SAT Math Notes. By Steve Baba, Ph.D FREE for individual or classroom use. Not free for commercial or online use.

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1 SAT Math Note B Steve Baba, Ph.D FREE for individual or claroom ue. Not free for commercial or online ue. For SAT reading ee m ite: for a free lit of 5000 SAT word with brief definition. Integer Poitive & negative whole number and ZERO. -3, -2, -1, 0, 1, 2, 3 Negative Number Left of zero on number line. Smaller Larger i GREATER than 4-1/4 i GREATER than 1/2 Order of Operation PEMDAS (Pleae ecue m dear aunt Sall) Parenthei Eponent Multiplication/Diviion left to right Addition/Subtraction left to right 3 2! (3) 2 = (3)(3) = 9 2 Becaue a+b = b+a and a b = b a, don t worr about order of addition or multiplication, but becaue a b! b a, and a b! b a watch ubtraction and diviion order in trick word problem. Odd/Even Operation There are rule: Odd number + Even number = Odd number ALWAYS. Odd + Odd = Even Even + Even = Even But it eaier to remember b uing an even or odd number = 5 (odd number) = 4 (even number) = 4 (even number) SAME IDEA, but not ame reult for multiplication: 3 2 = 6 (even number) 3 1 = 3 (odd number) 2 2 = 4 (even number) SAT often combine everal of the above rule: (odd+odd+even) odd Ue an even and an odd number to determine if reult i alwa even or odd: ( ) 3 = 24 (even) Multipling Poitive and Negative Number a b c d All Poitive Alwa Poitive All Negative i not alwa negative ince two or an EVEN number of negative number CANCEL each other negativit out. If ALL a, b, c, and d are negative, the product i poitive i POSITIVE ONE Negative number or an other ODD number of negative Negative Dividing i the ame a multiplication. The SAT often ha thee poitive/negative quetion backward. If the reult of a b c d i negative then? (one OR three of a, b, c, d i negative) Prime Number A number diviible b ONLY itelf and 1. Prime number: 2 (the onl EVEN prime number) 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,.. 1 i NOT a prime number Prime Factor (Tree) Factor 100: All Factor Tree give the ame prime factor, but NOT all factor. 100 can alo be factored a: giving the ame prime factor a above, but mied the nonprime factor 25 and 50. Both tree mied 4 and 20. Find ALL (nonprime) factor b multipling prime factor. 2 2 = 4 and = 20 and 5 5 = 25 and = 50 Or ue brute force and divide 100 b 2,3,4,5,6,7,8,9,then 10. (11 and higher i covered b checking 9 and lower) Leat Common Multiple (LCM) LCM of 10 and 12: = 120, a multiple (good enough for adding fraction) but not necearil the leat. Lit multiple of each: 10, 20, 30, 40, 50, 60, 70 12, 24, 36, 48, i Leat Common Multiple. On multiple-choice quetion, LCM can be found b working backward from anwer: a) 120 b) 80 c) 60 d) 36 e)10 b dividing each anwer b 10 and 12 and chooing the leat. Greatet Common Factor (of 75 and 100) Find ALL (including nonprime) factor of both. 75: 3, 5, 15, : 2, 4, 5, 10, 20, 25, 50 OR find the prime factor the have in common and multipl: 5 5 (both 75 and 100 have TWO 5 in factor tree) OR on multiple choice quetion work backward from anwer. a) 75 b) 50 c) 30 d) 25 e) 5 Onl 25 and 5 are factor of 75 and 100, and 25 i larger. Between v. Including And other tricking wording of between or including (incluive, counting the firt..) Integer BETWEEN 2 and + 2 (-1,0, 1) i not the ame a integer > -2 and < 2 (-2, -1, 0, 1, 2), which include 2 and 2. Fraction, Adding/Subtracting Common denominator (bottom) needed = + = OR can be done on calculator (one divided b 4 ), but if anwer are in fraction, it eaier to ta with fraction. Fraction, Multipling NO common denominator needed. Multipl acro = = = = Look for opportunitie to cancel (cro out): = Fraction, Dividing No common denominator needed. FLIP econd or bottom fraction then MULTIPLY = = flipped Mied number (3 ½ ) mut be converted to proper fraction (7/2) before operation. (3=6/2 add to ½) Page 1 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com

2 Fraction, Squaring, Cubing Same a multipling. Multipl b elf (½ ) 2 = = Note that ¼ i LESS than ½, while for number greater than 1 the quare i larger. (½ ) 3 i = Average: Arithmetic Mean Sum of Term Number of Term Average 5, 5, 10, 20: = 10 4 Mode: Mot frequentl occurring number. Mode of 5, 5, 10 and 20 i 5. Median: Number in middle when number ordered from mallet to larget. Median of 10, 11, 17, 19 and 20 i 17. Median of an EVEN number of term. Since there i no ingle middle number, the median i half wa between the two middle number or the average of the two middle number. Median of 10, 13, 19 and 20? The two middle number are 13 and 19. Halfwa between or the average i 16. Weighted Average A cla of 3 tudent ha an average grade of 70. The other cla of 5 tudent ha an average of 80. What i the average for the chool? (It NOT 75.) Aume ALL 3 tudent in firt cla got eactl 70. Aume ALL 5 five in econd cla got eactl 80. Compute uual average: Sum of Term = Number of Term = = Difficult weighted average quetion ue variable (a, b) for the number of tudent: a 70 + b 80 a + b Ma (ometime) / Mut (alwa) be true X i a poitive integer. X 2 > X MAY be true if X=2. But MUST be true i FALSE, ince X could equal 1. One fale eample (a counter eample) prove a MUST (be true) FALSE. One true eample prove a MAY (be true) TRUE. Inequalitie (X > 6) Like equalitie (X = 6) anthing done to one ide of the equation, do to the other ide, EXCEPT when multipling or DIVIDING b a NEGATIVE, witch inequalit ign. (8 > 6) Multipl both ide b 1 i NOT: (-8 > - 6), but i (-8 < -6). Percent - Part from Whole What (part) i 15% of 60 (whole)? 15% = 15/100 or.15 % = /100 or move decimal point two pace to convert: X = 15/ OR X = X = 9 Percent are the ame a fraction quetion: What (part) i 3/20 of 60 (whole)? Part = Fraction Whole Percent - Part from Whole, but ver large or mall percentage What (part) i.15% of 60 (whole)? Note the decimal point.15% =.15/100 or.0015 X =.15/ OR X = X =.09 What i 300% of 60? 300% = 300/100 or 3 X = 300/ OR X = 3 60 = 180 Percent Miing Percent 16 i what percent of 80? (part = 16, whole = 80) 16 = X/ X = 20 OR olve for decimal 16 = D 80 D =.2, and convert to percent b moving decimal point.. 2 = 20% Percent Miing Whole (working backward) 16 i what 20% of what? (part = 16, percent = 20%) 16 = 20/100 X X = 80 Percent - Increae What i 10% more than 90? Man alternate wording like: After a 10% increae from 90? X = 110/ = 99 ADD the original 100% AND the additional 10%. Note the part i more than the whole if increaed. Percent - Decreae What i 15% le than 20? Man alternate wording like: A $20 hirt on ale for 15% off (the full price) cot? X = 85/ = 17 Page 2 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com But the original 100% MINUS the decreae i the percent (85% = 100% - 15%) Multiple (uuall 2) percent change A tore bu cake wholeale for $10, and add 50% to get the freh-cake retail price. If the cake doe not ell in a week, the tore reduce the freh-cake retail price b 50% and ell a week-old cake. A week-old cake cot? (It NOT $10) Solve a TWO eparate problem. From the firt entence (underlined), olve for the freh-cake retail price. Thi i a imple percent increae problem. X = 150/100 $10 = $15 Then reduce the $15 b 50%. The $15 i now the new whole (ometime call new bae ). Thi econd part i jut a imple (50%) percent decreae problem. X = 50/100 $15 = $7.5 Change the whole or bae when doing multiple percent change. Ratio - Part to Part, no whole The ratio of apple to orange i 3 to 2. There are 15 apple. How man orange? Keep apple on top 3 15 X = NOT 2 X 15 keep orange on bottom

3 Cro-multipl to olve for X if anwer not obviou. X = 10 You can put all apple on top or all apple on bottom, but don t mi in one equation. Ratio Inche to Mile On a map 2/3 of an inch repreent 10 mile. 5 inche on map i? keep inche on top 2/3 5 =, X = X keep mile on bottom. Can alo be olved b finding 1 inch = 15 mile and multipling b 5 (inche). Ratio - Part to Part, and Total The ratio of apple to orange i 3 to 2. There i a total of 50 apple and orange. How man orange? keep apple on top = = = keep orange on bottom Find a ratio that add up to 50. On multiple choice problem work backward from anwer. Onl one anwer work. Can alo be done with algebra: Let 3 be number of apple. Then 2 i number of orange = 50, where i the multiple of the original ratio. Multiple Ratio The ratio of apple to orange i 3 to 2. The ratio of orange to pear i 3 to 4. What i the ratio of apple to pear? It NOT 3 to 4. Do one ratio at a time: Aume 18 apple. An number work, but pick a multiple of 3 that will divide evenl to avoid fraction. keep apple on top 3 18 = 2 X, Solve for X = 12 keep orange on bottom With 18 apple there are 12 orange. Now orange on top 3 12 = 4 Y Solve for Y = 16 keep pear on bottom. With 18 apple, there are 16 pear or 18/16 or 9/8. Direct Proportion Speed (X) Mile in 30 min (Y) In general = k, k i a contant k = ½ in thi eample Y = ½ X Mile in 30 min = ½ Speed Can alo be olved a ratio problem without finding k. At 40 MPH, what i ditance in 30 minute? Keep peed on top =, X = X keep ditance on bottom Invere Proportion Speed (X) Minute to Travel 60 Mile (Y) k = = = 3600 In general = k, k i a contant a increae, decreae keeping k contant. Rearranging: = k/ and = k/ k = 3600 in thi eample Common Invere Proportion: If double, mut half to keep k contant. If triple, mut be 1/3 to keep k contant. If goe up z time, mut be 1/z to keep k contant. Mot invere proportion can be done without calculating k, uing the above common invere proportion. Rate (MPH), Ditance Rate Time = Ditance 20 MPH 2 Hour = 40 mile Average MPH, Rate Fat, 40 MPH in morning driving to chool. Slow, 20 MPH in afternoon traffic. What i average MPH? Do NOT average 20 and 40 for 30. Aume the chool i 40 mile awa. 80 mile round trip. One hour in morning. Two hour in afternoon. 80 mile/3 hour=26 2/3 MPH FOIL multiplication Firt, outer, inner, lat (a + b) (c + d) = firt outer inner lat ac + ad + bc + bd FOIL (a+b) (a+b) firt outer inner lat a 2 + ab + ba + b 2 = a 2 +2ab + b 2 FOIL (a-b) (a-b) firt outer inner lat a 2 - ab - ba + b 2 = a 2-2ab + b 2 FOIL (a+b) (a-b) firt outer inner lat a 2 ab + ba b 2 = a 2 b 2 Difference of Two Square Page 3 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com Multipling b Zero 0 time anthing i 0. If a b = 0 then a and/or b (one or both) i zero. Thi i ued in factoring If (-3)(-5) = 0, (-3) and/or (-5) = 0, = 3 or = 5 Factoring Polnomial FOIL backward zero here = 0 Gue firt term that multipl to 2 : ( + ) ( + ) = 0 Gue lat term that multipl to 2: ( + 2) ( + 1) = 0 Tet to ee if outer + inner multiplication add to 3: = 3. It doe, but if not tr gueing other firt or lat term. ( + 2) ( + 1) = 0 = -2 or = -1 On multiple choice quetion: ou can work backward from the anwer without uing FOIL: a) 3 b) 2 c)1 d) 0 e) -1 b tring each in the original = 0 Oppoite Angle are equal. = and = = 180- On one ide of a line the angle (+) add up to 180 (half a 360 circle). Given one angle i 100 : 100 mut equal 80 to add up to 180 along a line. X mut equal 100 becaue it oppoite of 100 AND alo becaue + on one ide of a line mut equal 180.

4 Parallel Line: = 180- Viualize placing parallel line on top of each other. All X and Y are equal. Given an one angle, all other can be found. Iocele Triangle Two equal angle () Two equal ide () oppoite the equal angle Equilateral Triangle Are alwa Similar Triangle Have ame angle, but one i larger or maller than other. All ide are proportional. Ue ratio to olve. Thi Triangle i half the ize of the larger imilar triangle Similar triangle have the ame three angle. A imilar triangle inide a larger imilar triangle: 6 feet Flagpole Height? triangle twice hortet ide " 2 60 #hortet 30 ide 3 Congruent Same hape (angle) AND ame ize (length). Contrat with imilar hape with have the ame hape (angle) but not ame ize (length). One imilar triangle can be larger than other. Polgon: Interior Angle (number of ide 2) 180 Triangle (3 ide) = 180 Rectangle (4 90 ) = 360 Same for quare or ANY 4 ided figure. Pentagon (5 ide) = for each additional ide N gon (n ide) = (n-2) 180 Abolute Value Make poitive if negative A tudent ha 15 dirt hirt and 5 clean hirt in hi dorm room. Randoml picking a hirt in the dark, what i the probabilit of picking a clean hirt? (It not 5/15, the ratio of clean to dirt hirt) Firt find the total number of outcome, which i 20 (15 dirt + 5 clean). OK Outcome 5 1 = = Total Outcome 20 4 Coordinate Both and are poitive Poitive " %, + +,+ (,) (,) #Negative Poitive & %, % +, % (,) (,) ' Negative Both and are negative Line = m + b Two Perpendicular Line: = Three equal ide () Three equal angle. All 60 becaue ever triangle i 180, and 180 /3 = 60. Area of a triangle ½ bae height: which i half the area of a rectangle (bae height) or (length width) bae height Side of triangle i NOT the height unle it a right (90 ) triangle: bae height and ide of triangle 6/10 = Flagpole Height/50 Solve for Flagpole Height= 30 Pthagorean Theorem For right (90 ) triangle onl triangle hown above: (leg) 2 + (leg) 2 = (Hpotenue) 2 (3) 2 + (4) 2 = (5) = triangle hown above: (leg) 2 + (leg) 2 = (Hpotenue) 2 (6) 2 + (8) 2 = (10) = triangle (an Iocele Triangle) Two equal angle Two equal leg (ide) !!= if poitive, - if i originall negative!5!= 5 and!-5!= 5 Abolute value i ued for within problem: Adam (a = Adam age) doe not date women (w = date age) more than two ear older or ounger than himelf.!a-w!$ 2 which i the ame a!w-a!$ 2 Plug in number for age to tet:!17-15!$ 2 ame a!15-17!$ 2 Probabilit = Number of OK Outcome Total Number of Outcome = 1/2-2 = 2 + 1, in general = m + b ' ' lope -intercept When = 0 (on the ai), = b (the -intercept) A point on a line ( and ), and either lope (m) or the - intercept (b) can be ued to find the other (m or b) uing =m + b. Perpendicular line cro at 90 (right) angle and the lope of one (2 in thi cae or m in general) i the negative reciprocal (one over) of the other lope (-1/2 in thi cae or 1/m in general). Page 4 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com

5 Slope: Rie/Run increae in /increae in "lope = ' Rie ' 1 & Run If the line i clearl graphed, often it poible to eail count the rie and run between an two point for lope. Given an two point (1,3) and (0,1) lope i rie/run or: The firt the econd = The firt the econd 3 1 = Either point could be the firt point or the econd, but the reult i the ame. Slope, Negative, Poitive Slope, flatter Slope + 1 Slope 1 Slope + 1/2 Slope 1/2 Shifting graph With an function adding (ubtracting) OUTSIDE the function move the graph up (down). Take the implet function: = 2, the line previoul ued. Adding 2 AFTER/OUTSIDE THE FUCTION 2 move the line up 2 to the new - intercept of 2. Subtracting 2 move the line down 2 to the new -intercept of -2: = = 2 = Again take the function = 2 but add or ubtract before performing the function: Original: = 2 New: = 2(+2) One might gue (incorrectl) that adding 2 move the line up 2 or mabe to the right 2. But the curve hift left b two. X = 2 in the new function give the ame reult a = 0 in the original. X = 0 in the new function give the ame reult a = 2 in the original. = 2(+2) -2 # # # = 2 Subtracting (not adding) inide the function hift to the right. The SAT often tet for thee counterintuitive hift. One can alo do thee pointb-point b picking a value for, finding, and plotting to ee which wa the curve hift. Some can be done on a calculator if the formula i given. Ditance between 2 Point (Pthagorean Theorem) Given an two point ((1,2) and (3,1) chooe a third point to make a right triangle b taking the from one point and the from the other point. Either (1,1) or (3,2) make a right triangle, but (1,1) i hown below. (1,2) (1,1) (3,1) the leg of the triangle are the change in and the change in. Graphing the triangle ma be kipped. (leg) 2 + (leg) 2 = (Hpotenue) 2 (1) 2 + (2) 2 = (h) 2 5 = (h) 2 _ (5 = h Midpoint of a line egment. The midpoint of (1,1) to (3,7) i half wa between the X (halfwa between or average of 1 and 3 i 2) and halfwa between the Y (halfwa between or average of 1 and 7 i 4). The midpoint i (2,4). Counting Conecutive Integer (or conecutive ticket.) Ticket number 9 through 15 were old toda. How man? It NOT 15-9 or 6. For mall number one can count 9, 10, 11, 12, 13, 14, 15 for 7 ticket old. Subtract (15-9) AND add 1 to count the firt ticket old for 7. Eponent Multiplication ame bae, add eponent a 3 a 2 = (a a a) ( a a ) = a 5 = a 3+2 Can alo be olved, a a backup method or check, b letting a=2 and olving. Eponent Diviion ame bae, ubtract eponent a 4 a 2 = a a a a = a a cancel all ecept two top a a 2 = a 4-2 Eponent Raiing Power Multipl eponent (a 3 ) 2 = (a a a) ( a a a ) a 6 = a 3) 2 a 3 a 2! (a 3 ) 2 a 5! a 6 Negative Eponent 1 1 a 1 = a 2 = a a 2 In general a b = 1/ a b (put under 1 and drop the negative) negative eponent follow the rule for diviion a 2 a 4 = a a = a a a a cancel all ecept two bottom a 1 1 = = a a a 2 a 2 4 = a 2 Eponent (ab) 2 (ab) 2 = (ab) (ab) = a 2 b 2 In general, the eponent can be ditributed: (ab) k = a k b k Page 5 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com

6 Eponent Square root of both ide a 2 = b 4 rewriting a: (a a ) = (b b b b) it obviou that a = (b b) OR take the quare root of both ide (half the eponent) a = b 2 Thi work for cube root or an other root. Fractional Eponent Are quare/cube root a 1/2 = quare root of a a 1/3 = cube root of a a 1/n = n th root of a Fractional eponent are ueful for reducing: a 3 = b 9 (a 3 ) 1/3 = (b 9 ) 1/3 Uing the power raied rule to multipl eponent give: a = b 3 Permutation: ordering Jane ha 3 dree. (make the dree A, B, and C). Wearing a different dre on three different night, how man poibilitie? For ea problem with a mall number of outcome, the poibilitie can be written: ABC, ACB, BAC, BCA, CAB, CBA OR there are 3 option for the firt night (A,B, or C), 2 option for the econd night (the two remaining dree) and 1 option for the lat night (the one remaining dre). Multipl = 6. (Thi i three factorial or 3!) Oddball election A different quetion ma have unlimited (re)election of choice. If Jane can rewear the dree multiple time, then he could wear the ame dre three time (AAA, BBB or CCC) wear a dre twice (AAB, BBA.). Becaue of repeated election, there are 3 poibilitie for the firt dre, AND 3 poibilitie for the econd dre and 3 poibilitie for the third dre. Multipl = 27. Hard SAT quetion ma add oddball condition uch a Jane can t wear dre A on the firt night. Do a above but with onl two poibilitie for the firt night. Multipl = 18. Combination: Chooing unordered group Again, Jane ha 3 dree, but want to take 2 of the 3 on a trip. How man poibilitie are there? For ea problem with a mall number of outcome, poibilitie can be written: AB, AC, BA, BC, CA, CB But before ou anwer i, note that AB and BA are the ame combination. Likewie (AC and CA) and (BC and CB). Cro out the duplicate. OR there are 3 option for the firt dre and, 2 option for the econd dre (the two remaining dree). Multipl 3 2 = 6. But there are two ordering of each combination. Divide b 2. (2!) In general divide b the number of permutation (ordering) of the choen (maller) group, which i it factorial. Set, Double counting 5 tudent pla che. 4 tudent pla football. 2 tudent pla both che and football. How man tudent? It not 5+4 = 9, becaue thi double count the tudent who pla both. It = 7. Add et, ubtract interection. Circle * (pi) = 3.14 approimatel Diameter = 2 Radiu Page 6 M Advanced SAT Math Seminar, now on DVD, how how to olve hard SAT math with thee note. FreeVocabular.com diam Circumference = * D = * 2R length around entire circle Remember it 3.14 time the diameter not the radiu. If ou take 3.14 time the radiu, drawn above outide the circle for eaier comparion, ou can ee that ou will onl get halfwa around the circle Area = * R 2 Remember it the radiu quared, not the diameter quared. If ou quare the diameter, drawn above outide the circle for eaier comparion, ou get a quare bo larger then the circle. Arc and Sector of Circle are jut fraction of circle. A 60 B Sector (wedge, lice) are fraction of the entire circle area. Arc are fraction of the total circle circumference. But intead of aing 1/6 of a circle, quetion will a 60. A total circle i /360 = 1/6. To find the length of an arc, find the circumference of the total circle and multipl b the fraction (1/6 or 60/360 in thi eample). To find the area of a ector, find the area of the total circle and multipl b the fraction. Simplifing Square Root _ (50 = (25 2 = 5(2 (a 2 b = a(b Volume = Length width height It doe not matter which ide i called height or width a long a ou multipl all three. For a cube all three ide are the ame. Volume = (ide) 3 Clinder: height Volume of Clinder = (Area of top circle) height The top circle and bottom circle are the ame ize. Solving 2 equation: a + 2b = 3 2a + 6b = 10 Multipl both ide of firt equation b 2 and ubtract from the econd equation. 2a + 6b = 10 2a + 4b = b = 4 b = 2 Replace b in an equation to olve for a. Check with a and b in the other equation. Or in firt equation, iolate a: a = 3-2b and ubtitute (3-2b) for a into the econd equation: 2(3-2b) + 6b = b + 6b = 10 2b = 4 b = 2 Biector plit into equal part each half the original ize. PDF file of thee Math Note i on m ite for free: in addition to 5,000 free SAT vocabular word. When printing, tr unchecking the fit to page option.

Fair Game Review. Chapter 7 A B C D E Name Date. Complete the number sentence with <, >, or =

Fair Game Review. Chapter 7 A B C D E Name Date. Complete the number sentence with <, >, or = Name Date Chapter 7 Fair Game Review Complete the number entence with , or =. 1. 3.4 3.45 2. 6.01 6.1 3. 3.50 3.5 4. 0.84 0.91 Find three decimal that make the number entence true. 5. 5.2 6. 2.65 >