BEGINNING ALGEBRA (ALGEBRA I)

Transcription

1 /0 BEGINNING ALGEBRA (ALGEBRA I) SAMPLE TEST PLACEMENT EXAMINATION Downlod the omplete Study Pket: Students who hve tken yer of high shool lger or its equivlent with grdes of C or etter re eligile to tke this emintion. There re totl of 0 questions on the emintion. The test is timed for minutes. No lultors re llowed. Smple questions from eh of the nine res elow re on the k of this sheet. Students who reeive stisftory sore my enroll in the following ourses: Mth 0 Intermedite Alger Mth Intermedite Alger for Sttistis Mth Elementry Alger The following topis re overed y the emintion:. Arithmeti. Polynomils. Liner Eqution nd Inequlities. Qudrti Equtions. Grphing. Rtionl Epressions. Eponents nd Squre Roots 8. Geometri Mesurement. Word Prolems

2 Typil questions from eh of the ompeteny res of the Elementry Alger Test. Arithmeti (0.) (A) 0.00 (B) 0.0 (C) 0. (D) 0. (E).. Polynomils One of the ftors of ² - is (A) + (B) + (C) (D) (E). Liner equtions nd Inequlities If 8, then (A) (B) (C) (D) -. Qudrti Equtions Wht re the possile vlues of suh tht ² - 0? (A) - only (B) 0 only (C) only (D) 0 nd (E) (E) - nd only. Grphing On the numer line elow, whih letter est lotes? P Q R S T º I I I I I º 0 (A) P (B) Q (C) R (D) S (E) T. Rtionl Epressions w + w (A) w +. Eponent nd Squre Roots (B) w² (C) w w² (D) w + w² If > 0 then (A) 8 (B) 8 8 (C) (D) (E) 8 (E) w w² 8. Geometri Mesurement A In the right tringle shown to the right, wht is the length of AC? (A) 8 (B) (C) 8 (D) 8 (E) B C. Word Prolem If is to s y is to 8, wht is the vlue of when y? (B) (C) (D) (E) (A) ANSWERS:.B.B.C.D.B.C.B 8.B.C

3 Elementry Alger Dignosti Test Prtie Topi : Arithmeti Opertions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Frtions Simplifying Frtions: Emple: Redue /: (Note tht you must e le to find ommon ftor in this se in oth the top nd ottom in order to redue.) to : Redue:... Equivlent Frtions: Emple: ) / is the equivlent to how mny eighths? to : Complete:??.. 0 Emple: / nd 8/ First find LCM of nd : LCM 0, so 8, nd How to Get the Lowest Common Denomintor (LCD) y finding the lest ommon multiple (LCM) of ll denomintors: to : Find equivlent frtions with the LCD:. nd. nd 8 8. Whih is lrger, / or /? (Hint: find LCD frtions) Adding, Sutrting Frtions: If denomintors re the sme, omine the numertors: Emple: to : Find the sum or differene (redue if possile): ? 8 If denomintors re different, find equivlent frtions with ommon denomintors, then proeed s efore: ) ) to : Simplify:.. Emple: + 8 Multiplying Frtions: multiply the tops, multiply the ottoms, redue if possile. to : Simplify: Dividing Frtions: nie wy to do this is to mke ompound frtion nd then multiply the top nd ottom (of the ig frtion) y the LCD of oth: ) ) 8 to : Simplify: B. Deimls Mening of Ples: in., eh digit position hs vlue ten times the ple to its right. The prt to the left of the point is the whole numer prt. Right of the point, the ples hve vlues: tenths, hundredths, et., so. ( 00) + ( 0) + ( ) + ( /0) + ( /00) + ( /000).. Whih is lrger:. or.? To Add or Sutrt Deimls, like ples must e omined (line up the points). ). 0.. ) ).0 (.) to : Simplify $. $.8 Multiplying Deimls )... )...0 ) (.0) to : Simplify: (.) Dividing Deimls: hnge the prolem to n equivlent whole numer prolem y multiplying oth y the sme power of ten. ) Multiply oth y 00 to get 0 0 ) Multiply oth y 000, get to : Simplify: ) 8 ) C. Positive Integer Eponents nd Squre Roots of Perfet Squres Mening of Eponents (powers): to : Find the vlue: ( ). (.). (). ( 0.) 8... ( ). is non-negtive rel numer if 0 mens, where 0. Thus, euse. Also,. Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00..0

4 Elementry Alger Dignosti Test Prtie Topi : Arithmeti Opertions to : Simplify: D. Frtion-Deiml Conversion Frtion to Deiml: divide the top y the ottom. ). 0 ) ) + + ( ) to : Write eh s deiml. If the deiml repets, show the repeting lok of digits: Non-repeting Deimls to Frtions: Red the numer s frtion, write it s frtion, redue if possile: ) 0. four tenths ). three nd seventy-si hundredths 00 to 8: Write s frtion: E. Perent Mening of Perent: trnslte perent s hundredths : Emple: 8% mens 8 hundredths 8 or.08 or 00 To Chnge Deiml to Perent Form: multiply y 00: move the point ples right nd write the perent symol (%), ) 0.0.% ) ¼. % to 0: Write s perent: To Chnge Perent to Deiml Form, move the point ples left nd drop the % symol. ) 8.% 0.08 ) % 0. to : Write s deiml:. 0%. 0.0% To Solve Perent Prolem whih n e written in this form: % of is. First identify,, : to : If eh sttement were written (with the sme mening) in the form % of is, identify,, nd :. % of 0 is.. 00 is 0% of 00. out of is % Given nd, hnge % to deiml form nd multiply (sine of n e trnslted multiply ). Given nd one of the others, divide y the other (first hnge perent to deiml, or if nswer is, write it s perent). ) Wht is.% of $000? (% of is :.% of $000 is? _).% $000 $0 (nswer) ) prolems right out of 80 is wht perent? (% of is :? % of 80 is ) % (nswer) ) 0 people vote in n eletion, whih is 0% of the registered voters. How mny re registered? (% of is : 0% of? is 0) 0% (nswer) to 8: Find the nswer:. % of is wht?. Wht perent of 0 is? 8. % of wht is 0? F. Estimtion nd Approimtion Rounding to One Signifint Digit: ). rounds to ) 0.0 rounds to 0.0 ) 80 rounds to either 800 or 00 to : Round to one signifint digit To Estimte n Answer, it is often suffiient to round eh given numer to one signifint digit, then ompute. Emple: Round nd ompute: is the estimte. to : Selet the est pproimtion of the nswer: (, 0, 00, 000, 0000) (0.0, 0., 0.,, 0, 0) (,, 8, 0, 00). (.80) (,, 0, 000, 000) Answers. / 8.. /. 8. / /, /. 8/. /, /. 8/ 8. / (euse. 0/8 < /8. /. 0.. not rel #. / /... / /. /. /. 0.. / / /. /00 0. /8. /0 /0. 8/ 8. ¼ /. /. 0% % $ %

5 Elementry Alger Dignosti Test Prtie Topi : Polynomils Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Grouping to Simplify Polynomils The distriutive property sys: ( + ) + ) ( y) y (,, y) ) + ( + ) (,, ) ) + ( + ) to : Rewrite, using the distriutive property.. ( ).. ( ) Commuttive nd ssoitive properties re lso used in regrouping: ) ) ) + y + y + y + y + y to : Simplify y + y y + y B. Evlution y Sustitution ) If, then () ) If nd, then ( ) ( ) ( ) ) If, then ( ) ( ) to : Given, y, z, Find the vlue: 0... z. ( + z). z 8. + z. y + z. z. y + z. + y C. Adding nd Sutrting Polynomils Comine like terms: ) ( + + ) ( ) ) ( ) + ( + ) ) ( + ) ( + ) to : Simplify: 0. ( + ) ( + ). ( ) + ( ). ( ) + ( + ). (y y ) (y y + ). ( ) ( ). ( + ) D. Monomil Times Polynomil Use the distriutive property: ) ( ) + ( ) + ( ) ) ( + ) + ) ( ) + to : Simplify.. ( ). ( ) 8. ( + ). ( ) 0. ( ). ( )( ). 8( + ) E. Multiplying Polynomils Use the distriutive property: ( + ) + Emple: ( + )( ) is ( + ) if: ( + ),, nd So, ( + ) + ( + ) + ( + )( ) + 8 Short ut to multiply ove two inomils: FOIL (do mentlly nd write nswer. F: First times First: ()() O: multiply Outers : ()( ) 8 I: multiply Inners : ()() L: Lst times Lst ()( ) Add, get Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

6 Elementry Alger Dignosti Test Prtie Topi : Polynomils ) ( +)( + ) + + ) ( )( + ) + ) ( )( + ) ) ( ) + ) ( ) ( )( ) + ) ( + )( ) + to : Multiply.. ( + ). ( ). ( + )( ). ( +)( ). ( )( ) 8. ( ). ( ½ ) 0. ( )( + ). ( )( + ) F. Speil Produts These produt ptterns (emples of FOIL) should e rememered nd reognized: I. ( + )( ) II. ( + ) + + III. ( ) + ) ( ) + ) ( + ) ) ( +8)( 8) to : Mth eh pttern with its emple.. I:. II:. III: to : Write the nswer using the pproprite produt pttern:. ( + )( ). (y ). ( + ) 8. ( + )( ). ( )( ) 0. ( y). ( + y). ( +y)( y) G. Ftoring Monomil Ftors: + ( + ) ) ( ) ) y + y y( + ) Differene of Two Squres: ( + )( ) Emple: to : Ftor ompletely: y y y y y Answers: y y y y ( + )( ) Trinomil Squre: + + ( + ) ( ). +. Emple:. + ( ) Trinomil: Emple: ) ( )( + ) ) ( +)( ). + ¼ y y y + y. + y + y. y. ( + ). ( + ). ( + )( ). ( ). (y ) 8. ( )( + ). ( )( + ) 0. y( y). ( )( + ). ( ). ( + ). ( + ). y(y ). ( )( + ). ( )( )

7 Elementry Alger Dignosti Test Prtie Topi : Liner equtions nd inequlities Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Solving One Liner Eqution in One Vrile: Add or sutrt the sme thing on eh side of the eqution, or multiply or divide eh side y the sme thing, with the gol of getting the vrile lone on one side. If there re one or more frtions, it my e desirle to eliminte then y multiplying oth sides y the ommon denomintor. If the eqution is proportion, you my wish to ross-multiply. to : Solve: To solve liner eqution for one vrile in terms of the other(s), do the sme s ove: ) Solve for F : C (F ) Multiply y : C F Add : Thus, F C + C + F ) Solve for : + 0 Sutrt : 0 ) Solve for : + Sutrt : Divide y : to : Solve for the indited vrile in terms of the other(s): y y +. P + h 8. + y 0. y. y 0 y B. Solution of One-Vrile Eqution Reduile to Liner Eqution: some equtions whih don t pper liner n e solved y using relted liner eqution. + ) Multiply y : + Solve: (Be sure to hek nswer in the originl eqution.) + ) + 0 to : Solve nd hek: 0.. Think of s nd ross-multiply: + + But doesn t mke the originl eqution true (it doesn t hek), so there is no solution Emple: to 0: Solve: Sine the solute vlue of oth nd is, n e either or. Write these two equtions nd solve eh: or or Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the. y + y Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

8 Elementry Alger Dignosti Test Prtie Topi : Liner equtions nd inequlities C. Solution of Liner Inequlities Rules for inequlities: If >, then: If <, then: + > + + < + > < > (if > 0) < (if > 0) < (if < 0) > (if < 0) > (if > 0) < (if > 0) < (if < 0) > (if < 0) Emple: One vrile grph: Solve nd grph on numer line: (This is n revition for: {: }) Sutrt, get Divide y, Grph: 0 to 8: Solve nd grph on numer line:. >. <. <. >. +. > +. < 8. + Answers:. /. /. /. /.. / / 0.. / (F h)/. (y + )/. y. (y )/ 8. y/. / 0.. /. /... no solution. {,}. no solution 8. {,}. {/} 0. {, }. > 0. < ½ 0. / 0. > 0. > D. Solving Pir of Liner Equtions in Two Vriles: the solution onsists of n ordered pir, n infinite numer of ordered pirs, or no solution. to : Solve for the ommon solution(s) y sustitution or liner omintions:. + y. y y 8 + y 0. + y. y + y y. y. + y 8 + y. y. y y y 0. <. > (, ) 0. (, ). (8, ). (, ). (8/, /). (/, 0). no solution. Infinitely mny solutions. Any ordered pir of the form (, ), where is ny numer. Emple: (, ).

9 Elementry Alger Dignosti Test Prtie Topi : Qudrti equtions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A : qudrti eqution n lwys e written so it looks like where,, nd re rel numers nd is not zero. ) Add : + Sutrt : 0 + or + 0 ) Rewrite: 0 [Think of + 0 0] So:, 0, to : Write eh of the following in the form + + 0, nd identify,, : B. Ftoring Monomil Ftors: + ( + ) ) ( ) ) y + y y( + ) Differene of Two Squres: ( + )( ) Emple: ( + )( ) to 0: Ftor ompletely: y y y y y C. Solving Ftored Qudrti Equtions: the following sttement is the entrl priniple: If 0, then 0 or 0 First, identify nd in 0 : Emple: ( )( + ) 0 Compre this with 0 ( ) nd ( + ) Trinomil Squre: + + ( + ) + ( ) to : Identify nd in eh of the following:. ( ) 0 Emple: + ( ). ( ) 0 Trinomil: ) ( + )( ) ) ( +)( ). ( )( ) 0. 0 ( )( + ) Then, euse 0 mens 0 or 0, we n use the ftors to mke two liner equtions to solve: Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

10 Elementry Alger Dignosti Test Prtie Topi : Qudrti equtions ) If ( ) 0, then () 0 or ( ) 0 so, 0 or Thus, there re two solutions: 0 nd. ) If ( )( + ) 0, then ( ) 0 or ( + ) 0 so, or ) If ( + ) 0, then + 0 so,, nd. Note: there must e zero on one side of the eqution to solve y the ftoring method. to : Solve:. ( + )( ) 0. ( + ) 0. 0 ( ) 8. 0 ( +)( ). ( )( ) 0 0. ( ) 0. ( + )( ) 0 D. Solve Qudrti Equtions y Ftoring: Arrnge the eqution so zero is on one side (in the form + + 0), ftor, set eh ftor equl to zero, nd solve the resulting liner equtions. ) Solve: Rewrite: 0 Ftor: ( ) 0 So, 0 or ( ) 0 Thus 0 or ½ ) Solve: 0 Ftor: 0 ( )( + ) Then 0 or + 0 So, or to : Solve y ftoring:. ( ) ( + )( ). 0. ( + )( ) ( + ) Another Prolem Form: If prolem is stted in this form: One of the solutions of is d, solve the eqution s ove, then verify the sttement. Emple: One of the solutions of 0 0 is: A. B. ½ C. ½ D. E. Solve 0 0 y ftoring: ( ) 0 So, 0 or 0 Thus, 0 or ½ Sine ½ is one solution, nswer C is orret.. One of the solutions of ( )( + ) 0 is: A. / B. / C. 0 D. / E. /. One solution of 0 is: A. B. C. / D. / E. Answers: ( + ). ( + ) 8. ( + )( ). {0, /}. ( ) 8. { /, } 0. (y ). {}. ( )( + ) 0. {/}. ( )( + ). {, 0, }. y( y). {0, }. ( )( + ). {0, }. ( ). {0, ½}. ( + ). {, 0}. ( + ). {. } 8. y(y ). {, }. ( )( + ) 8. {. } 0. ( )( ). { /, /}. 0. { /, /}.. {}.. {, ½}. +. {, }. {, }. B. {, 0}. B (Note on to : ll signs ould e the opposite)

11 Elementry Alger Dignosti Test Prtie Topi : Grphing Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Grphing Point on the Numer Line to : Selet the letter of the point on the numer line with the given oordinte.. 0. ½... ½.... A B C D E F G 0 8 to 0: Whih letter est lotes the given numer: P Q R S to : Solve eh eqution nd grph the solution on the numer line: B. Grphing Liner Inequlity (in one vrile) on the Numer Line to 0: Solve nd grph on numer line:. > 8. <. <. > > +. < Emple: + 0 Rules for inequlities: If >, then: If <, then: + > + + < + > < > (if > 0) < (if > 0) < (if < 0) > (if < 0) > (if > 0) < (if > 0) < (if < 0) > (if < 0) Emple: One vrile grph: Solve nd grph on numer line: (This is n revition for: {: } ) Sutrt, get Divide y, Grph: 0 T ) > nd < The two numers nd split the numer line into three prts: <, < <, nd >. Chek eh prt to see if oth > nd < re true: > < prt vlues?? oth true? < no yes no < < yes yes yes (solution) > yes no no Thus the solution is < < nd the line grph is: 0 ) or < ( or mens nd/or ) < t lest prt vlues?? one true? yes yes yes (solution) < no yes yes (solution) > no no no So, or < ; these ses re oth overed if <. Thus the solution is < nd the grph is: 0 to : Solve nd grph:. < or >. 0 nd >. > nd C. Grphing Point in the Coordinte Pln If two numer lines interset t right ngles so tht: ) one is horizontl with positive to the right nd negtive to the left, ) the other is vertil with positive up nd negtive down, nd ) the zero points oinide, they they form oordinte plne, nd ) the horizontl numer line is lled the -is, ) the vertil line is the y-is, ) the ommon zero point is the origin, d) there re four y qudrnts, II I numered s shown: III IV To lote point on the plne, n ordered pir of numers is used, written in the form (, y). The -oordinte is lwys given first. to : Identify nd y in eh ordered pir:. (, 0). (, ). (, ). (0, ) To plot point, strt t the origin nd mke the two moves, first in the -diretion (horizontl) nd then in the y-diretion (vertil) indited y the ordered pir. Emple: (, ) Strt t the origin, move left (sine ), then (from there), up (sine y ). Put dot there to indite the point (, ) Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

12 Elementry Alger Dignosti Test Prtie Topi : Grphing 8. Join the following points in the given order: (, ). (, ), (, 0), (, ), (, ), (, 0), (, ), (, ), (, ).. Two of the lines you drw ross eh other. Wht re the oordintes of this rossing point? 0. In wht qudrnt does the point (, ) lie, if > 0 nd < 0? to : For eh given point, whih of it oordintes, or y, is lrger? D. Grphing Liner Equtions on the Coordinte plne: the grph of liner eqution is line, nd one wy to find the line is to join points of the line. Two points determine line, ut three re often plotted on grph to e sure they re olliner (ll in line). Cse I: If the eqution looks like, then there is no restrition on y, so y n e ny numer. Pik numers for vlues of y, nd mke ordered pirs so eh hs. Plot nd join. Emple: Selet three y s, sy, 0, nd. Ordered pirs: (, ), (, 0), (, ) Plot nd join: Note the slope formul 0 gives. ( ) 0 whih is not defined: vertil line hs no slope. Cse II: If the eqution looks like y m +, where either m or (or oth) n e zero, selet ny three numers for vlues of, nd find the orresponding y vlues. Grph (plot) these ordered pirs nd join. Emple: y Selet three s, sy, 0, nd. Sine y must e, the pirs re (, ), (0, ), (, ). The slope formul ( ) 0 gives. 0 0 nd the line is horizontl. Emple: y Selet s, sy 0,, : If 0, y 0 If, y If, y Ordered pirs: (0, ), (, ), (, ) ( ) Note the slope is, 0 And the line is neither horizontl nor vertil. to : Grph eh line on the numer plne nd find its slope (refer to setion E elow if neessry):. y.. y 0. y. y. y + 8. y E. Slope of Line Through Two Points to : Find the vlue of eh of the following:... ( ).. ( ) 0. y y hs slope. / The line joining the points P (, y ) nd P (, y ) 8 to : Find the slope of the line joining the given points: 8. (, ) nd (, ).. (0, ) nd (, ) 0. (, ) nd (, ). Answers:. D 0. IV. E.. C. y. F. y. B.. G.. B 8. Q.. T 0. S... /. / >. < ½. none. Emple: A(, ), B(, ) ( ) Slope of AB. > >. <. ½ 0. >. < or >. / 0. /. >.. <. / 0 y none. 8. /.. / /. /. (0, )

13 Elementry Alger Dignosti Test Prtie Topi : Rtionl Epressions Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk. mth teher y + or y someone else who understnds this topi. A. Simplifying Frtionl Epressions: Emple: ) (note tht you must e le to find ommon ftor in this se in oth the top nd ottom in order to redue frtion.) ) (ommon ftor: ) to : Redue: y. 0. y ).. 8 ( ) y.. y + Emple: to : Simplify: y 0 y. y y. y B. Evlution of Frtions ( )( ) ( )( ) 0 y y y y y y y ( ) Emple: If nd, find the + vlue of. + Sustitute: ( ) to : Find the vlue, given,, 0,, y, z :. 8. y... y y.. 0. z z C. Equivlent Frtions ) is the equivlent to how mny eighths? ) ) )? 8 8? +? + ( + ) How to get the lowest ommon denomintor (LCD) y finding the lest ommon multiple (LCM) of ll denomintors. 8 ) nd : First find the LCM of nd : ) nd : 0) to : Solve nd nd hek: LCM +, so + 0. ( ). to : Complete: ( + )( ) ( )( ) + ( )( + ) ( )( + ) LCM 0, so +.?..?. y +? to : Find equivlent frtions with the lowest ommon denomintor: 8. nd.. nd. 0. nd. +? LCM, so, nd + ( )( ) ( + )( ) ( )( ), 8, nd 0 0 nd 0? ( + ) ( + )( )? ( + )( ) nd nd + nd + Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

14 Elementry Alger Dignosti Test Prtie Topi : Rtionl Epressions D. Adding nd Sutrting Frtions: If denomintors re the sme, omine the numertors: to 8: Find the sum or differene s indited (redue if possile): + y y If denomintors re different, find equivlent frtions with ommon denomintors, then proeed s efore (omine numertors): ) ) ( + ) ( )( + ) ( ) ( )( + ) + ( )( + ) ( )( + ) to : Find the sum or differene E. Multiplying Frtions: Multiply the tops, multiply the ottoms, redue if possile: ) Emple: 0 0 y y y y y ( + ) ( + )( + )( ) + ( )( + )( ) ) to : Multiply, redue if possile ( + ) y 8. d y F. Dividing Frtions: A nie wy to do this is to mke ompound frtion nd then multiply the top nd ottom (of the ig frtion) y the LCD of oth: ) ) ) d 0 to : Simplify: d y y d d d d y y y y y ( ).. ( ) Answers:. ¼.. d. / 8.,.,... ( + ). 0.,. ( + ) ( + ) 8. /.,. y ( + ) ( ). y., ( + )( ) ( + )( ) + + y.. + ( + ), ( + ) /. / ( + ).. 0. / / + ( ).. /. /8 ( + ) / / / 8... /. / none ( ).. ( + )( ). y ¼

15 Elementry Alger Dignosti Test Prtie Topi : Eponents nd squre roots Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi.. 8. (0.).. (.) + ( ) I. IV. II. A. Positive Integer Eponents mens use s ftor times. ( is the eponent or power of.) to : Find the vlue..... ( )... ) mens, nd hs vlue of. ) ( ). Emple: Simplify: to 8: Simplify:... ( )( )( ) 8. ( y) B. Integer Eponents V. III. ( ) VI. 0 (if 0 ) VII. to 8: Find : ( ) to : Find the vlue: ( + ). ( ) 0. ( ). ( y ) C. Sientifi Nottion y y ( ) ) if the zeros in the ten s nd one s ples re signifint. If the one s zero is not, write.80 0, if neither is signifint:.8 0 ) ) 0 00 ) Note tht sientifi form lwys looks like 0 n where < 0, nd n is n integer power of 0. to : Write in sientifi nottion:.,000, to 8: Write in stndrd nottion: To ompute with numers written in sientifi form, seprte the prts, ompute, then reomine. ) (. 0 )() (.)() ) ) to : Write nswer in sientifi nottion: ( 0 ). (. 0 ) (. 0 )(. 0 ) 8. 0 Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

16 Elementry Alger Dignosti Test Prtie Topi : Eponents nd squre roots D. Simplifition of Squre Roots if nd re oth non-negtive ( 0 nd 0 ). ) ) ) If 0, If < 0, 8 0 Note: mens (y definition) tht ), nd ) to : Simplify (ssume ll squre roots re rel numers): E. Adding nd Sutrting Squre Roots 0 to : Simplify: ) + ) F. Multiplying Squre Roots ) ) ) to : Simplify: to 8: Find the vlue of : G. Dividing Squre Roots Emple: ) ), if 0 nd > to 8: Simplify: if 0 nd 0. ( )( ) ( )( ) ( ) ( ) ( ) (or ) 8 If frtion hs squre root on the ottom, it is sometimes desirle to find n equivlent frtion with no root on the ottom. This is lled rtionlizing the denomintor to : Simplify: Answers: /8. / / / y y / / /. 8. / /. 8 y / /

17 Elementry Alger Dignosti Test Prtie Topi 8: Geometri Mesurement Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Interseting lines nd Prllels: If two lines interset s shown, d djent ngles dd to 80 o. For emple, + d 80 o. Non-djent ngles re equl: For emple,. If two lines, nd, re prllel nd re ut y third line, forming ngles w,, y, z s shown, then z, w z, w + y 80 o so z y 80 o. w Emple: If nd, find the mesure of., so, + 80 o, so + 80, giving 80, or. Thus o. to : Given o. Find the mesures of the other ngles: y z t y z w. t. z. Find :. y. w B. Formuls for perimeter P nd re A of tringles, squres, retngles, nd prllelogrms Retngle, se, ltitude (height) h: P + : h A h If wire is ent in shpe, the perimeter is the length of the wire, nd the re is the numer of squre units enlosed y the wire. Emple: Retngle with nd h 8: P + h units A h 8 sq. units A squre is retngle with ll sides equl, so the formuls re the sme (nd simpler if the side length is ): P s A s Emple: Squre with side m hs P s m A s m (sq. m) A prllelogrm with se nd height h hs A h If the other side length h is, then P + Emple: Prllelogrm hs sides nd, nd is the length of the ltitude P + + h h A h Emple: P units.8 0 A ½ h ½ (0)(.8) sq units ) A irle with rdius r 0 hs d r 0 nd et irumferene C πr π 0 0π units ) If π is pproimted y. C 0π 0( ) 0 units pproimtely. ) If π is pproimted y., the pproimte C 0(.) units perpendiulr to the side. P units A h 0 sq. units In tringle with side lengths,, nd h is the ltitude to side, to : Find P nd A for eh of the following figures:. Retngle with sides nd 0.. Retngle, sides. nd. 8. Squre with side mi.. Squre, side yd. 0. Prllelogrm with sides nd, nd height 0 (on side ). Emple: If r 8, then A πr π 8 π sq. units Emple: A o with dimensions, nd hs wht volume? V lwh u. units Emple: A ue hs edge m. V e m (u. m) Emple: A ylinder hs r 0 nd h. The et volume is V πr h π 0 00π u. units If π is pproimted y, V u. units If π is pproimted y., V 00(.) u. units. Prllelogrm, ll sides, ltitude. Α sphere (ll) with. Tringle with sides,,, rdius r hs volume r nd is the height r on side. V π. The tringle shown: Emple: The et volume of sphere with rdius in. is V πr π π() 88π in C. Formuls for Cirle Are A nd Cirumferene C A irle with rdius r (nd dimeter d r ) hs distne round (irumferene) r C πd or C πr (If piee of wire is ent into irulr shpe, the irumferene is the length of wire.) The re of irle is A πr. to : Find C nd A for eh irle:. r units. r 0 feet. d km D. Formuls for Volume V A retngulr solid (o) with length l, width w, nd height h, hs volume V lwh. A ue is o with ll edges equl. If the edge is e, the volume V e. A (right irulr) ylinder with rdius r nd ltitude h hs V πr h. to : Find the et volume of eh of the following solids:. Bo, y 8 y. 8. Bo, y y. Cue with edge Cue, edge 0.. Cylinder with r, h 0.. Cylinder, r, h.. Sphere with rdius r.. Sphere with rdius r. E. Sum of the Interior Angles of Tringle: the three ngles of ny tringle dd to 80 o. Emple: Find the mesures of ngles C nd A : C (ngle C) is mrked to show its mesure is 0 o. B + C + 0, so A 80 o. Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00. l e w r h h

18 Elementry Alger Dignosti Test Prtie Topi 8: Geometri Mesurement to : Given two ngles of tringle, find the mesure of the third ngle:. 0 o, 0 o. o, o 8. 8 o, 8 o. 0 o, o. 8 o, o F. Isoseles Tringles An isoseles tringle is defined to hve t lest two sides with equl mesure. The equl sides my e mrked: or the mesures my e given: 0 to : Is the tringle isoseles? 0. Sides,,.. Sides,,.. Sides 8, 8, 8. The ngles whih re opposite the equl sides lso hve equl mesures (nd ll three ngles dd to 80 o ). Emple: Find the C B mesure of A nd C, given B o : A + B + C 80, nd A A B, so C 0 o.. Find mesure of A A nd B, C if C 0 o. B. Find mesure of B B nd C, If A 0 o. A C C 8. Find mesure of A. 8 8 A 8 B. If the ngles of tringle re 0 o, 0 o, nd 0 o, n it e isoseles? 0. If two ngles of tringle re o nd 0 o, n it e isoseles? If tringle hs equl ngles, the sides opposite these ngles lso hve equl mesures. Emple: Find the A B mesures of B, AB nd AC, given this 0 o figure, nd C 0 o ; C B 0 o (euse ll ngles dd to 80 o ). Sine A B, AC AB. AB n e found with trig---lter.. Cn tringle e isoseles nd hve 0 o ngle?. Given D E 8 o D nd DF. Find the mesure of F nd length of FE: E F G. Similr tringles: If two ngles of one tringle re equl to two ngles of nother tringle, then the tringles re similr. Emple: ABC nd FED re similr: C The pirs of orrespond- A B F E ing sides re AB nd FE, BC nd BD, nd AC nd FD.. Nme two similr D tringles nd list E the pirs of orresponding sides. A B C If two tringles re similr, ny two orresponding sides hve the sme rtio (frtion vlue): Emple: The y rtio to, or, z is the sme s y nd. Thus,, z y z, nd y z. Eh of these equtions is lled proportion. to : Write proportions for the two similr tringles: 0.. d 8 f e Emple: Find. Write nd solve proportion:, so, to : Find Find nd y : 0 y H. Pythgoren theorem In ny tringle with 0 o (right) ngle, the sum of the squres of the legs equls the squre of the hypotenuse. (The legs re the two shorter sides; the hypotenuse is the longest side.) If the legs hve lengths nd, nd the hypotenuse length is, then + (In words, In right tringle, leg squred plus leg squred equls hypotenuse squred. ) D Emple: A right tringle hs hypotenuse nd one leg. Find the other leg. Sine leg + leg hypotenuse, + + to : Eh line of the hrt lists two sides of right tringle. Find the length of the third side: leg leg hyp to : Find... If the sum of the squres of two sides of tringle is the sme s the squre of the third side, the tringle is right tringle. Emple: Is tringle with sides 0,, right tringle? 0 +, so it is right tringle. to : Is tringle right, if it hs sides:., 8. 0,, 8.,, Answers. o. yes. o. yes. o. n t tell.. o eh. o. 0 o, 0 o P A 8. 0 o. 0 un. 0 un. no. un. un 0. no 8. mi mi. yes:. yd, / yd.. o, 0. 0 u. 0 un. ABE, ACD. 8 un. un AB, AC. 0 un. 0 un AE, AD. un. un BE, CD C A. 0π un. π un.. 0π ft. 00π ft. π km π km. d f + f + e.. / 8. 0/. / / /. 0π 0. 0/, /. π. 8. π/.. π/.. 0 o. o.. o 8. o.. 8 o 0. no.. yes. yes. yes 8. no. yes

19 Elementry Alger Dignosti Test Prtie Topi : Word Prolems Diretions: Study the emples, work the prolems, then hek your nswers on the k of this sheet. If you don t get the nswer given, hek your work nd look for mistkes. If you hve troule, sk mth teher or someone else who understnds this topi. A. Arithmeti, perent, nd verge:. Wht is the numer, whih when multiplied y, gives?. If you squre ertin numer, you get. Wht is the numer?. Wht is the power of tht gives?. Find % of.. is wht perent of 88?. Wht perent of is 88?. is 80% of wht numer? 8. Wht is 8.% of $000?. If you get on 0-question test, wht perent is this? 0. The 00 people who vote in n eletion re 0% of the people registered to vote. How mny re registered? to : Your wge is inresed y 0%, then the new mount is ut y 0% (of the new mount).. Will this result in wge whih is higher thn, lower thn, or the sme s the originl wge?. Wht perent of the originl wge is this finl wge?. If the ove steps were reversed (0% ut followed y 0% inrese), the finl wge would e wht perent of the originl wge? to : If A is inresed y %, it equls B.. Whih is lrger, B or the originl A?. B is wht perent of A?. A is wht perent of B?. Wht is the verge of 8,, 8,, nd? 8. If two test sores re 8 nd 0, wht minimum sore on the net test would e needed for n overll verge of 80?. The verge height of people is 8 inhes. Wht is the new verge height if 8-inh person joins the group? B. Algeri Sustitution nd Evlution 0 to : A ertin TV uses wtts of power, nd opertes on 0 volts. 0. Find how mny mps of urrent it uses, from the reltionship: volts times mps equls wtts wtts kilowtt (kw). How mny kilowtts does the TV use?. Kw times hours kilowtt-hours (kwh). If the TV is on for si hours dy, how mny kwh of eletriity re used?. If the set is on for si hours every dy of 0-dy month, how mny kwh re used for the month?. If the eletri ompny hrges 8 per kwh, wht mount of the month s ill is for TV power? to : A plne hs ertin speed in still ir, where it goes 0 miles in three hours.. Wht is its (still ir) speed?. How fr does the plne go in hours?. How fr does it go in hours? 8. How long does it tke to fly 000 miles?. How long does it tke to fly y miles? 0. If the plne flies ginst 0 mph hedwind, wht is its ground speed?. If the plne flies ginst hedwind of z mph, wht is its ground speed?. If it hs fuel for. hours of flying time, how fr n it go ginst the hedwind of 0 mph.. If the plne hs fuel for t hours of flying time, how fr n it go ginst the hedwind of z mph? C. Rtio nd proportion: to : is to y s is to.. Find y when is.. Find when y is. to : s is proportionl to P, nd P when s.. Find s when P.. Find P when s. 8 to : Given y. 8. Write the rtio :y s the rtio of two integers.. If, find y. 0 to : nd y re numers, nd two s equl three y s. 0. Whih of or y is the lrger?. Wht is the rtio of to y? to : Hlf of is the sme s one-third of y.. Whih of nd y must e lrger?. Write the rtio :y s the rtio of two integers.. How mny s equl 0 y s? D. Prolems Leding to One Liner Eqution. is three-fourths of wht numer?. Wht numer is ¾ of?. Wht frtion of is? Copyright 8, Ron Smith/Bishop Union High Shool, Bishop, CA Permission grnted to opy for lssroom use only. Not to e sold. One of series of worksheets designed to provide remedil prtie. Coordinted with topis on dignosti tests supplied to the Mthemtis Dignosti Testing Projet, Gyley Center Suite 0, UCLA, 0 Hilgrd Ave., Los Angeles, CA 00.

20 Elementry Alger Dignosti Test Prtie Topi : Word Prolems 8. / of / of / of numer is. Wht is the numer?. Hlf the squre of numer is 8. Wht is the numer? 0. 8 is the squre of twie wht numer?. Given positive numer. Two times positive numer y is t lest four times. How smll n y e?. Twie the squre root of hlf of numer is. Wht is the numer? to : A gthering hs twie s mny women s men. W is the numer of women nd M is the numer of men.. Whih Is orret: M W or M W?. If there re women, how mny men re there?. If the totl numer of men nd women present is, how mny of eh re there?. $,000 is divided into equl shres. Bs gets four shres, Bill gets three shres, nd Ben gets the one remining shre. Wht is the vlue of one shre? E. Prolems Leding to Two Liner Equtions. Two siene fition oins hve vlues nd y. Three s nd five y s hve vlue of, nd one nd two y s hve vlue of. Wht is the vlue of eh? 8. In miing gm of % nd y gm of 8% solutions to get 0 gm of % solution, these equtions re used: y 0.0(0), nd + y 0. How mny gm of % solution re needed? F. Geometry. Point X is on eh of two given interseting lines. How mny suh points X re there? 0. On the numer line, points P nd Q re two units prt. Q hs oordinte. Wht re the possile oordintes of P? to :. If the length of hord AB is X nd the length of CB is B, wht is AC? A C. If AC y nd CB z, how O long is AB (in terms of y nd z )? to : The se of retngle is three times the height. Find the height if the se is 0.. Find the perimeter nd re.. In order to onstrut squre with n re whih is 00 times the re of given squre, how long side should e use? to : The length of retngle is inresed y % nd its width is deresed y 0%.. Its new re is wht perent of its old re?. By wht perent hs the old re inresed or deresed? 8. The length of retngle is twie the width. If oth dimensions re inresed y m, the resulting retngle hs 8 m more re. Wht ws the originl width?. After retngulr piee of knitted fri shrinks in length one m nd strethes in width m, it is squre. If the originl re ws 0 m, wht is the squre re? 0. This squre is ut into two smller squres nd two non-squre retngles s shown. Before eing ut, the lrge squre hd re ( + ). The two smller squres hve res nd. Find the totl re of the two non-squre retngles. Show tht the res of the prts dd up to the re of the originl squre. Answers:.. / :..08. y..%. :. 0% $.. 0%. / lower.. % 0. /. sme (%).. B.. %. M W. 80%... 8 men 8. women. 8. $ mps. :. 0 0 kw y:. 0 kwh 8. gm. kwh.. $.08 0., +. 0 mph.. 0 miles. y + z. 0 miles. 0/ 8. 0/ hr.. P 0/. y/0 hr. A 00/ mph. 0 times the. 0 z mph originl size. 000 mi.. %. (0 z)t mi.. % derese. / 8. 0/. / : ( + )