Math Fundamentals Memorization initiated 08/31/2013

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A M
Created by A M about 7 years ago
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Graduate Test Prep Flashcards on Math Fundamentals Memorization initiated 08/31/2013, created by A M on 08/31/2013.

Resource summary

Question Answer
1. 1/100 0.01
2. 1/50 0.02
3. 1/25 0.04
4. 1/20 0.05
5. 1/10 0.1
6. 1/9 0.11 ~ 0.111 = 11.1%
7. 1/8 0.125
8. 1/6 0.16 ~ 0.167
9. 1/5 0.2
10. 1/4 0.25
11. 1/3 0.333
12. 1/2 0.5
13. 3/8 0.375
14. 2/5 0.4
15. 3/5 0.6
16. 5/8 0.625
17. 2/3 0.667
18. 4/5 0.8
19. 5/6 0.833
20. 7/8 0.875
21. 5/4 1.25
22. 4/3 1.33
23. 7/4 1.75
24. Isosceles triangle: ratio of sides 1:1: root 2
25. 30:60:90 triangle ratio of sides 1:root 3:2
26. Diagonal of a square s root 2
27. Diagonal of a cube s root 3
28. Common Pythagorean triplets 3-4-5 9-12-15 12-16-20 5-12-13 10-24-26 8-15-17
29. For an unbiased die, the no. of possible outcomes. 6 ^ n, where n is the number of dice rolled; when order matters and repetition is allowed.
30. For an unbiased coin toss, the number of possible outcomes. 2^n where n is the number of times the coin is flipped, order matters and repetition is allowed.
31. Vertex of a parabola - b/2a
32. Slope of a horizontal line 0
33. Slope of a vertical line Undefined
34. slopes of perpendicular lines Are negative reciprocals of each other.
35. For a circle, circumference/diameter = Pi
36. Divisibility by 4 Divisible by 2, twice, or if the TWO-DIGIT number at the end is divisible by
37. Divisibility by 6 Integer is divisible by both 2 and 3.
38. Divisibility by 8 The integer is divisible by 2 THREE times in succession, or if the THREE digit number at the end is divisible by 8.
39. Divisibility by 9. SUM of the integer's DIGITS is divisible by 9.
40. Integer + Integer Integer
41. Integer x Integer Integer
42. To find all the prime factors of a number Use a factor tree. With larger numbers, start with the smallest primes and work your way up to the larger primes.
43. If you add or subtract multiples of N The result is a multiple of N.
44. If N is a divisor of x and of y then N is a divisor of x + y. e g. if 7 is a divisor of 21 and of 35 then 7 is also a divisor of 21+35.
45. Odd +/- Odd Even
46. Odd +/- Even Odd
47. Odd x Odd Odd
48. Odd x Even Even
49. Even x Even Even (and divisible by 4). ****
50. As the exponent increases, the value of the expression decreases when... The base of an exponential expression is a positive proper fraction (in other words, a fraction between 0 and 1).
51. Negative exponent Think reciprocal.
52. When something with an exponent is raised to another power, Multiply the 2 exponents together.
53. - x squared cannot be simplified further
54. (-x) squared x squared
55. All sets of consecutive integers Are sets of consecutive multiples.
56. All sets of consecutive multiples. Are evenly spaced sets.
57. All evenly spaced sets are fully defined if the following parameters are known: (1) The first (smallest) or last (largest) number in the set. (2) The increment (always 1 for consecutive integers). (3) The number of items in the set.
58. For all evenly spaced sets a. The arithmetic mean and median are equal to each other. b. The mean and the median of the set are equal to the average of the FIRST and the LAST terms. c. The sum of the elements in the set equals the arithmetic mean times the number of items in the set. ** arithmetic mean = average
59. For consecutive integers, the number of terms in the set is equal to (Last - First + 1)
60. For consecutive multiples, the formula for the number of elements in the set is equal to (Last - First) / Increment + 1
61. On ANY number line, numbers get bigger As they move from left to right.
62. The distance between the tick marks on a number line is given by (Upper - Lower) / # of intervals
63. For a number line, 4 tick marks correspond to 3 intervals
64. When questions provide incomplete information about the relative positions of points on a line segment, Make sure that you account for the lack of information by drawing multiple number lines.
65. To construct number lines efficiently and accurately, while remembering to keep track of different possible scenarios Always start with the most restrictive pieces of information first.
66. ** The ratio of men to women in a room is 3:4 may be written as 3x + 4x when the total number is given. This is particularly useful with 3 part ratios e.g. A recipe calls for amounts of lemon juice, wine and water in the ratio 2:5:7. Therefore total amount is 2x + 5x + 7x = 14x.
67. Car A and Car B are driving directly towards each other. Shrinking distance between; add the rates (a+b) and multiply by time.
68. Car A is chasing Car B and catching up. Shrinking distance between; subtract the rates (a - b) and multiply by time.
69. Car A is chasing car B and falling behind. Growing distance between; b - a and multiply by time.
70. Left.......and met/ arrived Sue left her office at the same time as Tara left hers. They met sometime later. Sue and Tara traveled for the same amount of time t.
71. A runs a race 30 secs faster than B A's time is less i.e. if B's time is t secs then A's time is t - 30 secs.
72. Sue and Tara left at the same time, but Sue arrived home about 1 hour before Tara did. Sue traveled for 1 hour less than Tara. Tara's time is t, Sue's is t - 1.
73. Sue left the office 1 hour after Tara, but they met on the road. Sue traveled for one hour less (t-1) than Tara (t).
74. The Kiss: Car A and Car B start driving toward each other at the same time. Eventually they meet each other. Add the rates (a+b), the time is the same for both cars (t) [unless one car starts earlier than the other], multiplying (a+b) x t will give you the total distance covered. a x t = A's distance, b x t = B's distance.
75. The QUARREL: Car A and Car B start driving away from each other at the same time... A's distance is a x t, B's distance is b x t, and the total distance covered is (a + b) x t. The math is the same as the kiss.
76. The CHASE: Car A is chasing car B. How long does it take for A to catch up to Car B? Please note that the cars start at the same time. Change in the gap between the cars = (a - b) x t. Note that the rates and the distances need to be subtracted, unless one car starts earlier than the other. Time of travel is the same for both cars.
77. The ROUND TRIP: Jan drives home to work in the morning, then takes the same route in the evening. Distance traveled is the same d, therefore total distance covered is 2d. Going time and return times will vary, add the times together when dealing with the entire round trip. You can pick a smart number - a value that is a multiple of all the given rates or times.
78. The FOLLOWING FOOTSTEPS: Jan drives home to the store along the same route as bill. Distance traveled by both is the same, the rates and the times vary. Pick a smart number if necessary.
79. The HYPOTHETICAL: Jan drove home from work. If she had driven home along the same route 10 miles per hour faster...... Actual rate is r, Hypothetical rate is r + 10. Distance traveled in both cases is d. Rates vary for the same distance.
80. If an object moves the same distance twice, but at different rates, then the average rate ......will NEVER be the average of the two rates given for the two legs of the journey. In fact, because the object spends more time traveling at the slower rate, the average rate will be closer to the slower of the two rates than to the faster. You may pick a smart number as one strategy.
81. POPULATION PROBLEMS: Some population typically increases by a common factor every time period. Solve with a POPULATION CHART: Make a table with a few rows, labeling one of the middle rows as "NOW". Work forward, backward, or both (as necessary in the problem), obeying any conditions given in the problem statement about the rate of growth or decay. In some cases, you might pick a Smart number for a starting point in your Population Chart.
82. For SD problems You do not need to know any formulae. Just pay attention to what the average spread is doing.
83. If you see a problem focusing on CHANGES in the SD ( i.e. when a set is transformed) Ask yourself whether the changes move the data closer to the mean, farther from the mean or neither. Only the SPREAD matters. The bigger the gaps, the higher the SD.
84. Variance Square of SD.
85. Adding a constant to each data point in a set i.e. increase by 7 .... ....will not affect any of the gaps between the data points, and thus will not affect how far the data points are from the mean.
86. Increase each data point by a factor of 7 i.e. each data point is multiplied by 7.... will increase the SD because this transformation will make all the gaps between the points 7 times as big as they were. Thus each point will fall 7 times as far from the mean. The SD will increase by a factor of 7.
87. A set is divided into 4 quartiles by 3 Quartile markers. Q1 is the median of the first half of the set, Q2 is the median of the entire set and Q3 is the median of the second half of the set.
88. For a normal distribution (Gaussian distribution).. The mean and median are equal, or almost exactly equal. The data is exactly, or almost exactly, symmetrical around the mean/median. Roughly 2/3rd of the sample will fall within 1 standard deviation of the mean. Roughly 96% of the sample will fall within 2 standard deviations of the mean. Only about 1/1000 (0.1%) of the curve is 3 or more standard deviations below the mean; the same is true above the mean. The GRE will not distinguish between random variables that are normally distributed versus ones that are nearly normally distributed.
89. Fundamental Counting Principle If you must make a number of separate decisions, then MULTIPLY the number of ways to make each INDIVIDUAL decision to find the number of ways to make ALL the decisions.
90. The number of ways of putting n distinct objects in order, if there are no restrictions is n factorial
91. Combinatorics If a GRE problem requires you to choose from two or more sets of items from separate pools.. count the arrangements SEPARATELY - perhaps by using a different anagram grid each time. Then multiply the numbers of possibilities for each step.
92. Probability Number of desired or successful outcomes/ Total number of possible outcomes Assuming that all outcomes are equally likely.
93. To determine the probability that event X and event Y will both occur.. MULTIPLY the two probabilities together. The events MUST be independent for this to work.
94. To determine the probability that event X or event Y will occur... ADD the two probabilities together; probability increases as one or the other may occur. X and Y must be mutually exclusive events.
95. If on a GRE problem, "success" contains multiple possibilities - especially if the wording contains phrases such as "at least" or "at most".. .. then consider finding the probability that success DOES NOT HAPPEN. If you can find this "failure" probability more easily (call it x), then the probability you really want to find will be 1 - x.
96. Be aware of both explicit constraints (restrictions actually stated in the text) and hidden constraints (restrictions implied by real-world aspects of a problem). For instance, in a problem requiring the separation of 40 people into 6 groups, hidden constraints require the number of people in each group to be a positive whole number.
97. In most cases, you can maximize or minimize quantities (or optimize schedules, etc.) by choosing the highest or lowest values of the variables that you are allowed to select.
98. For overlapping sets, remember that.. people/items who fit in both categories lessen the number of people/items in just one category. Thus, all other things being equal, the more people/items in "both", the fewer in "just one" and the more in "neither".
99. For Quantitative comparison questions, to try to prove D.. Use -1, 0 and 1. Use positive numbers greater than 1 and fractions between 0 and 1. Use negative numbers less than -1 and fractions between 0 and -1.
100. For Quantitative comparisons, Use the Invisible Inequality... Add or subtract to both quantities. Multiply or divide both quantities by a positive number. Square or square root both quantities if they are positive.
101. Use Quantitiy B as a Benchmark.. When Quantity B is a number (no variables )
102. Try to prove D when.. a variable has NO CONSTRAINTS and, also when a variable has CERTAIN PROPERTIES (e.g. x is negative)
103. If a quadratic appears in one or both quantities: a. FOIL the quadratic, b. eliminate the common terms, and c. compare the quantities.
104. If a Quadratic appears in the common information: a. factor the equation and find BOTH solutions, and b. plug both solutions into the quantities.
105. If a Quantitative comparison question with a strange symbol formula contains numbers... Plug in the numbers and evaluate the formula.
106. If a Quantitative Comparison question with a strange symbol does not contain numbers... Plug the given variable directly into the formula to compare the quantities.
107. It is impossible for an absolute value to have a value less than 0.
108. If you need to maximize an absolute value You need to make the number inside as far away from 0 as possible.
109. Sometimes, inequalities are used to.. order variables from least to greatest. In some cases, they may give the sign of the variables and/or give their order from least to greatest. To compare the 2 quantities, use the Invisible Inequality to: 1. eliminate the common terms, and 2. try to discern a pattern if one is present. If there is a pattern, the answer will be (A), (B) or (C). If there is no pattern, the answer will be (D).
110. When simplifying complex fractions, look to: 1. Split the numerator when the denominator is one term, and 2. turn division into multiplication by the reciprocal ( e.g., 2/(2/3)=2 . 3/2)
111. When fractions contain exponents and you have to plug in numbers for the exponents... Always plug in 0 and 1 first to save yourself time.
112. When dealing with percents, always pay attention to the size of the original value. Thus, 20% of a small number is less than 20% of a larger number.
113. The following 3-step process for tackling Geometry QC questions will be emphasized: 1. Establish what you NEED TO KNOW. 2. Establish what you KNOW. 3. Establish what you DON'T KNOW. To arrive at the correct answer consistently, you must act as though there is enough information, while accepting that the answer may ultimately be (D).
114. When Quantity B is a number: For Geometry QC questions, keywords such as area, perimeter and circumference are good indications that.. you can set up equations to solve for a previously unknown LENGTH. Keep the end in mind as you work.
115. When quantity B is a number for a geometry QC: Many geometry QC's will provide enough information to reach a definite conclusion. To solve for the value that you NEED TO KNOW: Establish What you Know: 1. SET UP EQUATIONS to find the values of previously unknown LINES and ANGLES. 2. MAKE INFERENCES to find additional information.
116. For Geometry QC questions, remember Don't trust the picture.
117. For geometry QC problems On questions for which both quantities contain UNKNOWN VALUES... Don't expect to find the exact values for either quantity. Instead, look for RELATIVE SIZE. To judge relative size, establish what you DON'T KNOW and take those values to EXTREMES. Identify how changes to these unknown values change the values in the quantities.
118. Both quantities unknown For a Geometry QC question, if a diagram presents a common shape, such as a triangle or a quadrilateral.. it is often helpful to CREATE VARIABLES to represent unknown angles or lengths. Use the properties of the shapes to create equations and express both quantities in terms of the same variables.
119. QC For ANY word geometry question, the first step in establishing what you KNOW is the same DRAW THE PICTURE. Note that it will sometimes be necessary to REDRAW the picture if you need to PROVE (D). As you establish what you DON'T KNOW, try to determine what about your diagram can change, and always ask, " What can change that will affect the relative size of the values in the quantities?"
120. Word geometry QC USING NUMBERS is a useful strategy when a Word geometry question.. references specific dimensions of shapes ( e.g. length, width, radius) but does not provide any actual numbers. The conclusion you reach will be valid, as long as you 1) pick numbers that match any restrictions in the common information or in the quantities themselves, 2) test several valid cases, trying to prove (D), and 3) determine the reason for any pattern that indicates that the answer might be (A), (B) or (C).
121. QC A very popular theme related to Number properties is... TRYING TO PROVE (D).
122. QC The most important dichotomy in QC's is Positive/Negative distinction. 3 possible clues: 1. x > 0 2. pq > 0 3. (-x^4) an expression containing both a negative sign and an exponent.
123. QC On questions that involve variables and exponents, try to PROVE (D). Try numbers GREATER THAN 1 and numbers BETWEEN 0 AND 1.
124. QC To compare the sums of sets of consecutive integers, ELIMINATE OVERLAP in order to make a direct comparison.
125. Whenever you see a word problem on Quantitative comparisons, make sure you have the information you need before doing any computation. If you don't have enough info, the answer is (D).
126. QC On a ratios problem.. Try to prove (D). Choose one scenario in which the actual values are the same values as the ratio and choose another scenario in which the numbers are much larger (but still pick numbers that are easy to work with).
127. QC In any question that involves two groups that have some kind of average value.... use the principle of weighted averages. If two groups have an equal number of members, the total average will be the average of the two groups. If one group has more members, the total average will be closer to the average of that group.
128. The Basic process of solving a Data Interpretation Question 1. Scan the graph/s. 15-20 secs. What type of a graph is it? Is the data displayed in percentages or absolute quantities? Does the graph provide any overall total value?
129. The Basic process of solving a Data Interpretation Question 2. Figure out what the question is asking. What does it ask you to do? Calculate a value? Establish how many data points meet a criterion?
130. The Basic process of solving a Data Interpretation Question 3. Find the graph(s) with the needed information. Look for the keywords in the question.
131. The Basic process of solving a Data Interpretation Question 4. IF you need to establish how many data points meet a criterion, keep track as you go by taking notes.
132. The Basic process of solving a Data Interpretation Question 5. IF you need to perform a computation, translate the question into a mathematical expression BEFORE you try to solve it.
133. The Basic process of solving a Data Interpretation Question 6. IF one of the answer choices is " cannot be determined ", check that you have ALL the information you need.
134. The Basic process of solving a Data Interpretation Question 7. Use the calculator when needed, but keep your eye out for opportunities to use time-saving estimation techniques. Does the question use the word "approximate"? Are the numbers in the answer choices sufficiently far apart?
135. Shortcut tips and strategies when attempting data interpretation graphs. Use a piece of paper or even your finger to make a straight edge to make comparisons and reading graphs (horizontal and vertical) easier. Any given amount is a larger percentage of a smaller number than it is of a bigger number. Be careful of the difference between charts that show percentages and charts that show actual quantities. Changing average problems are very popular on the GRE. Remember the average change estimation shortcut. Try visual estimation before performing calculations. Sometimes it is easier to calculate the percentage that does NOT satisfy a condition rather than calculate a percentage directly.
136. For multiple data interpretation graphs... Use good scrap paper organization methods because more charts mean more opportunities to become confused and waste time.
137. cube L cubed 6 L
138. cuboid l x b x h 2 (lb + bh + lh)
139. right circular cylinder pi r squared h 2 pi r ( r + h)
140. sphere 4/3 pi r cubed 4 pi r squared
141. cone one third pi r squared h pi r ( r + l) l is the slant height, not the vertical height
142. set of integers .... -3, -2, -1, 0, 1, 2, 3...... An integer is a number that can be written without a fractional or decimal component. For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers. The set of integers is a subset of the real numbers, and consists of the natural number (1, 2, 3, ...), zero (0) and the negatives of the natural numbers (−1, −2, −3, ...).
143. lcm of two non-zero integers a and b is the greatest positive integer that is a divisor of both a and b. e.g. lcm of 30 and 75 is 2x3x5 ; 3x5x5 = 2x3x5x5 = 150
144. hcf of 2 non-zero integers a and b is the greatest positive integer that is a divisor of both a and b. e.g. hcf of 30 and 75 is 2x3x5; 3x5x5 = 3x5 = 15
145. hcf of 2 non-zero integers a and b is the greatest positive integer that is a divisor of both a and b. e.g. hcf of 30 and 75 is 2x3x5; 3x5x5 = 3x5 = 15
146. a prime number is an integer greater than 1 that has only two positive divisors: 1 and itself
147. composite number an integer greater than 1 that is not a prime number is called a composite number
a prime number is an integer greater than 1 that has only two positive divisors: 1 and itself.
148. for odd order roots there is exactly one root for every number n, even when n is negative e.g. cube root of 8
149. for even order roots there are exactly two roots for every positive number n and no roots for any negative numbers.
150. Domain of x is the set of all permisable inputs. Without explicit restriction, the domain is assumed to be that set of all values of x for which f(x) is a real number.
151. The equation of a circle is given by (x-a)squared + (y-b) squared = r squared where the center of the circle is the point (a,b) and the radius of the circle is r.
152. consider the absolute value function defined by h(x) = lxl by using the definition of absolute value, h can be expressed as a piecewise-defined function. x, x is greater than or equal to 0 -x, x is less than 0 the graph of this function is V-shaped and consists of 2 linear pieces, y=x and y=-x, joined at the origin.
153. In general, for any function h(x) and any positive number c, the graph of h(x) + c is the graph of h(x) shifted upward by c units.
154. In general, for any function h(x) and any positive number c, the graph of h(x) - c is the graph of h(x) shifted downward by c units.
155. In general, for any function h(x) and any positive number c, the graph of h(x+c) is the graph of h(x) shifted left by c units.
156. In general, for any function h(x) and any positive number c, the graph of h(x-c) is the graph of h(x) shifted right by c units.
157. In general, for any function h(x) and any positive number c, the graph of ch(x) is the graph of h(x) stretched vertically by a factor of c if c > 1. or shrunk vertically by a factor of c if 0 < c < 1.
158. parallelogram area is bxh perimeter is 2(l+b)
159. trapezium area is 1/2 (sum of the parallel sides) * h perimeter is the sum of all the sides.
160. congruency 2 sides + included angle, 3 sides, 2 angles + included side.
161. Measure of the arc of a circle is the measure of the central angle of the arc.
162. Isosceles triangle has at least 2 congruent sides, the angles opposite to the equal sides are congruent.
163. Effect of outliers on range,mean and median mean affected, range directly affected, median is fairly unaffected by unusually high or low values relative to the rest of the data.
164. Percentiles are mostly used for very large lists of data ordered from least to greatest. Instead of dividing the data into 4 groups, the 99th percentiles P1, P2, P3...P99 divide the data into 100 groups. Consequently, Q1 = P25, Q2 = M = P50, Q3 = P75. Because the number of data in a list may not be divisible by 100, statisticians apply various rules to determine values of percentiles.
165. Measures of position include Quartiles and percentiles.
166. Measures of dispersion include range, interquartile range and standard deviation.
167. A measure of dispersion that is not affected by outliers is the interquartile range. It measures the spread of the middle half of the data i.e. Q3 - Q1.
168. Measures of central tendency Mean, median, mode
169. A measure of spread that depends on each number in the list is the standard deviation. (unlike the range and the interquartile range).
170. The std deviation is computed by calculating the mean of the n values, finding the difference between the mean and the n values, squaring each of the differences, finding the average of the n squared differences, taking the non negative square root of the avg. squared differences.
171. Sample standard deviation is calculated by dividing the sum of the squared differences by n-1 instead of n (to distinguish it from population std. deviation)
172. Standardization is the process of subtracting the mean from each value and then dividing the result by the standard deviation. In any group, most of the data are within about 3 std deviations above or below the mean.
173. Arithmetic with remainders You can add,subtract or multiply remainders as long as you correct excess remainders in the end.
174. Mathematical relationship between dividend, divisor, quotient and remainder. x = Q.N + R where, x is the dividend Q is the quotient N is the divisor R is the remainder
175. For mensuration area problems, the only thing that matters is that the base and the height are perpendicular to each other.
176. The Euclidean distance between two points of the plane with Cartesian coordinates (x1,y1) and (x2,y2) is the square root of (x2-x1 squared + y2-y1 squared)
177. Total after Percentage change Original (1 +/- Percent change/100)
178. 2/9 0.2 (2 recurring)
179. 4/9 0.4 (4 recurring)
180. 23/99 0.23 (23 recurring)
181. 1/11 or 9/99 0.09 (09 recurring)
182. 3/11 or 27/99 0.27 (27 recurring)
183. Mean ( working definition) sum total of all the values divided by the number of values
184. 13 squared 169
185. 14 squared 256
186. 15 squared 225
187. 16 squared 256
188. 17 squared 289
189. 18 squared 324
190. 19 squared 361
200. area of a rhombus The product of the diagonals divided by 2 i.e. half the product of the diagonals. Also, base into height.
201. For quantitative comparison questions containing a variable pay attention to expressions that result in fractions that might actually solve to be 0 or that might be undefined.
202. For lines with negative slopes in the xy plane you see that for each line that does not pass through the origin the x and y intercepts are either both positive or both negative. Conversely, you can see that if the x and y intercepts of a line have the same sign then the slope of the line is negative.
203. For a line that does not pass through the origin, if the x intercept is twice the y intercept you can conclude that both intercepts have the same sign and the slope of line is negative.
204. When dealing with QC questions that have variables and inequalities.... pay special attention when cross multiplying. In some cases, you might have to look for variables that for which you know the sign so that you may confidently divide or multiply both sides of the equation.
205. If ab > 0, then a/b > 0
206. When using inequalities (variables) with specified ranges, consider using LT, GT, LTET, GTET for clarity and to minimize errors.
207. To solve for the values of a recursive sequence you need to be given... the recursive rule and ALSO the value of one of the items in the sequence. Whenever you look at a recursive formula, articulate it's meaning in your mind. If necessary, also write out one or two specific relationships that the recursive formula stands for. ** If you do not know the value of any one term, then you cannot calculate the value of any other.
208. When solving quadratic equations, if you have trouble determining the factors... write out all the factors, prior to applying the formula.
209. Rules of exponents when the bases x and y are non zero real numbers and the exponents are integers.
210. x^a = x^b then a = b ( x>0; x not equal to 1) This is true for all positive numbers x, except x = 1, and for all integers a and b.
211. If the cube root of x is 9 the what is the value of x? x = 9^3 = 279
212. If confused about the negative sign on a number when solving equations or inequalities, it may help to imagine that the negative sign is because of multiplication by -1. Keep this in mind when multiplying and dividing and also when squaring equations or numbers.
213. As positive proper fractions are multiplied... their value decreases. For example, 1/2 cubed is less than 1/2 squared. Refer guide 4 page 94.
214. A Proper Fraction has a numerator less than the denominator. An improper fraction has a numerator equal to or greater than the denominator.
215. Remainder must always be less than the divisor.
216. When you divide an integer by a positive number N, the possible remainders range from 0 to (N-1). Thus there are N possible remainders. Negative remainders are not possible, nor are remainders equal to or larger than N.
217. If x/y has a remainder of 0 and z/y has a remainder of 3, then what is the remainder of xz/y? 0. Since x/y has a remainder of 0.
218. 4/5 has a remainder of 4.
219. Can 0 be categorized as odd or even? Yes, zero is the first even number.
220. If f and g are prime numbers, what is f + g? Cannot be determined from the information given because 2 is a prime number.
221. The 10ths digit of the product of two even integers divided by 4 vs The 10ths digit of the product of an even and and odd integer divided by 4. Cannot be determined from the information given. The tenths digit of the product of an even and an odd integer could be divisible by 4. However, it could also not be divisible by 4.
222. When is |x-4| equal to 4-x? When x is than or equal to 4. DO NOT OVERLOOK THE SCENARIO IN WHICH THE VALUE OF THE ENTIRE EXPRESSION IS 0.
223. The average of any set of consecutive integers with an EVEN number of items is NEVER an even integer.
224. If a>b and ab<0 then, a>0 and b<0
225. Another way to think of |x - 3.5| is the distance on a number line from x to 3.5.
226. Is 6/5n greater or less than n? Cannot be determined if you do not know whether n is positive or negative. THIS IS EXTREMELY IMPORTANT ON THE GRE. DO NOT ASSUME THAT THE NUMBER IS POSITIVE UNLESS SPECIFIED.
227. To simplify complex fractions a good first step is to SPLIT THE NUMERATOR. DIVIDE EACH TERM IN THE NUMERATOR BY THE DENOMINATOR.
228. To switch from an improper fraction to a mixed number figure out the largest multiple of the denominator that is less than or equal to the numerator. 5/4 = 4/4 + 1/4
229. To compare fractions and to estimate computations involving fractions USE BENCHMARK VALUES. Remember, try to make your rounding errors partially cancel each other out by rounding some numbers up and others down.
230. When dealing with percents, always pay attention to the size of the original value. 20% of a small number is less than 20% of a larger number.
231. 0! 1
232. If two numbers have a finite sum (396 + 404 = 800 and 398 + 402 = 800) their product will get larger as the numbers get closer together. 4 x 4 is greater than 3 x 5, 99 x 101 is greater than 97 x 103 and so on for any similar example that you can think of.
233. In geometry, for a finite perimeter, the area of a shape is maximized by making the shape as "regular" as possible. That is, the more equilateral the shape, the greater the area.
234. Third Side Rule The third side of a triangle must be less than the sum of the other two sides and greater than their difference.
235. Knowing the signs of what you are multiplying or dividing is enough to know the sign of the answer. BUT, when adding or subtracting, you need to know the relative sizes of what you are adding or subtracting.
236. When adding or subtracting, to know the sign of the answer, you need to know the relative sizes of what you are adding or subtracting. When multiplying and dividing, knowing the signs of what you are multiplying or dividing is enough to know the sign of the answer.
237. For a rectangle (e.g. TV) with a fixed perimeter or even diagonal (both the perimeter and the diagonal depend on width and height) the area is maximized when the aspect ratio is 1. The closer the aspect ratio is to 1, the larger the area will be.
238. For the Third Side Rule: If the two sides of a triangle are x and y, then the possible values of the third side must lie between x+y and x-y. This will give you a range of values to choose from. Make a brief table.
239. For a triangle with 2 GIVEN sides, the area is maximized if these 2 sides are placed at right angles to each other.
240. For mensuration problems please remember to include pi as part of the answer wherever it is necessary. DO NOT FORGET. READ THE QUESTION CAREFULLY AND COPY VALUES CORRECTLY. ESPECIALLY AREAS, DIAMETERS, PERIMETERS, RADII etc...
241. 21 squared 441
242. 22 ^ 2 484
243. 23 ^ 2 529
244. 24 ^ 2 576
245. 25 squared 625
246. For an isosceles right triangle or a 30-60-90 triangle, if you are given the area, then you can calculate the lengths of ALL the sides because you already know the ratios. Area = 1/2 x (short sides) ^ 2 for isosceles right Area = 1/2 x (short side) x (root 3 times short side) for the 30-60-90 triangle.
247. For a circles problem, if asked to calculate the length of a sector, then calculate this with respect to the central angle. Avoid making the mistake of using the inscribed angle for your ratio calculations with respect to the total circumference of the circle.
248. For circles, arcs may be defined or labeled unusually as seen on page Geometry 106 problem 10. Refer to how arc AXB is defined. It is the same as sector AXC.
249. When doing calculations on paper LABEL EACH STEP CLEARLY to ensure that you do not mix up numbers and quantities.
250. When adding or subtracting (especially for simultaneous and other linear equations) make sure that you include the signs and the magnitudes of the variables CAREFULLY. ONLY IF TIME PERMITS, RECHECK.
251. Multiplying the numerator of a positive, proper fraction by a number greater than 1 increases the numerator. As the numerator of a positive fraction increases, its value increases.
252. Divisibility by 7 No rule specified.
253. Equation for a parabola y = ax^2 + bx + c for a regular (vertical) parabola x = ay^2 + by + c for a sideways (horizontal) parabola
254. The "vertex" form of a (regular, vertical) parabola with its vertex at (h, k) is: regular: y = a(x – h)2 + k OR regular: 4p(y – k) = (x – h)2 The important thing to notice is that the h always stays with the x, that the k always stays with the y, and that the p is always on the unsquared variable part.
255. The "vertex" form of a (sideways, horizontal) parabola with its vertex at (h, k) is: sideways: x = a(y – k)2 + h OR sideways: 4p(x – h) = (y – k)2 The important thing to notice is that the h always stays with the x, that the k always stays with the y, and that the p is always on the unsquared variable part.
256. State the vertex and focus of the parabola having the equation (y – 3)^2 = 8(x – 5). Comparing this equation with the conics form, and remembering that the h always goes with the x and the k always goes with the y, I can see that the center is at (h, k) = (5, 3). The coefficient of the unsquared part is 4p; in this case, that gives me 4p = 8, so p = 2. Since the y part is squared and p is positive, then this is a sideways parabola that opens to the right. The focus is inside the parabola, so it has to be two units to the right of the vertex: vertex: (5, 3); focus: (7, 3)
257. State the vertex and directrix of the parabola having the equation (x + 3)^2 = –20(y – 1). The temptation is to say that the vertex is at (3, 1), but that would be wrong. The conics form of the equation has subtraction inside the parentheses, so the (x + 3)2 is really (x – (–3))2, and the vertex is at (–3, 1). The coefficient of the unsquared part is –20, and this is also the value of 4p, so p = –5. Since the x part is squared and p is negative, then this is a regular parabola that opens downward. This means that the directrix, being on the outside of the parabola, is five units above the vertex. vertex: (–3, 1); directrix: y = 6
258. Every terminating decimal shares this characteristic If, after being fully reduced, the denominator of a fraction has any prime factors besides 2 or 5, then its decimal will not terminate (it will repeat). If the denominator only has factors 2 and/or 5, then the decimal will terminate.
259. If both the numerator and denominator of a fraction are irrational numbers this means they are decimals which never exhibit a repeat pattern and therefore cannot be expressed as fractions with integers.
260. To find the units digit of a product or a sum of integers only pay attention to the units digit of the numbers you are working with. Drop any other digits.
261. Which integer values of b would give the number 2002/10^-b a value between 1 and 100? CAUTION Please be cautious while moving decimal points and making sure that you include ALL possible values.
262. Distance between 2 points (x1,y1) and (x2,y2) on a coordinate system is given by the formula whole square root of [(x2-x1) ^ 2 + (y2 - y1) ^ 2]
263. x intercept of a line is obtained when y = 0
264. y intercept of a line is obtained when x = 0
265. When comparing fractions a shortcut used is cross-multiplication Set up the fractions next to each other. Cross-multiply the fractions and put each answer by the corresponding numerator (NOT THE DENOMINATOR) Smaller number, smaller fraction; larger number, larger fraction. Try 7/9 compared to 4/5.
266. For geometry problems that use the same figure but varying values... REDRAW THE FIGURE TO AVOID CONFUSION. This should only take a few seconds.
267. (1/2) ^ - 1/2 2 ^ 1/2
268. (- 1/2) ^ - 1/2 VERIFY For even order roots, for any negative number n, there are no roots. What about even order negative fractions? Check.
269. 3 ^ 5.5 3 ^ 11/2
270. A great way to solve successive percent problems is to choose real numbers and see what happens. Usually, 100 will be the easiest real number to choose for percent problems.
271. The fastest way to success with percent problems WITH UNSPECIFIED AMOUNTS is to pick 100 as a value.
272. Probability of an event happening Number of ways it can happen (i.e. the number of ways in which one may achieve a successful outcome)/ Total number of ALL the equally possible outcomes (total number of ALL possible ways that are available)
273. For probability tree diagrams maximize the angle between the branches after anticipating the total number of branches expected. This will ensure an organized worksheet. Only populate the relevant branches, not all of them, to save time and reduce clutter.
274. For probability tree diagrams make sure that the sum of all he branches is 1.
275. 45% of the children in a school have a dog, 30% have a cat, and 18% have a dog and a cat. What percent of those who have a dog also have a cat? Venn diagram approach for this type of probability question is adequate. 40%
276. probability questions, with Replacement: the events are Independent (the chances don't change)
277. For probability questions, without Replacement: (e.g. marbles in a bag) the events are Dependent (the chances change)
278. To determine whether events are dependent or independent A Tree Diagram: is a wonderful way to picture what is going on.
279. The probability of event B given event A equals the probability of event A and event B divided by the probability of event A. Venn diagram approach
280. Probability of event A and event B equals the probability of event A times the probability of event B given event A P(A and B) = P(A) x P(B|A)
281. For permutations use n ^ r when n is the number of things to choose from, and you choose r of them (Repetition allowed, order, rank, position, assignment, uniqueness matters)
282. For permutations without repetition use n! divided by (n - r)! where n is the number of things to choose from, and you choose r of them (No repetition, order matters)
283. Combinations without Repetition n! divided by r!(n-r)! where n is the number of things to choose from, and you choose r of them (No repetition, order doesn't matter)
284. Combinations with Repetition (n+ r - 1)! divided by r!(n-1)! where n is the number of things to choose from, and you choose r of them (Repetition allowed, order doesn't matter)
285. For combinatorics with repetition you may verify your answer using ANAGRAM GRIDS.
286. If seven people board an airport shuttle with only three available seats, how many different seating arrangements are possible? (Assume that three of the seven will actually take the seats.) Draw an anagram grid.
287. If three of seven standby passengers are selected for a flight, how many different combinations of standby passengers can be selected? DRAW AN ANAGRAM GRID.
288. For permutations with repetition, to reduce the number of available choices you may at times need to divide the total number of available permutations by the factors that are repeated: consider the case when multiple factors need to be accounted for. In how many different ways can the letters in the word "LEVEL" be arranged?
289. For Venn diagrams, if a value is neither given nor able to be calculated, then simply assume that it is 0.
290. % Profit/Loss [(Income - Expenditure)/Expenditure] x 100
291. When applying the rules of direct and inverse proportion to fractions make sure that you are extra cautious. Ensure that you have the correct values in the numerator and the denominator. These appear quite frequently in rate, time, work problems.
292. relative speed and similar rate problems make sure that the rates and the times have the same units.
293. Yana and Gupta travel for the same amount of time till the time they meet between x and y. So, the distance covered by them will be the same as the ratio of their speeds. Ratio of time taken Yana : Gupta :: x + 4 :: x + 9 => Ratio of speeds of Yana : Gupta :: x + 9 :: x + 4 or 1 : x+4/x+9
294. 3/4 of a man's usual speed means he takes 4/3 of his usual time to cover the same distance, i.e. he takes 4/3 - 1 = 1/3 time extra.
295. For a boat traveling in a stream simply add the speeds of the boat and the stream if going downstream. Subtract the speed of the stream if the boat is going upstream.
296. If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of 'a'? 363
297. Find the largest five digit number that is exactly divisible by 7, 10, 15, 21 and 28. 99960. Given multiple choices.
298. Greatest common divisor or highest common factor is found by listing down factors common to all numbers in their respective lowest power and multiplying them.
299. Test of Divisibility by 11 If the digits at odd and even places of a given number are equal or differ by a number divisible by 11, then the given number is divisible by 11. 6390484584
300. If a number 'n' can be expressed as ap * bq, where a and b are prime factors of n, the number of factors of n = (p + 1)(q + 1)
301. How many numbers are exactly divisible by 49 between 1 and 5000? 5000/49 = 102 numbers.
302. For the Pythagorean theorem, remember that 1 is a square of itself and may be one side of a right angled triangle i.e. one of the Pythagorean triplets.
303. Least common multiple of 2 numbers a and b is calculated by finding the prime factorization of both a and b then taking the product of the sets of primes with the highest exponent value among a and b.
304. For percentage problems with unknown values Read through the problems in their entirety prior to beginning any major calculations. You may decide to opt for an alternative approach e.g. choosing smart numbers versus a variable x.
305. Rs.432 is divided amongst three workers A, B and C such that 8 times A’s share is equal to 12 times B’s share which is equal to 6 times C’s share. How much did A get? Note that this is not the same as the ratio of their wages being 8 : 12 : 6 In this case, find out the L.C.M of 8, 12 and 6 and divide the L.C.M by each of the above numbers to get the ratio of their respective shares. The L.C.M of 8, 12 and 6 is 24. Therefore, the ratio A:B:C :: 24/8 : 24/12 : 24/6 => A : B : C :: 3 : 2 : 4
306. When computing the result of successive percent changes and individual percent changes the multiplier concept can save you time in computation. e.g. reduction of 5% and then increase by 10% on P would be computed as 0.95 x 1.10 x P.
307. The final value of principal P plus interest at a rate of r, compounded annually for t years is given by F = P(1+r)^t, where r is expressed as the decimal equivalent of the annual percentage interest rate.
308. Squaring on the online calculator using 'x=' 1.2x=x=x1000
309. 1 - 0.01 0.99
310. 1 - 0.02 0.98
311. 1 - 0.03 0.97
312. 1 - 0.04 0.96
313. 1 - 0.05 0.95
314. a/9, ab/99, abc/999 ..... a recurring, ab recurring, abc recurring......
315. If asked to convert a recurring number into a fraction You may assign this value to a variable x. e.g. for a 2 digit (ab) recurrance, 100x = ab.ababababab 100x = ab + x ab = 100x - x
316. root 3 1.7320508
317. root 2 1.4142136
318. When asked to compare slopes think Rise/run In order to help visualize this, you may draw a right angled triangle. The steeper line "falls" faster, has a larger absolute value of rise/run.
319. For negative slopes, a steeper slope means rise/run has a larger absolute value, but the slope of the flatter line is the larger quantity. The slope of the flatter line is less negative.
320. When a quadrilateral is inscribed inside a circle opposite angle must add up to 180.
321. circumference of a circle equals pi times the diameter of the circle i.e. pi times d or pi times 2r.
322. Variables versus smart numbers Variables: more assurance for multiple scenarios. Just make sure that the calculations are simple enough and that the quantities are easily comparable.
323. When an integer is raised to an integer power, it falls into one of 4 possible cases (imp)*** A -ve integer ^ an even power - +ve A +ve integer ^ an even power - +ve A -ve integer ^ an odd power - - ve A +ve integer ^ an odd power - +ve True for both +ve and -ve exponents. Test it out with nos.
324. When 2 even numbers are multiplied the product is always divisible by 4.
325. Are fractions odd or even numbers? NO.
326. 3! 6
327. 4! 24
328. 5! 120
329. 6! 720
330. 7! 5040
331. 8! 40320
332. 9! 362880
333. 10! 3628800
334. If you are given a set of numbers, of which one is a variable, and you need to calculate the median or change in the median, then you may need to compute the median without the variable first, and then list case by case how the median changes with a change in the value of the variable.
335. After reading the question it may help to scan the answer choices prior to calculation to see if any of the choices may be eliminated by a simple process e.g. substitution.
336. Rhombus Opposite sides and opposite angles are equal. All sides are also equal.
337. Parallelogram Opposite sides are equal. Opposite angles are equal.
338. n is an integer and |2n+7| is less than or equal to 10, may also be written as -10 is less than or equal to 2n+7 which is less than or equal to 10.
339. When looking for representative cases for say x^2 - ax + b = 0 you can begin with the solutions rather than trying to work directly with substituted values for the eqn. e.g. (x-1)(x-2)
340. If a is not the square of an integer, its square root cannot possibly be the square of an integer (in fact, the square root of a cannot even be an integer itself, since a is not a perfect square).
341. 1/2 (y - 1) = |x - 4| may also be written as y = 2|x-4| + 1 The "notch" of the absolute value function will be located at the value of x for which the absolute value reaches the minimum possible value of 0.
342. Any number (+ve) divided by a proper fraction becomes larger
343. Any number (+ve) multiplied by a proper fraction becomes smaller
344. GDP/GDP per capita number of people GDP per capita simply means GDP per person
345. Debt as a percentage of GDP Total Debt/Total GDP
346. Debt per capita Total debt/Total number of people
Test approach Attempt the questions in order. Make quick, decisive judgements. Don't dwell on incompletes or unknowns, make a selection and move on. Expect to run into problems of this type, it might happen.
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