is given by the formul, P ( y). y. The volume V cm of rectngulr prism with squre bse of side cm nd height h cm is given by the formul, V h. h Chpter 5 simple lgebric frctions 9
6 B Chnging the Subject of Formul Do you know wht the subject of the formul A ( b)h is? I know! It s the letter A. The subject of formul is single term tht ppers only on the lefthnd side of formul. We cn find the vlue of subject, in the bove cse, A, by substituting the vlues of, b, nd h in the formul. However, if we wnt to find the vlue of b, given the other vlues in the formul, then we hve to mnipulte the terms. Alterntively, we cn rewrite the formul by epressing b s the subject which is epressing b in terms of A,, nd h. This process is clled chnging the subject of formul. Emple 4 Given the formul d (u v)t, epress v s the subject of the formul, find the vlue of v when t 4, u 54, nd d 0. Solution d (u v)t d (u v)t Multiply both sides by. d t u v Divide both sides by t. v d t u Subtrct u from both sides. When t 4, u 54, nd d 0, we hve 0 v Discuss Compre the solution for Emple 4 with tht of Emple. Wht do you notice? Try It! 4 Given the formul s ut t, epress u s the subject of the formul, find the vlue of u when 6, s 0, nd t 7.
7 In Nutshell Simple Algebric Frctions Algebric epressions such s, b c 9, nd pq r re lgebric frctions, where the vlues of the denomintors cnnot be zero. An lgebric frction cn be simplified if there is common fctor in its numertor nd its denomintor. e.g., 9 ( )( ) ( )( ), where Formuls An eqution connecting or relting vribles is formul. The vlue of vrible in formul cn be found by substitution. The vrible tht ppers s single term only on the lefthnd side of formul is clled the subject of the formul. The subject of formul cn be chnged using lgebric mnipultions. Multipliction nd Division of Algebric Frctions P Q R S P Q R S P Q R S P Q S R P S Q R Addition nd Subtrction of Algebric Frctions Find the LCM of the denomintors of the lgebric frctions in n epression. Convert ech frction in the given epression to n equivlent frction with the LCM s the denomintor. e.g., b y 4 6y y b y y b, y where 0, y 0. ( )( ) ( ) ( ) 5, ( ) ( ) where,. Frctionl Equtions p b q c b ( p)( q) p q c ( p)( q) ( q) b( p) c ( p)( q) Simplify frctionl eqution by multiplying both of the eqution by their common denomintor. Check nd reject ny derived roots tht cuse the divisionbyzero error in the originl eqution. 6
8 REVIEW EXERCISE 5. Simplify ech of the following lgebric frctions. 8u 0u 5 u 9 c bc 9 6b (c) (d) k 6 k 7 k k 9 7. Simplify ech of the following. 6 y y 6 4 b 8 b 8 (c) t 9 t 4 t 7 4t t (d) p p 4q q p q. Simplify ech of the following s single frction in its simplest form (c) (d) 6 ( 6) Simplify ech of the following s single frction in its simplest form. y 4 49 y y 5. Simplify t 6 t 9. Hence simplify 6 t t 9 (t 0t ). 6. Solve ech of the following equtions. 4 8 (c) (d) 7 7. Mke the letter in the brckets the subject for ech of the given formul. y 5 (h k) [k] z u v u 5v [v] (c) p [p] (d) H 4r m n [m] 8. The cost $C of mking circulr brooch of rdius r cm is given by the formul, C 5 8r 00 n, where n is the number of brooches mde. If 00 brooches of rdius cm re mde, find the cost of mking ech brooch. Epress r s the subject of the formul. (c) If 400 brooches re mde nd the cost of mking ech brooch is $55.50, find the rdius of ech brooch. 9. A formul is given by y c, where c 0 c. Mke the subject of the formul. When c 5 nd y 4, find the vlue of, rounded to significnt figures. Chpter 5 simple lgebric frctions 7
9 0. Jim nd Peter ech ern $76 week from their prttime jobs. Jim is pid $ per hour nd Peter gets $ per hour more thn Jim. Write down, in terms of, the number of hours tht Jim nd Peter ech work in week. Given lso tht Jim works 6 hours more per week thn Peter, form n eqution in nd find the vlue of. (c) Find the number of hours tht Jim works in week.. Two good friends, Betty nd Susn, ech rn the 4 km of mrthon rce. Susn rn the rce t n verge speed which ws km/hr less thn Betty s verge speed. Given tht the difference between the times tken by the two girls to complete the rce ws minutes, form n eqution to represent this sitution, find, in hours nd minutes, the time Betty took to complete the rce.. The denomintor of frction is more thn times of its numertor. If is subtrcted from the numertor nd 5 is subtrcted from the denomintor, the new frction is equivlent to. Find the originl frction. 5 Comple Frctions A comple frction is frction in which the numertor or the denomintor or both re frctions. The following re three emples of comple frctions. Only one of them is simplified correctly. Identify the correct one nd point out the mistkes mde in the other two.. b c c d b d d bc bd d bc cd.. y y y 4 y y y y b c bd cd 4 y y ( y )( y ) y y Write in your journl. Eplin wht you would do to dd or subtrct lgebric frctions with qudrtic epressions in the numertor nd/or denomintor.. Discuss why it is necessry to chnge the subject of formul. 8
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