Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: Ratios and Proportions Correct answer: 8 Cross-multiply: (4 + x) = (12 + x)(.6) Distribute: 4 + x = 7.2 + .6x Subtract .6x: 4 + .4x = 7.2 Subtract 4: .4x = 3.2 Divide by .4: x = 8 Common mistake: 4 This is the result if you forget to add the quantity x to the denominator as well as the numerator. 4 12 6 + + = x x . Formula/Concept: The Triangle Inequality: ⏐B – A⏐ < C < ⏐B + A⏐ Correct answer: 19 (B – A) < C < (B + A) Substitute for A and B: (12 – 8) < C < (12 + 8) Simplify: 4 < C < 20 Common mistake: 20 This is the common error that can be made if you mistakenly set 4 ≤ C ≤ 20. Formula/Concept: Weighted Averages If two numbers being averaged have different “weights,” you must remember to account for that when finding the average. Correct answer: 85 Common mistake: 86 This is the result if you mistakenly take the “simple” average of the two scores, 82 and 90, rather than taking their weighted average. 10 82 6 90 16 820 540 16 85 () + () = + = A B C Formula/Concept: Percent change e.g. To increase a value by 20%, multiply by 1.20 To decrease a value by 10%, multiply by 0.90 Correct answer: 92 1999 = $100 (100)(1.25) = 125 2000 = $125 (125)(.8) = 100 2001 = $100 (100)(.8) = 80 2002 = $80 (80)(1.15) = 92 2003 = $92 Common mistake: 100 This is result if you simply add up the percent changes instead of calculating the changes as above: 25 - 20 - 20 + 15 = 0. Formula/Concept: Average formula: Correct answer: 2 Multiply by 3: x + (x + 2) + (2x + 8) = 18 Combine like terms: 4x + 10 = 18 Subtract 10 4x = 8 Divide by 4: x = 2 Common mistake: 5 This is the result if you mistakenly think that there are 5 numbers instead of 3. xx x++ () ++ () = 228 3 6 Formula/Concept: Integer arithmetic Correct answer: 28 Remember that the sum of the numbers from -12 to 12 is 0, because the negative integers “cancel” the positives. 13 + 14 + 15 = 42, so n must be 15. To find the number of integers in the set, just subtract the first from the last and add 1. Common mistake: 27 This is the result if you forget that 0 is an inte- ger or simply subtract the least from the great- est to count the integers in the set. sum average # = Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math If the area of square ABCD in the figure above is 100 ft 2 , then what is the cir- cumference of inscribed circle O? In 1984, a share of stock in Black’s Oil Trust cost $3. By 2000, it had increased to $15 per share. What is the percent increase in the price of the stock from 1984 to 2000? If (x + 4)(x – 4) = 65, then what is the value of x 2 ? A B CD O If the third Friday in January occurs on the 15th, what is the date of the fourth Wednesday in January? At a department store, all shirts are priced at s dollars, but if you buy one shirt at full price, you can buy any num- ber of additional shirts at a $2 discount per shirt. What is the cost of buying x shirts at this sale? l A E C D x° 60° 47° 44° 51° B Note: Figure not drawn to scale. In the figure above, line segments AB –– , EB –– CB –– , and DB –– intersect line l at point B. What is the value of x? Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: Using Patterns Correct answer: 27th Make a calendar: Common mistake: 20th This is the result if you mistakenly assume, without drawing a calendar to confirm, that the third Wednes- day occurs before the third Friday. In this particular month, the first Wednesday is after the first Friday. Formula/Concept: A linear angle measures 180؇. Correct answer: 85؇ 44؇ + x؇ + 51؇ = 180؇ Combine like terms: 95؇ + x؇ = 180؇ Subtract 95؇: x؇ = 85؇ Common mistake: 47؇ This is the result if you mistakenly think that an- gles CBD and ABE are vertical angles. But vertical angles must involve intersecting lines, not line seg- ments. There are no vertical angles in this figure. Formula/Concept: Translating words into expressions Correct answer: s + (x – 1)(s – 2) The first shirt costs s dollars, and each addi- tional shirt costs (s – 2) dollars. Don’t multiply the (s – 2) by x, because this would account for the first shirt twice. Only (x – 1) shirts are priced at (s – 2) dollars. Common mistake: s + (x)(s – 2) This is the result if you forget that only x – 1 shirts are discounted, rather than all (x) shirts. M T W Th F Sa Su 456 78 910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 123 Formula/Concept: Circumference = 2πr = πd Correct answer: 10 Area of square = (side) 2 Substitute 100 for area: 100 = (side) 2 Take square root: 10 = side of square dia- meter d = side of square Use circumference C = πd = π(10) = 10π formula Common mistake: 25π This is the result if you confuse the area formula with the circumference formula. Formula/Concept: Correct answer: 400% Common mistake: 500% This is the result if you mistakenly find what per- cent 15 is of 3 instead of finding the percent change from 3 to 15. Percent change = − ×= 15 3 3 100 400%% Percent change = − × final original original 100% Formula/Concept: FOILing (a + b)(a – b) = a 2 – ab + ab + b 2 Correct answer: 81 (x + 4)(x – 4) = 65 FOIL: x 2 – 4x + 4x – 16 = 65 Combine like terms: x 2 – 16 = 65 Add 16: x 2 = 81 Common mistake: 9 This is result if you mistakenly solve for x instead of x 2 . Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math If w divided by 1 ⁄4 is equal to 32, then what is the value of w? At Holston Hospital, a “team” consists of 1 resident and 2 medical students. If there are 4 residents and 5 medical students in the emergency department, how many different teams could be formed? If the average of 5, 6, 8, x, and 8 is 6, then what is the median of the set? Point W is on line segment XY such that . If WY = 12 then what is XY? XW WY = 3 4 For all values of x, let x = . Which of the following is equal to – ? (A) (B) (C) (D) (E) x − 2 4 When 34 is divided by 6, the remainder is n. What is the remainder when n is divided by 2? 6 34 10 8 14 24 26 Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: Ratios and Proportions Correct answer: 21 Plug in 12 for WY: Cross-multiply: 4XW = 36 Divide by 4: XW = 9 Solve for XY: XY = 9 + 12 = 21 Common mistake: 9 This is the result if you solve for XW instead of XY. XW 12 3 4 = XW WY = 3 4 Formula/Concept: New Symbols/Functions Correct answer: E Common mistake: A This is the result if you do not notice that the answer choices are also in boxes. Formula/Concept: Remainders Correct answer: 0 Common mistake: 2 This is the result if you find the quotient rather than the remainder. ) ) 634 30 5 4 24 4 2 0 40 −− RR 34 34 2 4 32 4 810 10 2 4 8 4 2 34 10 8 2 6 26 26 2 4 24 4 6 = − == = − == −=−= = − == Formula/Concept: To divide by a fraction, multiply by its reciprocal. Correct answer: 8 Write an equation: w ، 1 ⁄4 = 32 Multiply by the reciprocal: w ؋ 4 = 32 Divide by 4: w = 8 Common mistake: 128 This is the result if you divide 32 by 1 ⁄4 instead of multiplying it by 1 ⁄4. Formula/Concept: The Fundamental Counting Principle Correct answer: 4 ؋ 5 ؋ 4 ، 2 = 40 The number of options for choosing a resident is 4, since there are 4 residents. The number of different pairs of interns is 10, because there are 5 options for intern A and then 4 options for intern B, but since choosing AB is the same as choosing BA, we must divide this set by 2. Common mistake: 100 or 80 This is the result of 4 ؋ 5 ؋ 5 or 4 ؋ 5 ؋ 4. Formula/Concept: Median = middle number Mean = average Correct answer: 6 Multiply by 5: 5 + 6 + 8 + x + 8 = 30 Combine like terms: 27 + x = 30 Subtract 27: x = 3 Find the median: 3, 5, 6, 8, 8 Common mistake: 8 This is the result if you confuse mode with median. 568 8 5 6 ++++ = x Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math What is the area of the triangle in the figure above? The slope of line l is – 1 ⁄2. If two points on line l are (2, 4) and (x, –2), what is the value of x? There are 25 students in Ms. Jamison’s 4 th -grade homeroom class. If 12 of her students have a cat, 19 have a dog, and every student has at least 1 pet, how many students have both a cat and a dog? (5,6) (7,0) (14,0) x y A printer can produce 50 pages in 3 minutes. At this rate, how many pages can it print in 300 minutes? If f(x) = (x – 2) 2 , what is the range of this function over the domain –2 ≤ x ≤ 3? If b varies inversely as the square of c and directly as a, and b = 4 when c = 4 and a = 8, then what is the value of b when a = 18 and c = 6? Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: Ratios and Proportions Correct answer: 5,000 Set up a ratio: Cross multiply: 15,000 = 3x Divide by 3: 5,000 = x Common mistake: 18 This is the result if you set up the ratio improperly. 50 3 300pages minutes minutes pages = x 50 3 300 pages minutes pages minutes = x Formula/Concept: The range of a function is the set of all of the possible outputs, or “y-values.” Correct answer: 0 ≤ y ≤ 16 Plug in the integer values of the domain and find the range: f(-2) = 16; f(–1) = 9; f(0) = 4; f(1) = 1; f(2) = 0; f(3) = 1. This yields a range of 0 ≤ y ≤ 16. Or you could graph it: Common mistake: 1 ≤ y ≤ 16 This results if you try to find the range by plug- ging in only the two domain endpoints, ignoring the points in between. Formula/Concept: Direct and Inverse Variation Correct answer: 4 Cross-multiply: 64 = 8k Divide by 8: k = 8 Set up new equation: Common mistake: 1 ⁄2 This is the result if you inversely relate b to the square root of c instead of relating it to the square of c. x y 5 5-5 -5 b ka c k == () () 22 4 8 4 b a c b== () () == 8818 6 144 36 4 22 Formula/Concept: Area = 1 ⁄2(base)(height) Correct answer: 21 The base of the triangle is the distance from (7, 0) to (14, 0), which is 14 – 7 = 7. The height is the distance from the x-axis to (5, 6), which is 6 – 0 = 6. Area = 1 ⁄2(7)(6) = 21 Common mistake: 17.5 This is the common error that can be made if you use 5 as the height instead of 6. Formula/Concept: Correct answer: 14 Cross-multiply: 2(–6) = –1(x – 2) Simplify: –12 = –x + 2 Subtract 2: –14 = –x Divide by –1: 14 = x Common mistake: –1 This is the result if you mistakenly calculate “run over rise” for the slope: xx yy 21 21 − − slope yy xx x = − − = −− − = − 21 21 24 2 1 2 slope rise run yy xx == − − 21 21 Formula/Concept: Venn Diagrams Correct answer: 6 Let x represent the number of students with both cats and dogs: (12 – x) + (19 – x) + x = 25 Combine like terms: 31 – x = 25 Subtract 31: –x = –6 Divide by –1: x = 6 Common mistake: 25 or 31 This is the result if you confuse “a cat and a dog” with “a cat or a dog.” Cats Dogs 12 – x 19 – xx Math Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Math Smart Cards Math Smart Cards Math If 5 – 2⏐x⏐> –9, find the range of possible values for x. –2, 0, 2, –2, 0, 2, –2, 0, 2, . . . The sequence above continues accord- ing to the pattern shown. What is the sum of the first 31 terms of this sequence? The sum of a set of 7 integers is 67. If each of these numbers must be less than 14, what is the smallest possible value of any one number in the set? If -1 < w < 0, then which of the follow- ing statements is true? (A) w < w 2 < w 3 (B) w 2 < w < w 3 (C) w < w 3 < w 2 (D) w 3 < w 2 < w (E) w 3 < w < w 2 A jar contains only red and white mar- bles. If the probability of randomly selecting a white marble from the jar is 1 ⁄4 and there are 15 red marbles in the jar, how many white marbles are there? Jim takes the same train to and from work each day. One winter day, during the morning commute, the train aver- aged 80 miles per hour. During the evening commute, due to ice on the track, the train averaged only 48 miles per hour. If Jim spent 2 hours in the train that day, how many miles is the train ride to work? Math Math Math Math Math Math Math Math Math Math Math Math Formula/Concept: Powers of Fractions Correct answer: C Plug in a number for w: Plug in – 1 ⁄2 for w: w = – 1 ⁄2 w 2 = (– 1 ⁄2) 2 = 1 ⁄4 w 3 = (– 1 ⁄2) 3 = – 1 ⁄8 Rank the values of w, w 2 , w 3 :– 1 ⁄2 < – 1 ⁄8 < 1 ⁄4 Common mistake: A or E You might choose E if you forget that – 1 ⁄8 > – 1 ⁄2. You might choose A if you forget that squaring any negative number will result in a positive number. Formula/Concept: Ratios and Probability Correct answer: 5 If the probability of selecting a white marble is 1 ⁄4, then the probability of selecting a red marble is 1 – 1 ⁄4 = 3 ⁄4. Set up a ratio: Cross-multiply: 60 = 3(total) Divide by 3: 20 = total = white + red Solve for white: 20 = white + 15 Subtract 15: white = 5 marbles Common mistake: 20 This is the result if you answer for the total number of marbles instead of the number of white marbles. Formula/Concept: Distance = Rate × Time Correct answer: 60 (because he takes the same train, the distance is the same going in both directions.) To work: d = (80)(t) From work: d = (48)(2 – t) Set them equal: 80t = 48(2 – t) Distribute: 80t = 96 – 48t Add 48t: 128t = 96 Divide by 128: t = .75 hours Plug in .75 for t: d = 80(.75) = 60 miles Common mistake: 64 This is the result if you average the two speeds, 80 and 48, and then using that speed of 64 to calculate a dis- tance of 64 miles to work. red total total == 3 4 15 3 4 Formula/Concept: Solving absolute value inequalities Correct answer: –7< x < 7 5 – 2⏐ x ⏐ > –9 Subtract 5: –2⏐ x ⏐ > –14 Divide by -2: (swap the inequality) ⏐ x ⏐ < 7 Take away absolute value: –7 < x < 7 Common mistake: x > 7 or x < –7 This is the common error that can be made if you forget to swap the inequality when dividing or multiplying by a negative number. Formula/Concept: Sequence problems: finding the sum of a random number of terms Correct answer: –2 The pattern repeats every 3 digits, and the sum of each repetition is –2 + 0 + 2 = 0. The pattern occurs 31 ÷ 3 = 10 1 ⁄3 times, or 10 with remainder 1. The 10 full repetitions have a sum of 10(0) = 0. Since the 31st term is –2, the sum is 0 + –2 = –2. Common mistake: 0 This is the common error that can be made if you forget to add the 31st term after finding out that the pattern occurs 10 1 ⁄3 times. Formula/Concept: Numerical Reasoning Correct answer: –11 a + b + c + d + e + f + g = 67 If you want g to be as small as possible, then make the sum of the other numbers as large as possible: Substitute 13 for a through f: 13 + 13 + 13 + 13 + 13 + 13 + g = 67 Combine like terms: 78 + g = 67 Common mistake: 4 This is the result if you assume the integers must be different: 13, 12, 11, 10, 9, and 8. Math Smart Cards Roots Smart Cards Math Smart Cards Roots Smart Cards Smart Cards Math Smart Cards Roots Math Smart Cards Math Smart Cards Smart Cards Math Smart Cards Roots Smart Cards Math Smart Cards Math If the rectangular solid above has a volume of 162 cubic inches, what is the surface area of the solid? Which point corresponds to the result when the numbers corresponding to points E and B are multiplied? 9 6 h ABCDE –2 –1 0 1 2 n p W x v Y b ° c ° Z m l 66 ° 106 ° magnanimous Given the graph of y = f (x) shown above and the transformed graph on the right, what is the equation of the new function? 5 5 y x 5 5 y Instructions for studying with Root Smart Cards 1 st point: Define the word on the front of the card 2 nd point: Give the meaning of the root in bold. 3 rd point: List at least three words that contain the root. Note: Figure not drawn to scale. In the figure above, if l ʈ m and n ʈ p, what is the value of b + c? . result if you simply add up the percent changes instead of calculating the changes as above: 25 - 20 - 20 + 15 = 0. Formula/Concept: Average formula: Correct answer: 2 Multiply by 3: x + (x + 2). 1. Common mistake: 27 This is the result if you forget that 0 is an inte- ger or simply subtract the least from the great- est to count the integers in the set. sum average # = Math Smart Cards. the fourth Wednesday in January? At a department store, all shirts are priced at s dollars, but if you buy one shirt at full price, you can buy any num- ber of additional shirts at a $2 discount per