EOG REVIEW NOTES Number Systems

Transcription

1 EOG REVIEW NOTES Number Systems Adding Fractins ) Find a cmmn denminatr. (LCM) 2) Cnvert the fractins. (Equivalent Denminatrs) ) Add the numeratrs and keep the denminatr. ) Simplify. Eamples f Adding Fractins Adding Fractin Steps Adding Fractin Eamples Subtracting Fractins ) Find a cmmn denminatr. (LCM) 2) Cnvert the fractins. (Equivalent Denminatrs) ) Subtract the numeratrs and keep the denminatr. Brrw if necessary. ) Subtract the whle numbers. 5) Simplify. Subtracting Fractins with Whle Numbers Subtracting Fractins with Brrwing (Brrw) Subtracting Fractin Steps Subtracting Fractins with Whle Numbers Eample Multiplying Fractins ) Cnvert any mied numbers t imprper fractins. 2) Crss reduce if pssible. ) Multiply straight acrss. ) Simplify Dividing Fractins ) Cnvert any mied numbers t imprper fractins. 2) Keep, Change, Flip (KCF) Keep the first fractin Change the peratin t multiplicatin, Flip the secnd fractin t its reciprcal. ) Fllw multiplicatin rules K C F X 5 K C F Multiplying Fractin Steps Dividing Fractin Steps

2 EOG REVIEW NOTES Number Systems Simplifying Fractins Steps ) Find the GCF f the numeratr & denminatr. 2) Divide each number by the GCF. OR Use the upside dwn cake methd 5 Factrs :, 2,, 8, :, 2,, 5, 0, N D Eample (N) (D) /5 Equivalent Fractins Steps Multiply r divide the numeratr and denminatr by the same number t get an equivalent fractin Equivalent Eample?? 5 5? (5) 5 5 Simplifying Fractins Equivalent Fractins Cnverting Mied Numbers t Imprper Fractins Steps ) Multiply the denminatr and the whle number. 2) Use that answer and add it t the numeratr. ) Keep yur riginal denminatr. () (8) Eamples Cnverting Imprper Fractins t Mied Numbers Steps ) Divide the numeratr by the denminatr. (That's yur whle number) 2) The remainder is yur numeratr. ) Keep yur riginal denminatr. ) Simplify final answer. Eamples r Cnverting Mied Numbers t Imprper Fractins Cnverting Imprper Fractins t Mied Numbers Cnverting Fractins t Decimals ) Divide the numeratr by the denminatr. 0. Eamples: Cnverting Decimals t Fractins ) Read the number prperly. 2) Write eactly hw it sunds. ) Simplify Eamples: Cnverting Fractins t Decimals Cnverting Decimals t Fractins 2

3 EOG REVIEW NOTES Number Systems Adding Decimals. Line up yur decimal pints. 2. Add in 0's as placehlders 2. Add each number.. Drp yur decimal pint straight dwn. Eamples: Subtracting Decimals. Line up yur decimal pints. 2. Add in 0's as placehlders 2. Subtract each number.. Drp yur decimal pint straight dwn. Eamples: Adding Decimals Subtracting Decimals. Line up yur numbers. Multiplying Decimals 2. Multiply withut lking at the decimal pints.. Cunt hw many digits are t the right f each decimal pint.. Mve yu decimal t the left that many times. Eample: (2) () () Ttal Digits t right f decimal 0.5. Frmat yur numbers. Dividing Decimals 2. Mve decimal pint in the divisr and dividend t the right until the divisr is a whle number Divide the numbers Eample:.2. 0 Answer:.. Multiplying Decimals Dividing Decimals Eample Prblem Adding Integers Same Signs. Add the numbers 2. Use the Same Sign Different Signs.Subtract the smaller # frm the larger # 2.Use the sign f the larger number Subtracting Integers.Cpy the st number 2.Change the subtractin sign t additin.change the 2nd numbers sign t its ppsite.fllw the rules fr adding. (Cpy Change Change) Multiplying & Dividing. Multiply r Divide the number withut lking at the signs. 2. If there is an Even number f negatives yur answer is Psitive. If there is an Odd number f negatives yur answer is Negative. Adding and Subtracting Integers Multiplying & Dividing

4 EOG REVIEW NOTES Number Systems Abslute Value the distance frm zer. I I I units I 7 7 units Oppsites Tw numbers that are equal distance frm zer n the number line. Oppsite f is units units Multiplicative Inverse Prperty a a a a Any number multiplied by its reciprcal is always ne. Eample: 2 2 Refleive Prperty a a The sign reflects the same value n bth sides f the equatin. Eamples: Oppsites and Abslute Value Multiplicative Inverse Prperty & Refleive Prperty Distributive Prperty a(b + c) ab + ac Distribute what is utside f the parenthesis by what is inside the parenthesis. Eamples: ( + ) () + () + 2 Additive Inverse Prperty a + a 0 The sum f a number and its ppsite is always zer. Eamples: Distributive Prperty Additive Inverse Prperty Assciative Prperty (a + b) + c a + (b + c) (a b) c a (b c) Cmmutative Prperty a + b b + a a b b a N matter hw the numbers are gruped, the answer will always be the same. Numbers may be added r multiplied tgether in any rder. Eamples: ( + ) ( + 5) ( ) 5 ( 5) Eamples: Assciative Prperty Cmmutative Prperty

5 EOG REVIEW NOTES Epressin and Equatins Distributive Prperty Ntes Different Methds Same Results B Methd Arrw Methd ( + ) ( + ) Remember: Multiply sides Drp #'s & Operatin Multiply () + () + 2 X + X 2 X X X 2, X X X X "Like" terms: all have the same variable, same pwer:,, 2 all are cnstants (numbers):, 2.5, ½, 5 all have the same variable, same pwer: 2y 2, y 2, y 2 Cmbining like terms: used t simplify an epressin r an equatin. circle r b the like terms in each epressin r equatin use the number in frnt f the variable (cefficient) and it s sign t cmbine the term Distributive Prperty Cmbining Like Terms Circle, b, r underline the like terms in each epressin r equatin. Grup the peratin with the term. Use the number in frnt f the variable (cefficient) and it s sign t cmbine the term. Cmbining Like Terms Eample Prblem Multi Step Equatins ) Distribute Eample Prblem Cmpleted ( ) ) Cmbine Like Terms ) Variable t same side 7 2 ) Cmbine 7 2 cnstant terms ) Slve & Check > Inequalities Graph the Variable > Greater Than r equal t > Greater Than Less Than r equal t Less than is less than r equal t is less than > is greater than r equal t is greater than 2 > Multi Step Equatins Eample Graphing Inequalities

6 EOG REVIEW NOTES Epressin and Equatins * Special Rule * If yur variable term is negative yu flip yur inequality sign. > *Helpful Hints* If yur variable is n the left side then yur inequality symbl matches the arrw pint. > 2 2 I I I > I I I > 2 2 Arrw represents values f the variable. Inequalities Helpful Hints Translating Wrds t Equatins Writing wrd statements as inequalities. ) Label the parts f the sentence. 2) Write the inequality. Eample: The sum f a number and 7 is less than Put the peratin where the "and" is. I I I > Inequality Chart Wrds t Math Eample 2

7 EOG REVIEW NOTES Ratis and Prprtins Simple Interest: I Prt I Interest paid (in dllars) P Principal (the amunt f mney brrwed) r rate (change the percent t a decimal) t time (in years) Interest: an amunt that is cllected r paid fr the use f mney Simple Interest: Mney paid n principal Principal: the amunt f mney depsited r brrwed Rate f Interest: percent charged r earned fr the principal What is a Rati? A rati is a cmparisn f tw quantities with the same units. The ways t write a rati. Cln Fractin Bar Wrds 2: 2 2 t Remember that a rati must always be in simplest frm but have tw numbers. Simple Interest Frmula Ratis Types f Ratis. There are three different types f a rati. ) Part t Part White 2, Black 2:, 2 t Black, White 2 :2, t 2 2) Part t Whle White 2, Ttal 0 5 2:5, 2 t 5 Black, Ttal 0 5 :5, t 5 ) Whle t Part Ttal 0 5, White 2 5:2, 5 t 2 Ttal 0 5, Black 5:, 5 t *Hint Unit Rate is hw many units f the first quantity (numeratr) crrespnds t ne unit f the secnd quantity (denminatr). Eamples: $/lb, mi/hr, ft/sec, $/z, mi/gal Jayda takes hurs t deliver 89 newspapers n her paper rute. What is the rate per hur at which she delivers the newspaper? Newspapers Hurs is n bttm because it says per hur. Hurs 89 Answer: Newspapers Hur Types f Ratis Unit Rates If ckies cst $.59, the hw much des it cst per ckie? Wrds Rati Divide Write Final Answer Cst $0.5 per ckie Ckies Hw t determine if each pair f ratis frms a prprtin. Eamples: * Remeber the the wrd "per" shws yu what is being cmpared (Cst per Ckie) Yes 5 8 N Yes 7 2 N 5 Unit Rate Eample Hw t determine if each pair f ratis frms a prprtin.

8 EOG REVIEW NOTES Ratis and Prprtins Hw t set up and slve prprtin wrd prblems. Eample: ckies cst $. If yu have $5, hw many ckies can yu buy? Wrd Statement (What are yu cmparing) Cst Ckies Prprtin 5 Tp numbers represent cst and bttm numbers represent ckies Ckies Hw t slve a prprtin with a missing value. Eamples: 8 n 8 n n 8 n n Crss Multiply Slve One Step Equatin 5 r 8 5 r 8 8r r.75 r Hw t set up and slve prprtin wrd prblems. Hw t slve a prprtin with a missing value. Percent Wrd Prblems Part t Whle Eample: Paul makes a grss salary f $0 each week. If 8% f his salary is withheld fr taes and scial security, hw much is withheld frm his weekly check? Hw much wuld be withheld in a mnth? 8 00 % Part 00 Whle 0 Ta: *amunt f mney added t the ttal cst f a bill * % ta 00 cst Eample: Yur cell phne needs a new battery that will cst $0.00 plus ta. If the sales ta is 7%, hw much will yur ttal be? Weekly: Mnthly: Part t Whle Ta Tip: *amunt f mney added t the cst f a service *calculated befre ta * % tip 00 cst Tip Eample: Yu and yur friend went t lunch. The bill was $5.00. If yu left the server a 5% tip, then hw much tip did yu leave? Discunt: *amunt f mney subtracted frm an item r a ttal * % discunt 00 cst Discunt Eample: Yur favrite stre is having a sale. The item yu buy was riginally $0.00, and it is 0% ff. Hw much is the item? Tip Discunt 2

9 EOG REVIEW NOTES Ratis and Prprtins Mark Up An increase in the cst f an item: T make a prfit, stres have t charge mre fr an item than what they paid fr it (whlesale amunt) Frmula: % 00 Eamples: Find the mark up and retail price f the fllwing items. Whlesale cst f a pen: $0.95 Markup: 0% Mark Up Whlesale Whlesale cst f a cmputer: $, Markup: 75% Whlesale + Mark Up Retail Price Mark Up Cmmissin The amunt f mney that an individual receives based n the level f sales he r she has btained. The sales persn is prvided a certain amunt f mney in additin t his/her standard salary based n the amunt f sales btained. Mark Up While wrking at his Uncle's Car Dealership, Dntae sld this $28,9 2 Mustang and earned a % cmmissin. Frmula: % 00 Cmmissin Whle Cmmissin Percent Change Cmmissin Percent Change Steps Percent f change is the amunt, stated as a percent, that a number increases r decreases.. Subtract t find the difference Percent Change Amunt f Change 00 Original Amunt 2. Divide difference by riginal number Percent Change Largest # Smallest # 00 Original Amunt Percent Change I Difference I 00 Starting Amunt Percent Change amunt f change riginal amunt. Multiply answer by 00 % f change amunt f change riginal amunt X 00 Percent Change

10 EOG REVIEW NOTES Gemetry T slve fr missing measurements in a scale drawing yu must set up a prprtin. ) Write yur wrd statement. 2) Set Up the prprtin ) Crss multiply ) Slve the ne step equatin. 5) Lk at the wrd statement fr yur units. Eample: The scale f a drawing is / 2 in ft Find the measurement fr 8 inches. in / 2 8 ft 2 / 2 / 2 / 2 8 Line segment whse endpints are the center f a circle and any pint n the circle. Radius Line segment whse endpints are any tw pints n a circle. Chrd Diameter Line segment that passes thrugh the center f a Diameter circle and whse endpints lie n the circle. Radius Circumference Part f a circle named Arc Area by its endpints Map Scale Prprtin Parts f a Circle Area and Circumference f Circles Use these t help yu remember Wrking Area and Circumference Backwards Circumference Cherry Pie's Delicius Area Apple Pies are t ) Write the frmula 2) Plug in the infrmatin ) Slve fr the variable ) Answer the questin asked Area and circumference Wrking Area and Circumference Backwards Finding the Area f the Shaded Regin ) Find the ttal area f the figure 2) Find the area f the nn shaded figures ) Find the difference between the areas ) Answer the questin asked Vlume f Prisms Vlume is the amunt f three dimensinal space an bject ccupies. Rectangular Prism V B h B Base Area h height f prism *Remember the base names the figure. Triangular Prism Area f Shaded Regins Vlume f Prisms

11 EOG REVIEW NOTES Gemetry Wrking Backwards with Vlume ) Read the prblem. 2) Write the Frmula fr the answer given in the prblem. ) Substitute the infrmatin given in the prblem. ) Slve fr the missing variable. 5) Make sure yu have answered what the prblem is asking. Surface Area Surface Area is the sum f all the areas f all the shapes that cver the surface f the bject. ) Label all sides. 2) Calculate area f each shape. (Write frmula, plug in numbers, then slve) ) Add all areas tgether. Wrking Backwards Steps Surface Area Fldable Triangles Fldable Types f Triangles Sum f the Interir Angles f a Triangle The sum f the interir angles f a triangle is always 80. A m A + m B + m C 80 O B C Eample: Find the missing value Fldable Sum f the Interir Angles f a Triangle 2

12 EOG REVIEW NOTES Gemetry Relatinship f Eterir Angles with Triangles The sum f the eterir angle and the interir angle is equal t 80 because they make a straight line. m w + m y 80 O Relatinship f Eterir Angles with Triangles The eterir angle must be equal t the sum f the remte interir angles. m w m + m z w y z w y z Relatinship f Eterir Angles with Triangles Relatinship f Eterir Angles with Triangles Acute Angle Right Angle Obtuse Angle Less than 90 Parts f an Angle Verte Cmmn ( ) Endpint Side: Ray Side: Ray Eterir: Outside f angle Interir: Inside f angle Eactly 90 Greater than 90 Naming Angles A C ABC r CBA B * Verte must always be in the middle Types f Angles Cmplementary Tw Angles are Cmplementary when they add up t 90 degrees (a Right Angle). Supplementary Tw Angles are Supplementary when they add up t 80 degrees. Vertical Vertical Angles are the angles ppsite each ther when tw lines crss. They are always equal. Adjacent Tw angles are Adjacent when they have a cmmn side and a cmmn verte (crner pint), and dn't verlap. Quick Review Types f Angles Cmplementary Tw Angles are Cmplementary when they add up t 90 degrees (a Right Angle). A B m A + m B 90 Challenge Questin: B 7 If m A equals 2 + and m B equals, slve fr. m A + m B 90 (2 + ) + ( ) If m A equals, Find m B. m A + B 90 ()+ B 90 Supplementary Tw Angles are Supplementary when they add up t 80 degrees. If m A equals 8, Find m B. m A + m B 80 A B (8)+ m B m A + m B 80 Challenge Questin: B 97 If m A equals + and m B equals 7 +, slve fr. m A + m B 80 ( + ) + (7 + ) Cmplementary Angles Supplementary Angles

13 EOG REVIEW NOTES Gemetry Vertical Vertical Angles are the angles ppsite each ther when tw lines crss. They are always equal. A If m A equals, Find m B. C D B m A m B () m B m A m B & m C m D Challenge Questin: If m A equals + 9 and m B equals 8, slve fr. m A m B ( + 9) (8 ) Adjacent Tw angles are Adjacent when they have a cmmn side and a cmmn verte (crner pint), and dn't verlap. A A B D B C Cmmn Side A A B B Cmmn Verte A is adjacent t D & B B is adjacent t A & C C is adjacent t D & B D is adjacent t A & C Vertical Angles Adjacent Angles

14 EOG REVIEW NOTES Statistics and Prbability Mean The sum f the numbers in a set f data divided by the ttal number f pieces f data. Average r Equal Share Eample: 8, 9, 2, 22 Sum Divide by 80/ Mean Median The number in the middle f a set f data when the data are arranged in rder. If there are tw numbers the median is their average. Eample: 0,,, 5, 9 * Mark ut the ends as pairs Eample # Median Eample #2: 0,,,, 5, 9 Find the average: (+)/2.5 Eample #2 Median.5 Mean Median Mde The number that ccurs mst ften. Range Difference between the maimum and the minimum. Eample: 75, 78, 80, 80, 90, 00, 0 Eample: 8, 9, 2, Mde 80 Range Mde Range Outlier A number in a data set that is significantly smaller r larger than the ther numbers. Variatin A measure f hw spread ut a set f data is. Eample: 55, 80, 75, 90, 85, 95, 00, 00 Eample: 75, 78, 80, 80, 90, 00, 0 Outlier 55 There is a cluster arund A gap is frm The range is. 2 Outlier Variatin

15 EOG REVIEW NOTES Statistics and Prbability Least Value 2,, 5,, 8, 9, 2, 2, 2 Lwer Quartile Median f lwer half Median f entire set Upper Quartile Median f upper half Greatest Value Least Value 2,, 5,, 8, 9, 9, 2, 2, Median f entire set Lwer Quartile Median f lwer half Upper Quartile Median f upper half Greatest Value *If the median falls between tw numbers yu have t average them. (Add and divide by 2) *If the median falls between tw numbers yu have t average them. (Add and divide by 2) Cmpleted B and Whisker Plt Cmpleted B and Whisker Plt Lwer Etreme The smallest number in the data set. st Quartlie The middle number f the st/lwer half f the data. 2nd Quartile The middle number f the data set. Als knw as the. median rd Quartile The middle number f the 2nd/upper half f the data. Upper Etreme The largest number in the data set. Interquartile Range (IQR) The range f the b (between the st and the rd quartiles.) Mean Abslute Deviatin The mean abslute deviatin r MAD f a set f data is the average distance between each data value and the mean. Steps t find the MAD:. Find the mean f the data. 2. Find the distance (abslute value) between each data value and the mean.. Add the distances tgether and divide by the number f data pints. This answer is the MAD. Identifying Infrmatin n a B and Whisker Plt MAD Ntes Eperimental Prbability Favrable Outcme Number f Trials Cnducted What des happen? Eample: Yu tss a die 0 times. Yu recrd the number. Yu want t find the eperimental prbability f getting a. If ccurred times, the prbability is 0 5 Theretical Prbability Favrable Outcme Ttal Pssible Outcmes What shuld happen? Eample: There are numbers n a die. Yu want t find the theretical prbability f getting a. The prbability f rlling a When tssing a die yu shuld get a ne sith f the time. Eperimental Prbability Theretical Prbability 2

16 EOG REVIEW NOTES Statistics and Prbability Determining Outcmes ) Divide the items int grups. 2) Determine hw many items are in each grup. ) Multiply. Fundamental Cunting Principle (FCP) If an event has m pssible utcmes and anther independent event has n pssible utcmes, then there are mn pssible utcmes fr the tw events tgether Outcmes A restaurant ffers dinner specials cnsisting f a main curse, ne vegetable and ne dessert.if there are 2 main curses, vegetables, and 2 desserts, hw many dinner specials are pssible? M V V 2 V M 2 V V 2 V D M V D D 2 M V D 2 D M V 2 D D 2 M V 2 D 2 D M V D D 2 M V D 2 D M 2 V D D 2 M 2 V D 2 D M 2 V 2 D D 2 M 2 V 2 D 2 D M 2 V D D 2 M 2 V D 2 FCP utcmes Outcmes Outcmes Simple Cmpund Simple and Cmpund Events