GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Of the five factor pairs, three yield integer values for x and y.


GMAT number properties note: Double-check that they fit the equation. To maximize the value of q, you would need to constrain p to NOT be a multiple of 10.


GMAT number properties note: In other words, q should count every factor of 10 in 200!, and p would be the product of the remaining factors of 200!. To count factors of 10 in 200!, you could start by counting the multiples of 10 between 1 and 200, inclusive.


GMAT number properties note: But this method would undercount the number of 10’s that are factors of 200! Because it would miss other pairs that can create 10, such as:.


GMAT number properties note: Therefore, this problem is really about counting the number of 2 × 5 factor pairs that can be made from the factors of 200. Of 2 and 5, there will be fewer 5’s overall, so count the number of 5’s found among the factors of 200!.


GMAT number properties note: Number of Multiples between 1 and 200, Inclusive Multiples of 5 ( = 51) Multiples of 25 ( = 52) 200! Has 2 and 5 as factors, which multiply to 10, creating another factor of 10.


GMAT number properties note: Has 6 and 15 as factors, which multiply to 90, creating another factor of 10.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Number of Multiples between 1 and 200, Inclusive Multiples of 125 ( = 53) 125 is the only one: 1 The higher multiples contribute more than one factor of 5 to the total.


GMAT number properties note: There are eight multiples of 25 in the range (namely, 25, 50, 75, 100, 125, 150, 175, and 200). Each of these is also a multiple of 5, so each has already been counted once, and thus each of the eight multiples of 25 contributes one additional factor of 5.


GMAT number properties note: Finally, 125 contributes a total of three 5’s to the count—but two have already been counted, leaving only one additional factor of 5 to include. Therefore, the prime factorization for 200!


GMAT number properties note: Includes 40 + 8 + 1 = 49 factors of 5. Since x ≠ 0, x y does not equal 0 y or 0.


GMAT number properties note: This statement indicates that x is equal to 2 or −2. The statement indicates nothing about y.


GMAT number properties note: Simplify using exponent rules, noting the common factors of 6 and 8 on each side of the equation. Since y = 0 and x ≠ 0 (as stated in the question stem), this information is sufficient to conclude that x y = x0 = 1.


GMAT geometry note: Watch divisibility, factors, remainders, parity, and prime structure. The question stem specifies that the triangle’s sides all have integer lengths.


GMAT geometry note: Perimeter is calculated by summing the three side lengths. In order for that sum to be odd, either all three numbers have to be odd or one of the three numbers has to be odd.


GMAT geometry note: If the triangle is a right triangle, then the Pythagorean theorem (a2 + b2 = c2) holds. If all three side lengths are odd, this would mean odd2 + odd2 = odd2.


GMAT geometry note: If one side length is odd and the other two are even, then there are two possibilities:. Every right triangle possibility is impossible, so the triangle cannot be a right triangle.


GMAT geometry note: If, as this statement indicates, 2A is not an integer, then A itself is not an integer. As a result, at least one of b and h is also not an integer.


GMAT geometry note: For right triangles, b and h are the two shorter sides of the triangle. The question stem specifies that all three sides are integers, so if the triangle were a right triangle, then b and h would both have to be integers.


GMAT geometry note: This is impossible, so the triangle cannot be a right triangle. O + O + O = odd O + E + E = odd odd2 + even2 = even2 (impossible: odd + even = odd) even2 + even2 = odd2 (also.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. The problem specifies that x has to be positive, so try some small positive integers to see how the expression works.


GMAT algebra note: If x = 1, then the expression becomes . If x = 2, then the expression becomes 1 + 1 = 2.


GMAT algebra note: What if x is a fraction, such as ? In that case, the expression becomes ...


GMAT algebra note: It’s even bigger than the integer options above. The expressions and are reciprocals of each other.


GMAT algebra note: As long as x is positive, there are two broad scenarios possible:. If one expression is less than 1, then the other has to be greater than 1.1.


GMAT algebra note: If one expression equals 1, then the other also equals 1.2. If the average of the cost of the three models is $900, then the sum is $2,700.


GMAT algebra note: Call the three models a, b, and c, in order from least expensive to most. The statement indicates that c = 1.25b, so a + b + 1.25b = 2,700.


GMAT algebra note: This isn’t enough to solve for specific values, but it might be enough to tell whether c > $1,000.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. By definition, a < b, so plug LTb (less than b) into the equation for a.


GMAT word problem note: From here, estimate: 13 goes into 2,600 a total of 200 times and it goes into 100 approximately 7 times. Therefore, LTb = (207)(4) = 828.


GMAT word problem note: If b is approximately 830, then 125% of b is definitely larger than 1,000. Therefore, the most expensive model does cost more than $1,000.


GMAT word problem note: Follow a similar path: c = b + 210. If b is a bit more than 830, then b + 210 is definitely over 1,000.


GMAT word problem note: The product of the slopes of two perpendicular lines is −1.


GMAT algebra note: Thus, the slope of a line perpendicular to k is −2. Use the desired slope to create a slope-intercept equation and look for a match among the answers.


GMAT algebra note: Alternatively, put each choice in slope-intercept form: y = mx + b, where the slope is m. Stop when you find the correct answer.


GMAT algebra note: The question stem asks a Yes/No question:. Manipulate the absolute value expression to represent this inequality on a number line.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This can be interpreted as, “The distance between x and 6 is less than 5.” On a number line, this is the region between 1 and 11.


GMAT algebra note: All possible values for x are positive, so the answer to the question is a definite Yes. Manipulate the inequality to get 0 on one side, then factor the resulting quadratic.


GMAT algebra note: The factored quadratic on the le side will equal 0 when x = 3 and 7. These are the boundary points.


GMAT algebra note: On a number line, check the regions on either side of these boundary points to determine the valid region(s) for x. Note that when 3 < x < 7, (x − 7)(x − 3) = (neg)(pos) = neg.


GMAT algebra note: Since both positive and negative values are possible for x, the answer is Maybe.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since you are looking for the minimum number of K-numbers that sum to 600, a practical place to start is with the largest K-number less than 600, or 222.


GMAT number properties note: There are between two and three multiples of 222 in 600, so subtract out the two whole multiples. Now, the next largest K-number is 22.


GMAT number properties note: Again, subtract as many whole multiples as possible. The next largest K-number is 2.


GMAT number properties note: Thus, 600 = 222(2) + 22(7) + 2(1), and the K-weight of 600 is 2 + 7 + 1 = 10. If the greater of the two integers is x, then the two integers can be expressed as x − 1 and x.


GMAT number properties note: The sum of the reciprocals would therefore be:. Alternatively, choose smart numbers.


GMAT number properties note: The smaller number would therefore be 3 − 1 = 2. The sum of the reciprocals would be.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. The sequence A is defined as follows:.


GMAT fractions, decimals, and percents note: A1 = 1, and An = An − 1 + (−1)n + 1(n2) for all integer values n > 1. What is the value of A15 − A13?


GMAT fractions, decimals, and percents note: If d represents the hundredths digit and e represents the thousandths digit in the decimal 0.4de, what is the value of this decimal rounded to the nearest tenth? D − e is a positive perfect square.(1) 133.


GMAT fractions, decimals, and percents note: If x is a positive integer greater than 1, and y is the smallest positive integer that is evenly divisible by every integer between 1 and x, inclusive, what is the value of x?


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. A chain is comprised of 10 identical links, each of which independently has a 1% chance of breaking under a certain load.


GMAT fractions, decimals, and percents note: If the failure of any individual link means the failure of the entire chain, what is the probability that the chain will fail under the load? If Ash plans to travel by bus from City A, to City B, to City C, and back to City A, what is the shortest possible duration, in minutes, of his entire trip?


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT quant note: If DP = PQ = QB = 1, what is the length of AB?


GMAT coordinate geometry note: In the rectangular coordinate system above, point A is not shown. If the area of triangle OAB is at least 16, which of the following could NOT be the coordinates of point A?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Workout Set 14 Solutions 31.


GMAT sequences and patterns note: Generate the first several values of the sequence, using the given relationship. Notice that a component such as (−1)n + 1 switches the sign of the additive term back and forth.


GMAT sequences and patterns note: Use a table to keep your work organized. N An = An − 1 + (−1)n + 1(n2) 2 1 + (−1)2 + 1(22) = 1 − 4 = −3 3 −3 + (−1)3 + 1(32) = −3 + 9 = 6 4 6 + (−1)4 + 1(42) = 6 − 16 = −10 5 −10 + (−1)5 + 1(52) = −10 + 25 = 15 6 15 + (−1)6 + 1(62) = −15 − 36 = −21 What is the pattern?


GMAT sequences and patterns note: The sign alternates between positive and negative. Ignoring the sign of the terms (taking absolute values) might help to determine the full pattern.


GMAT sequences and patterns note: N An = An − 1 + (−1)n + 1(n2) |An| Change from Previous Term.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. N An = An − 1 + (−1)n + 1(n2) |An| Change from Previous Term That is, the absolute value of each term equals the absolute value of the previous term plus n.


GMAT fractions, decimals, and percents note: Following this pattern, |A15| is 15 greater than |A14|, and |A14| is 14 greater than |A13|. Thus, |A15| − |A13| = +15 + 14 = 29.


GMAT fractions, decimals, and percents note: When rounding to the nearest tenth, use the hundredths digit, or d in this case. Thus, a rephrase of this question is, “Is d < 5 (or, equivalently, is d ≥ 5)?” (1) INSUFFICIENT:.


GMAT fractions, decimals, and percents note: Since d − e is positive, d > e. Since d and e are digits (i.e., 0, 1, 2.


GMAT fractions, decimals, and percents note: Three perfect squares less than or equal to 9: d − e = 1, 4, or 9.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. D e d − e = Perfect Square Is d < 5?d e d − e = Perfect Square Is d < 5?


GMAT statistics note: 9 0 9 − 0 = 9 ✓ No 1 0 1 − 0 = 1 ✓ Yes It is possible for d to round up or down, so this statement isn’t sufficient. Since d and e are positive, you can square each side of the inequality without worrying about flipping the sign: d > e4.


GMAT statistics note: Since d and e are digits 24 = 16, which is too large.) Test cases again. Whenever possible, reuse numbers that you tried in the last statement, assuming those numbers are allowed by the new statement.


GMAT statistics note: 9 0 3 > 0 ✓ No ✓ Yes It is again possible for d to round up or down, so this statement isn’t sufficient. Taking both statements together, e must be 0 or 1 and d − e must equal 9, 4, or 1.


GMAT statistics note: You’ve already tested the d = 9 and d = 1 scenarios for both of the statements individually, so you don’t need to retest them here. Even together, the two statements allow d to round up or down.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. List a few values of x and y to gain understanding.


GMAT number properties note: Doing so also prepares you to most easily interpret the statements. There is only one: x = 6 and y = 60.


GMAT number properties note: To be completely confident that this is the only valid case, approach the problem . However, x itself cannot have any common factors with x – 1, so 10 must be divisible by x – 1.


GMAT number properties note: So x – 1 equals 1, 2, 5, or 10, so x must be 2, 3, 6, or 11. Therefore, x = 2 and x = 3 have already been eliminated as possibilities and x = 6 has already been confirmed to work.


GMAT number properties note: The only remaining case that needs to be checked is x = 11. Since 110 is not divisible by every number smaller than 11 (e.g., 110 is not divisible by 3 or 4), x = 11 is not a valid case.


GMAT number properties note: The only possible answer is x = 6, so this statement is sufficient. There are two possible answers, so the statement is insufficient.


GMAT word problem note: Separate total outcomes from desired outcomes and count carefully. Qualitatively, many failure scenarios could occur.


GMAT word problem note: Given the complexity of the failure scenarios, it is easier to look at the opposite scenario. For each of the links, the probability that it will not fail is 1 − 0.01 = 0.99.


GMAT word problem note: The probability that all 10 will not fail is thus (0.99)10, since the probability that all 10 will not fail equals the product of the probabilities of the individual links not failing. Therefore, the probability that at least one link will fail equals 1 − (0.99)10.


GMAT word problem note: None of the links will fail. Exactly one of the links will fail.


GMAT word problem note: Exactly two of the links will fail. Probability that at least one link will fail = 1 − probability that No links will fail 35.


GMAT word problem note: Ash’s round trip consists of two parts: the time spent on the bus and the time spent waiting for the next bus. No matter which buses he takes, the time spent on the bus will always be equal: 25 + 35 + 60 minutes = 120 minutes.


GMAT word problem note: This question is really about minimizing the time that Ash spends waiting. Note the mismatch between the departure frequencies and the trip durations.


GMAT word problem note: Because buses leave each city every 20 or 30 minutes, and both of these time periods divide evenly into one h.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Then, to minimize waiting, Ash should take the first available buses in City B and City C.


GMAT word problem note: For each of the three possible times that Ash could leave City A, note the next available buses for each leg of his trip.


GMAT word problem note: Leave A Arrive in B Leave B Arrive in C Leave C Arrive in A Total Time 5:00 5:25 5:30 6:05 6:20 7:20 2 hr 20 min 5:20 5:45 6:00 6:30 6:40 7:40 2 hr 20 min 5:40 6:05 6:30 7:05 7:20 8:20 2 hr 40 min Therefore, the shortest possible duration of the entire trip is 2(60) + 20 = 120 + 20 = 140 minutes. Consecutive integers differ by exactly 1.


GMAT word problem note: For x and y to be consecutive, either x = y + 1 or x = y − 1. Rephrase the question as, “Does x equal either (y + 1) or (y − (1) SUFFICIENT:.


GMAT word problem note: There’s only one term with x but a couple with y, so it is easiest to solve for x. Since x and y are both positive, you can drop the absolute value signs and conclude that x = y + 1.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Therefore, x = 0 or x − y − 1 = 0.


GMAT algebra note: The question stem indicates that x is positive, so x cannot equal 0. As a result, it must be the case that x − y − 1 = 0, or x = y + 1.


GMAT algebra note: The two integers are consecutive. Solve for the expression in question by adding (−x2 − x) to both sides.


GMAT algebra note: The answer depends on the value of x, which is not given.


GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. Each statement provides a different expression equal to y − x2 − x.


GMAT coordinate geometry note: Set these two equal to each other and solve. Once the squared terms drop out, the equation is solvable.


GMAT coordinate geometry note: Stop when you know you can solve! This problem asks you to compare two “nested” functions: f(g(x)) and g(f(x)).


GMAT coordinate geometry note: To simplify a nested function, replace the variable term in the outer function with the entire inner function. To find the values of x for which f(g(x)) < g(f(x)), simplify the following inequality.


GMAT coordinate geometry note: One possibility is to multiply out both quadratics and then simplify as much as possible.


GMAT coordinate geometry note: In turn, this simplifies to –2 < x < 2. Another approach involves using the multiples of 9 in the initial inequality.


GMAT coordinate geometry note: Factor out a 9 on the le side of the inequality. Start by labeling the three line segments with length 1.


GMAT coordinate geometry note: Label the unknown length as x.


GMAT geometry note: From here, there are two ways to approach the problem. One approach is to use the Pythagorean theorem to label the lengths of the missing sides.


GMAT geometry note: Triangle ABP is a right triangle, so the following is true:. Similarly, APD is a right triangle, and the following is true:.


GMAT geometry note: Finally, since BAD is a right triangle, the Pythagorean theorem can be used to solve for the value of x.


GMAT geometry note: The other solution uses the fact that triangles APD and BPA are similar. To prove this, note that APD is a right angle.


GMAT geometry note: Therefore, angles PAD and ADP sum to 90 degrees. Therefore, ADP and BAP have the same degree measure.


GMAT geometry note: Since triangles APD and BPA have two pairs of matching angles, the third pair of angles must match as well, and the two triangles are similar. The sides line up as follows:.


GMAT geometry note: Triangle PAD Triangle PBA Shortest Side side PD side AP Middle Side side AP side BP Longest Side side AD side AB Since PD = 1, BP = 2, and AP = AP, you can set up a proportion to find the length of side AP.


GMAT geometry note: Finally, use the Pythagorean theorem to determine the length of AB. Side AB appears to be longer than BP, but shorter than BD.


GMAT geometry note: Its length should be between 2 and 3.


GMAT geometry note: The triangles they yield will have the same area. Therefore, neither can be the correct answer.


GMAT geometry note: The question is asking for triangles with area less than 16 square units, so first try drawing a triangle with an area of exactly 16 square units. If the base of the triangle is OB, it has a length .


GMAT geometry note: Use the area formula to find the height. Draw the height of the triangle perpendicular to OB through the origin in either quadrant II or quadrant IV.


GMAT geometry note: The vertex of the height is at either (–4, 4) or (4, –4). Any triangle with the same height will have the same area.


GMAT geometry note: Therefore, if A lies on the line y = x + 8 or on the line y = x – 8, the triangle will have an area of 16. If A lies anywhere between those two lines, then the triangle will have an area less than 16.


GMAT geometry note: Of the answers, only (1, 8) lies between the two lines.


GMAT geometry note: Rather than calculating the exact area of each triangle, notice that each one is fairly close to a triangle whose area is easier to calculate. The latter triangle has an area of exactly 16 square units.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If , where a and b are integers and a does not equal 1 or −2, which of the following could be the value of b?


GMAT number properties note: The currencies of three countries are called credits, units, and bells. If 12 credits can be exchanged for a total of exactly 8 units and 5 bells, what is the smallest integer number of units that can be exchanged for an integer number of bells, with no change le over?


GMAT number properties note: 15 credits can be exchanged for 4 units and 25 bells, with no change le over.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. 25 bells can be exchanged for 8 units, with no change le over.(2) 143.


GMAT number properties note: Each digit 1 through 5 is used exactly once to create a five-digit integer. If the 3 and the 4 cannot be adjacent digits in the integer, how many five-digit integers are possible?


GMAT number properties note: For positive integers a, b, and c, b is 120% greater than a, and c is 25% less than the sum of a and b. Which of the following could be.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. A soccer competition consisted of two rounds.


GMAT sequences and patterns note: In each round, the 20 teams competing were divided randomly into 10 pairs, and each pair played a single game. None of the games resulted in a tie, and of the teams that lost in the first round, 7 also lost in the second round.


GMAT sequences and patterns note: How many teams won both of their games? What is the units digit of the positive integer x?


GMAT sequences and patterns note: , where y is a positive integer.(1) , where z is a positive integer.(2).


GMAT geometry note: If x and y are positive integers, what is the value of x + y? For all n such that n is a positive integer, the terms of a certain sequence B are given by the following rules:.


GMAT geometry note: Bn = Bn − 1 + 5 if n is odd and greater than 1 Bn = −Bn − 1 if n is even B1 = 3.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. What is the sum of the first 65 terms in the sequence?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Workout Set 15 Solutions 41.


GMAT algebra note: Since the fraction equals 0, either ab + 6 = 0 or ab − 3 = 0. Since a and b are integers, and ab = −6 or ab = 3, it must be the case that b is a factor of either −6 or 3.


GMAT algebra note: If b = 1, then a = −6 or a = 3, which are both allowed.I. If b = 2, then a = −3 or a = .


GMAT algebra note: One of those (a = −3) is allowed.II. If b = 3, then a = −2 or a = 1, neither of which is allowed.III.


GMAT algebra note: Use three variables to represent the unknown values in this problem. Therefore, your goal is to find the relationship between the value of a unit and the value of a bell.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. This statement can be rewritten as the following equation, using the same variables as above.


GMAT algebra note: Therefore, 8 units are worth the same amount as 25 bells. Because the ratio 8: 25 cannot be reduced further, 8 is the smallest number of units that can be exchanged for an integer number of bells.


GMAT algebra note: For instance, 4 units could only be exchanged for 12.5 bells, which is not an integer. The answer to the question is 8.


GMAT algebra note: Combine this with the equation from the question stem. Because the goal is to find the relationship between b and u, focus on eliminating the other variable, c.


GMAT algebra note: First, multiply the equations by 5 and 4, respectively, so that the coefficient of c is 60 in both.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If 8 units can be exchanged for 25 bells, then because 8 and 25 have no common factors, no smaller integer number of units can be exchanged for bells, as described above.


GMAT number properties note: The answer to the question is 8, and this statement is sufficient. All of these line equations are of the form y = mx + b, where m is the slope and b is the y-intercept.


GMAT number properties note: Two of these lines have a slope of and are thus parallel to each other. The other two lines are parallel to one another with a slope of .


GMAT number properties note: Two of the lines have a y-intercept of 6 while the other two lines have a y-intercept of −6.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. In each quadrant, there is a triangle with the dimensions 6–8–10, a multiple of the common 3–4–5 right triangle.


GMAT sets, probability, and counting note: Alternatively, recognize that the quadrilateral is a rhombus (four equal sides of length 10) and use the formula for the area of a rhombus: , where D indicates the length of the diagonals. In a constrained combinatorics question such as this one, it is o en easier to consider the “violating” cases instead of the “okay” cases.


GMAT sets, probability, and counting note: # of permutations that obey the constraint = # of permutations total − # of permutations that violate the constraint At first glance, ignore the constraint that 3 and 4 cannot be adjacent to determine the total number of five-digit integers possible: # of permutations total = 5!


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Now, consider the basic ways that 3 and 4 might be adjacent to each other in a five-digit number: 3 4 x x x x 3 4 x x x x 3 4 x x x x 3 4 For each of these four base cases, there are two ways to order the 3 and 4 (3, 4 and 4, 3), as well as 3!


GMAT sets, probability, and counting note: Ways the other digits (1, 2, and 5) can be arranged in the x positions. Thus: # of permutations that violate the constraint = 4 × 2 × 3!


GMAT sets, probability, and counting note: = (4)(2)(3) Therefore, the number of permutations that do not violate the constraint equals 120 − 48 = 72. Start by translating the text into math.


GMAT sets, probability, and counting note: Note that b is 120% greater than a, not 120% of a.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. There are three variables and only two equations, so you won’t be able to solve in the traditional sense, but you may be able to determine something about these integers.


GMAT number properties note: Substitute for b in the second equation. Both a and c are integers, and 5 and 12 share no prime factors.


GMAT number properties note: So a must be divisible by 5 and c must be divisible by 12. Try a = 5 and c = 12 to see whether b is a positive integer in this case.


GMAT number properties note: This is a valid case because b is a positive integer. Each team either won or lost the first game and either won or lost the second game.


GMAT number properties note: Therefore, this problem can be approached using overlapping sets.


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Won first game Lost first game Total Won second game Lost second game 7 Total 20 Initially, it appears that there is not enough information to determine how many teams won both games.


GMAT quant note: However, note that there were no ties and each round consisted of exactly 10 games. Since each of the 10 games yielded one winner and one loser, there were exactly 10 teams that won in the first round and 10 teams that lost in the first round.


GMAT quant note: Similarly, there were 10 teams that won in the second round and 10 teams that lost in the second round. Based on this information, fill in the remainder of the matrix.


GMAT quant note: Won first game Lost first game Total Won second game 7 3 10 Lost second game 3 7 10 Total 10 10 20 Seven teams won both of their games.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, x is 1 greater than a multiple of 5.


GMAT sequences and patterns note: Since all multiples of 5 end in either 0 or 5, x must end in either 1 or 6. Alternatively, you could list numbers.


GMAT sequences and patterns note: Since y is a positive integer, 5y could be 5, 10, 15, 20, etc. Thus, x could be 6, 11, 16, 21, etc.


GMAT sequences and patterns note: The units digit of x could be 1 or 6. This indicates that x is odd, because 2z is an even number.


GMAT sequences and patterns note: Any odd single- digit integer is a possible units digit for x: 1, 3, 5, 7, or 9. The question stem asks for the value of the combo x + y.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. (1) INSUFFICIENT: (x + y − 1)!


GMAT number properties note: To see the possible values of (x + y − 1), list the factorials of the first few integers. Add x to both sides of y = x2 − x + 1 to create (x + y) on one side of the equation.


GMAT number properties note: The exact value of (x + y) is unknown, as it depends on the value of x2, which could be any positive integer. (1) AND (2) INSUFFICIENT: (x + y) ≤ 5 and x + y = x2 + 1 combine to indicate that x2 + 1 ≤ 5.


GMAT number properties note: There are two integer solutions: x = 1 or x = 2. = 120 (too large) If x = 1, then y = 12 − 1 + 1 = 1 and x + y = 1 + 1 = 2.


GMAT number properties note: If x = 2, then y = 22 − 2 + 1 = 3 and x + y = 2 + 3 = 5. The size of the angle depends on two things.


GMAT geometry note: The size of the circle The distance of C from the circle (inversely related to arc length ADB) Larger circle → Larger x Farther C → smaller x Larger arc ADB → smaller x (1) INSUFFICIENT:. The rubber band picture on the right indicates that for a circle of fixed size, x can still vary with the length of arc ADB (i.e., x varies with the placement of C).


GMAT geometry note: Draw some cases to prove that x can vary for a given arc ADB.


GMAT geometry note: Or nearly half of the circle, and the lines tangent to the circle at A and B will join at a smaller angle x. Thus, arc ADB is of the circumference of the circle.


GMAT geometry note: There is only one way to draw the lines tangent to the circle at A and B, so x must be one specific value. List the first few terms of the sequence according to the given rules.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Note that the pattern is a four-term repeat: 3, −3, 2, −2.


GMAT sequences and patterns note: Also note that the sum of this repeating group is (3) + (−3) + (2) + (−2) = 0. This repeating group will occur 16 times through term number 64.


GMAT sequences and patterns note: Because the sum of the four terms is 0, you don’t actually have to figure out how many times the pattern repeats.) Thus, the sum of the first 64 terms will be 0. This leaves the 65th term, which will have the same value as B1: 3.


GMAT sequences and patterns note: Therefore, the sum of the first 65 terms is 3.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. If the store made a total profit of 16% on sales of these three models of bicycles last month, and these three models represented the entirety of the store’s profits, what percent of its total profits last month came from sales of Model A bicycles?


GMAT statistics note: The store earned 20% of its profits last month from sales of Model B bicycles. The store earned 50% of its profits last month from sales of Model C bicycles.


GMAT statistics note: A beaker contains 100 milligrams of a solution of salt and water that is x% salt by weight such that x < 90. If the water evaporates at a.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Rate of y milligrams per hour, how many hours will it take for the concentration of salt to reach (x + 10)%, in terms of x and y?


GMAT number properties note: Set A consists of five consecutive positive integers. Set B consists of every integer that is in set A as well as the sum of every pair of two distinct integers from set A, discarding any numbers that already appear in set B.


GMAT number properties note: What is the greatest integer in set B? The range of set B is 13.(1) There are exactly 12 integers in set B.(2).


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. A survey included two questions, each of which could be answered with either Yes or No.


GMAT number properties note: A total of 100 respondents took the survey, and every respondent answered both questions. How many respondents answered both questions with a Yes?


GMAT number properties note: Of the respondents who answered Yes to the first question, 40% answered Yes to the second question. Of the respondents who answered Yes to at least one question, 30% answered Yes to both questions.


GMAT number properties note: If x is a positive two-digit integer, is x divisible by 17? X is divisible by 7.(1) x has exactly 3 unique prime factors.(2) 156.


GMAT number properties note: Green paint is made by mixing blue paint with yellow paint in a ratio of 2 to x. Turquoise paint is made by mixing green paint with blue paint in a ratio of y to 2.


GMAT number properties note: In terms of x and y, how many gallons of yellow paint are required to make 10 gallons of turquoise paint?


GMAT quant note: Watch divisibility, factors, remainders, parity, and prime structure. Erin writes a list of every sixth integer in increasing order, starting with the positive integer x.


GMAT quant note: Harry writes a list of every ninth integer in increasing order, starting with the positive integer y. Will any integer appear on both Erin’s list and Harry’s list?


GMAT quant note: X is a multiple of y.(1) x – y is a multiple of 3.(2) 158. The students in a certain college class took a midterm exam and a final exam, both of which were scored out of a total of 100 points.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. What percent of the students earned a higher score on the final than on the midterm?


GMAT word problem note: Of the students in the class, 28% scored at least 6 points higher on the final than on the midterm. Of the students who scored higher on the final than on the midterm, 40% scored at least 6 points higher on the final.


GMAT word problem note: A triathlon consists of three legs: a swim, a bike ride, and a run. A race organizer analyzes the results of a certain triathlon and observes that the ratio of swim distance to bike distance to run distance was 1 to 50 to 12, and that the ratio of average swim time to average bike time to average run time for all participants was 3 to 10 to 6.


GMAT word problem note: If the average speed of racers in each leg of the race does not change, but the race organizer changes the distance of each leg of the race so that the average swim time, average bike time, and average run time are now equal to each other, what fraction of the total distance of the race will be represented by the swim leg of the triathlon?


GMAT geometry note: An unpainted wall with a total area of 100 square feet needed to be painted with three coats of paint. Nelson began the job by painting an area of 60 square feet.


GMAT geometry note: Then, Jamaica painted a total area of 70 square feet. Finally, Bryan painted a total area of 80 square feet.


GMAT geometry note: A er Bryan finished painting, an area of q square feet of the wall was covered in three coats of paint. If the entire surface area of the wall had received at least one coat of paint, which of the following specifies all the possible values of q?


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Workout Set 16 Solutions 51.


GMAT statistics note: A store sold three different models of bicycles last month. For each of these models, the profit earned by the store represents a different percentage of the sales of that model.


GMAT statistics note: The question stem also states that the total profit was 16% of the total sales. One way to handle the question stem is to use variables A, B, and C to represent the total sales of Model A, B, and C bicycles, respectively.


GMAT statistics note: Then, the profit from Model A bicycles is 0.12A, the profit from Model B bicycles is 0.16B, and the profit from Model C bicycles is 0.20C. Also, the total dollar amount of sales is A + B + C, so the following is true and can be simplified:.


GMAT statistics note: You might also notice that 16% is exactly in the middle of the three percentages provided in the question stem. The problem is effectively a weighted average problem: if more dollars worth of Model A bicycles were sold, then the average profit would be pulled downwards, closer to 12%.


GMAT statistics note: If more dollars worth of Model C bicycles were sold, then the average profit would be pulled upwards, closer to 20%. Since the actual profit was in the middle, at 16%, the dollar value of the Model A bicycles sold and the dollar value of the Model C bicycles sold must be equal.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. The question asks about the share of profits that came from Model A bicycles.


GMAT statistics note: This question can be rephrased as, “What is ?” which can then be simplified as follows:. What is What is What is The variables would cancel entirely if you could determine a known ratio of A to B.


GMAT statistics note: (1) SUFFICIENT: 20% of the store’s profits came from Model B bicycles, which can be translated and simplified as follows, remembering that A = C and the total profit equals 0.16(A + B + C). Substituting A = 2B into the question causes all unknowns to cancel out, yielding an answer of , or 30%.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The store earned 50% of its profits from sales of Model C bicycles, which can be translated and simplified as follows, remembering that A = C and the total profit equals 0.16(A + B + C).


GMAT word problem note: The answer is again 30%, as shown above, so this statement is sufficient. Therefore, it is possible to use Smart Numbers.


GMAT word problem note: Therefore, the beaker initially contains 10% of 100 mg = 10 mg of salt. The question asks how many hours it will take for the concentration of salt to reach (x + 10)%, or 20%.


GMAT word problem note: The beaker will always contain 10 mg of salt, since the salt does not evaporate. The 10 mg of salt will represent 20%, or , of the total weight when the total solution weight has been reduced to 50 mg.


GMAT word problem note: To get to this point, 50 mg of water must evaporate. This will take a total of = 25 hours.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Alternatively, approach the problem with algebra.


GMAT fractions, decimals, and percents note: Initially, the beaker contains x mg of salt in a total solution of 100 mg. A er h hours, hy mg of the solution will have evaporated, so the solution will have a total weight of (100 – hy) mg, and the amount of salt will still be x mg.


GMAT fractions, decimals, and percents note: So the new concentration, as a percentage, will be . This percent should equal (x + 10)%.


GMAT fractions, decimals, and percents note: Set up an equation and solve for h, the number of hours.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Then, set B will initially include 1, 2, 3, 4, and 5, and more numbers will be added to it as follows, discarding any numbers that already appear in set B.


GMAT number properties note: 1 + 2 = 3 discard 1 + 3 = 4 discard 1 + 4 = 5 discard 1 + 5 = 6 will be added to set B 2 + 3 = 5 discard 2 + 4 = 6 discard 2 + 5 = 7 will be added to set B 3 + 4 = 7 discard 3 + 5 = 8 will be added to set B 4 + 5 = 9 will be added to set B.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Thus, if set A contains the integers from 1 to 5, set B consists of 1, 2, 3, 4, 5, 6, 7, 8, and 9.


GMAT number properties note: The question asks for the greatest integer in set B. The range of a set is the difference between the greatest and the smallest numbers in that set.


GMAT number properties note: The smallest number in set B is the smallest number from set A, since all of the additional numbers in set B are constructed by summing numbers together and will therefore be greater. The greatest number in set B is created by summing the two greatest numbers from set A.


GMAT number properties note: Those numbers are x + 3 and x + 4, and their sum is 2x + 7. Therefore, the range of set B is (2x + 7) – x.


GMAT number properties note: Set this equal to the given range of 13, and solve for x:. The smallest number in set A is 6, and both sets can be listed with certainty.


GMAT number properties note: There is only one answer to the question, so this statement is sufficient. There are exactly 12 integers in set B, and you will likely have to test cases to determine how this could happen.


GMAT number properties note: Above, it was determined that if set A contains the integers 1, 2, 3, 4, and 5, then set B contains 9 integers in total: the integers from 1 to 9, inclusive. This is not enough integers:.


GMAT number properties note: Find a case with fewer duplicates to discard. If set A contains the integers 5.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. This set B again contains 12 integers, but its greatest integer is 207.


GMAT number properties note: This is an Overlapping Sets problem. Create a Double-Set Matrix and fill out the known information.


GMAT number properties note: Yes to Question 1 No to Question 1 Total Yes to Question 2 No to Question 2 Total 100 The question asks for the number of respondents who gave two Yes answers. Try to determine the number that belongs in the top le of the matrix.


GMAT number properties note: The matrix can be filled out as follows:. Yes to Question 1 No to Question 1 Total Yes to Question 2 0.4x No to Question 2 0.6x Total x 100 – x 100 If x = 5, then the answer is 0.4(5) = 2.


GMAT number properties note: If x = 90, then the answer is 0.4(90) = 36. More than one value is possible for the answer, so this statement is not sufficient.


GMAT number properties note: Then, the number of people who answered Yes to at least one question is.


GMAT quant note: Percent problems usually relate part, whole, rate, and percent change. Since 30% of these people answered Yes to both questions, the matrix can be filled out as shown.


GMAT quant note: Yes to Question 1 No to Question 1 Total Yes to Question 2 0.3(100 – y) No to Question 2 y Total 100 If y = 10, the answer is 0.3(90) = 27. If y = 80, the answer is 0.3(20) = 6.


GMAT quant note: Since more than one value is possible for the answer, this statement is insufficient. Using the information in both statements, you can conclude that 0.3(100 – y) = 0.4x.


GMAT quant note: This simplifies to 0.4x + 0.3y = 30, but both x and y can still have multiple values. Accordingly, the matrix can be filled out with numbers in several different ways.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. The question stem includes a constraint that x is positive and has two digits.


GMAT number properties note: Use the constraint as you solve the problem. Since x is a positive two-digit integer, x is between 10 and 99, inclusive.


GMAT number properties note: The question asks whether x is divisible by 17. Since there are only a few integers in this range that are divisible by 17, jot them down for later reference: 17, 34, 51, 68, and 85.


GMAT number properties note: The question can be rephrased as, “Is x equal to 17, 34, 51, 68, or 85?” (1) SUFFICIENT:. Positive two-digit integer x is a multiple of 7.


GMAT number properties note: Since none of the numbers listed earlier (17, 34, 51, 68, 85) are multiples of 7, x is definitely not one of these numbers. Therefore, x is not a multiple of 17, and the answer is definitely No.


GMAT number properties note: Alternatively, use logical reasoning to prove that the statement is sufficient without doing too much arithmetic. Seven and 17 are both prime numbers, so the smallest number that is divisible by both of them is their product, (7)(17) = 119.


GMAT number properties note: However, you already know that x is smaller than 119, since x only has two digits. Therefore, x cannot be a multiple of both 7 and 17, so the answer is defini.


GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. It will be simpler to choose y first, since the value of y is more closely related to the 10 gallons of turquoise paint.


GMAT fractions, decimals, and percents note: Choose y = 8, so that the sum of y and 2 in the ratio of the paints used to make turquoise can be 8 + 2 = 10. In this case, it takes 8 gallons of green paint and 2 gallons of blue paint to make 10 gallons of turquoise paint.


GMAT fractions, decimals, and percents note: Because the total amount of green paint should be 8 gallons, choose x = 6, so that the of x and 2 in the ratio to make green paint is 2 + 6 = 8. In total, it will take 6 gallons of yellow paint to create the turquoise paint, so the answer to the question is 6.


GMAT fractions, decimals, and percents note: The correct answer will simplify to 6. Erin’s list starts with an unknown positive integer and then includes every sixth integer.


GMAT fractions, decimals, and percents note: So Erin’s list could be any of the following, depending on the value of x.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. X Erin’s Listx Erin’s List etc.


GMAT number properties note: Similarly, Harry’s list could be any of the following: y Harry’s List etc. If x = y = 1, then x is a multiple of y, and the number 1 appears on both lists.


GMAT number properties note: Therefore, the answer to the question is Yes. If x = 3, all of the numbers on Erin’s list are multiples of 3.


GMAT number properties note: If y = 1, none of the numbers on Harry’s list are multiples of 3. Therefore, x is a multiple of y, but no numbers appear on both lists, so the answer to the question is No.


GMAT number properties note: If x – y is a multiple of 3, then x – y = 3n, for some integer n. The integers on the two lists are as follows.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If n is a multiple of 3, then 3n is a multiple of 9, so 3n + y appears on both lists.


GMAT fractions, decimals, and percents note: If n is 1 more than a multiple of 3, then 3n + 6 = 3(n + 2) = 3(multiple of 3) is a multiple of 9, so y + 3n + 6 appears on both lists. Finally, if n is 2 more than a multiple of 3, then 3n + 12 = 3(n + 4) = 3(multiple of 3) is a multiple of 9 so y + 3n + 12 appears on both lists.


GMAT fractions, decimals, and percents note: Regardless of the value of n, some number appears on both Erin’s and Harry’s lists, so the answer to the question is Yes. According to the question stem, each student in the class earned a certain score on the midterm and a certain score on the final.


GMAT fractions, decimals, and percents note: A student’s final score could have been higher than, equal to, or lower than his or her midterm score. To answer the question, you’ll need to determine what percent of the students fell into the first of these three categories.


GMAT fractions, decimals, and percents note: It is possible that all of the remaining students scored between 1 and 5 points higher on the final than on the midterm, which would make the answer to the question 100%. It is also possible that all of the remaining students scored higher on the midterm than on the final, which would make the answer to the question 28%.


GMAT fractions, decimals, and percents note: This statement relates the percent of students who scored higher on.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 28(total students) = 0.4(students who scored higher on the final) Simplify this equation to calculate the percent of students who scored higher on the final.


GMAT word problem note: Since an exact answer to the question can be calculated, the statements together are sufficient. This problem provides two pieces of information.


GMAT word problem note: The first is the ratio of the swim distance to the bike distance to the run distance: 1 to 50 to 12. The second piece of information is the ratio of the average swim time to average bike time to average run time: 3 to 10 to 6.


GMAT word problem note: Use an unknown multiplier for each of these ratios. Distance Time Swim d 3t Bike 50d 10t Run 12d 6t.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Now, use an RTD (rate/time/distance) chart to calculate the average speed for competitors in each of the three legs, in terms of d and t.


GMAT word problem note: Rate Time Distance Swim 3t d Bike 10t 50d Run 6t 12d The question asks about a scenario in which the average swim time, bike time, and run time are equal, but the rates stay the same. In this scenario, the following will be the case:.


GMAT word problem note: Use the RTD formula to fill in the missing column of this chart. Rate Time Distance Swim t Bike t 5d Run t 2d.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The distances are in a ratio of to 5 to 2, or 1 to 15 to 6.


GMAT word problem note: The swim distance represents of the total distance. The problem does not specify the extent to which each person paints over the area that was already painted by the previous person.


GMAT word problem note: For instance, when Jamaica painted 70 square feet (sq) of the wall, she might have painted over the entire 60 sq already painted by Nelson, or she might have painted over only some of it. However, by the time Bryan finished painting, the entire wall had been painted with at least one coat of paint, and some of it might have been painted with as many as three coats.


GMAT word problem note: First, make q as large as possible. In order to do so, the amount of paint used on the remainder of the wall must be minimized.


GMAT word problem note: The area q will be as large as possible when the remaining 100 – q square foot area is covered in only a single coat.


GMAT geometry note: The entire amount of paint applied by all three painters is 60 + 70 + 80 = 210 sq. If q sq of the wall has three coats of paint, and the remaining 100 – q has only one coat, the following is true: 3(area with 3 coats) + 1(area with 1 coat) = total paint applied Turn this into an equation and solve for q.


GMAT geometry note: The greatest possible value of q is 55 sq. If the painting is done as shown below, then 55 sq of the wall will have three coats of paint and the rest will have only one coat.


GMAT geometry note: To find the smallest possible number of square feet with three coats, do the opposite:. Assume that the rest of the wall has as much paint as possible.


GMAT geometry note: In that case, the remaining 100 – q sq of wall will have two coats of paint, so the following will be true:. The smallest possible value of q is 10.


GMAT geometry note: Here is another way to visualize the minimization of the three-coat area.


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GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


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GMAT Math Flashcards from GMAT Prep Now_p147_c1


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GMAT Math Flashcards from GMAT Prep Now_p149_c1


GMAT Math Flashcards from GMAT Prep Now_p150_c1


GMAT Math Flashcards from GMAT Prep Now_p151_c1


GMAT Math Flashcards from GMAT Prep Now_p152_c1


GMAT Math Flashcards from GMAT Prep Now_p153_c1


GMAT Math Flashcards from GMAT Prep Now_p154_c1


GMAT Math Flashcards from GMAT Prep Now_p155_c1


GMAT Math Flashcards from GMAT Prep Now_p156_c1


GMAT Math Flashcards from GMAT Prep Now_p157_c1


GMAT Math Flashcards from GMAT Prep Now_p158_c1


GMAT Math Flashcards from GMAT Prep Now_p159_c1


GMAT Math Flashcards from GMAT Prep Now_p160_c1


GMAT Math Flashcards from GMAT Prep Now_p161_c1


GMAT Math Flashcards from GMAT Prep Now_p162_c1


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. We learn from this statement that either p or q is negative, but since we know from the question that p < q, p must be negative.


GMAT fractions, decimals, and percents note: To determine whether pqr < p, let's test values for p, q, and r. Our test values must meet only 2 conditions: p must be negative and q must be positive.


GMAT fractions, decimals, and percents note: We learn from this statement that either p or r is negative, but since we know from the question that p < r, p must be negative and r must be positive. However, q could be either positive or negative.


GMAT fractions, decimals, and percents note: If we look at both statements together, we know that p is negative and that both q and r are positive. Our test values must meet 3 conditions: p must be negative, q must be positive, and r must be positive.


GMAT fractions, decimals, and percents note: For instance, if p = -2, q = 10, r = 5.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. If p = -2, q = 0.3 and r = 0.4, pqr = -0.24, which is NOT less than p.


GMAT algebra note: Sides of a square, number of votes, etc. When you have a positive constraint, you can:.


GMAT algebra note: Eliminate negative solutions from a quadratic function Multiply or divide an inequality by a variable Cross-multiply inequalities: x/y < y/x  x2 < y2 Change an inequality sign for reciprocals: x<y. (1) T < S (2) F > G If one assumes the signs for all four variables to be positive, then T/S (an improper fraction) is NOT greater than F/G (an improper fraction).


GMAT algebra note: However, if negative signs are taken into account, we cannot conclusively ascertain whether T/S > F/G.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT number properties note: GMAT STRATEGY/TIPS 64 is the only number (besides 1) under 100 that is the square of an integer and the cube of an integer 16 and 81 are the only numbers (again, besides 1) that are both the square of an integer and the square of a square 30 and 42 are the smallest integers with at least three prime factors The only integers with exactly three factors are the squares of prime numbers When odd number n is doubled, 2n has twice as many factors as n When even number is doubled, 2n has 1.5 more factors as n All prime numbers above 3 are of the form 6n – 1 or 6n + 1, because all other numbers are divisible by 2 or 3 Xn – yn is always divisible by x – y and is divisible by x + y if n is even Finding the number of powers of a prime number P in n!


GMAT number properties note: 25/2 + 25/4 + 25/8 + 25/16  12+ 6 + 3 + 1  22 Finding the power of non-prime in n! Determine the prime factorization the number:.


GMAT number properties note: PF of 900 is 22*32*52 Step 2:. Find the powers of these prime numbers in n!


GMAT number properties note: (refer above) Power of 2 = 47, power of 3 = 22 and power of 5 = 12 Step 3:. We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Total – Neither = A + B – X Formula for elements who are AT LEAST one of the two groups:.


GMAT sets, probability, and counting note: A + B – X Formula for elements who are in ONLY one of the two groups:. A + B – 2X Only in group A = a Only in group B = b Both in group A & B = X Neither = N.


GMAT quant note: DOUBLE SET MATRIX – 2 MUTUALLY EXCLUSIVE OPTIONS FOR A DECISION For GMAT problems involving only two categorizations or decisions, the most efficient tool is the Double Set Matrix: a table whose rows correspond to the options for one decision and whose columns correspond to the options for the other decision.


GMAT quant note: The rows should correspond to the mutually exclusive options for one decision. Likewise, the columns should correspond to the mutually exclusive options for the other.


GMAT quant note: For instance, if a problem deals with students getting either right or wrong answers on problems 1 and 2, the columns should not be "problem 1" and "problem 2," and the rows should not be "right" and "wrong." Instead, the columns should list options for one decision-problem 1 correct, problem 1 incorrect, total-and the rows should list options for the other decision-problem 2 correct, problem 2 incorrect, total.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Total – Neither = A + B + C – (a+b+c+2x) Formula for elements who are AT LEAST one of the three groups:.


GMAT sets, probability, and counting note: A + B + C – (a + b + c + 2x) It is important to note that “a” represents elements that belong ONLY to A and C (not B) Total = A + B + C – (a+ x + b + x + c + x) + x Formula for elements in any two of the groups: a+ b + c Note:. Work from the inside out Only A = A – a – b – x Only B = B – b – c – x Only C = C – a – c – x x a c b.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.


GMAT statistics note: MEAN – THE AVERAGE MEAN = MEDIAN in a set of consecutive integers If each one of the given numbers is increased (or decreased) by K, their average is increased (or decreased) by k If each one of some given numbers is multiplied by K, their average is multiplied by K The sum of first “n” natural numbers is given by n (n + 1)/2 If the average of a few consecutive integers is 0, then either all the numbers are zero or there will be an odd number of integers DETERMINING THE MEAN Mean = Sum of terms/ number of terms For an evenly spaced set (i.e., series where the difference between any two consecutive numbers is the same – 3, 6, 9), or a set of consecutive integers Mean = (first term + last term) / 2 Mean = middle term (if the number of terms in the set is even, then the mean is the average of the middle two terms) IMPORTANT:.


GMAT statistics note: The mean of a set of consecutive integers will be an INTEGER only if the number of elements in the set is ODD or if the difference between/sum of the “consecutive” terms is divisible by 2.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. When you add the new term to the set, those 15 "extra points" are distributed evenly among all ten data points, increasing each by 1.5.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. When you add the new term to the set, those 15 "extra points" are distributed evenly among all ten data points, increasing each by 1.5 RESIDUALS Residuals = Data point – Mean For any set, the residuals sum to zero.


GMAT statistics note: Alternatively, the positive residuals ("overs") and negative residuals ("unders") for any set will cancel out If the mean of the set {97, 100, 85, 90, 94, 80, 92, x} is 91, what is the value of x? You could certainly solve this problem with the traditional formula for averages, but you would waste a lot of time on arithmetic.


GMAT statistics note: Instead, just use the given mean of 91 to compute the residuals for all the terms except x: +6, +9, -6, -1, +3, -11, + 1. These residuals sum to +1.Therefore, x must leave a residual of -1, since all the residuals sum to zero.


GMAT statistics note: As a result, x is one less than the mean, or 90.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. MEDIAN – THE MIDDLE TERM MEAN = MEDIAN in a set of consecutive integers.


GMAT statistics note: However, just because the mean and the median are equal does not mean that the set comprises of evenly spaced/consecutive integers Ultimately, if your mean and median are the same, what it tells you is that your data is arranged symmetrically around the median.


GMAT statistics note: For every bit over the mean on one side, there's a corresponding bit under the mean on the other side, balancing it out. When they're different, that tells you that this symmetry has been broken DETERMINING THE MEDIAN To find the median, list the numbers in ascending or descending order, and find the middle term.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. This symmetry has been broken DETERMINING THE MEDIAN To find the median, list the numbers in ascending or descending order, and find the middle term When number of terms N is large, then the following two formulae are useful for finding the middle term:.


GMAT statistics note: When N is odd, the middle term is (N+1)/2th term When N is even, the median will be the average of the middle two terms, i.e., average of the (N/2)th term and the (N/2)th + 1 term VALUE OF THE MEDIAN When the number of items in a set of integers is odd, the median is an integer When the number of terms in a set of integers is even, the median can only end with 0.5 as it is the average of the middle two terms.


GMAT statistics note: However, it is important to note that the median does not ALWAYS end in 0.5 when the number of terms in the set is even.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. S is the set of all integers from a to b, inclusive.


GMAT statistics note: Q is the set of all integers from b to c, inclusive. The median of set S is (3/4)*b.


GMAT statistics note: The median of set Q is (7/8)*c. If R is the set of all integers from a to c, inclusive, what fraction of c is the median of set R?


GMAT statistics note: Since we the sets contain consecutive integers, the median = (largest + smallest term)/2 Therefore, equate (a+b)/2 to (3/4)*b for Set S and (b+c)/2 to (7/8)*c for Set Q.


GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. WEIGHTED AVERAGE (weight1*data point 1) + (weight2*data point 2) (weight 1 + weight 2) SUM OF THE WEIGHTS The sum of the weights must equal to 1.


GMAT fractions, decimals, and percents note: However, if the sum of the weights does not equal 1, we need to divide by the sum of the weights.


GMAT fractions, decimals, and percents note: If we put a weight of 2 on 20 and a weight of 3 on 30, we have to divide by 5 to get weights that sum to 1, as shown below: [2(20) + 3(30)]/ 5 = (2/5)*20 + (3/5)* 30 Average = 26 ADVANCED PROPERTIES A weighted average of only two values will fall closer to whichever value is weighted more heavily For instance, if a drink is made by mixing 2 shots of a liquor containing 15% alcohol with 3 shots of a liquor containing 20% alcohol, then the alcohol content of the mixed drink will be closer to 20% than to 15%.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread.


GMAT statistics note: Containing 15% alcohol with 3 shots of a liquor containing 20% alcohol, then the alcohol content of the mixed drink will be closer to 20% than to 15% The ratio that determines how to weight the averages of two or more subgroups in a weighted average ALSO REFLECTS the ratio of the distances from the weighted average to each subgroup’s average If the average weight of the entire group was twice as close to the average weight of the men as it was to the average weight of the women, there must be twice as many men as women In case two groups have an equal number of elements, then the weighted average of the two groups combined will be the average of the individual averages of the two groups.


GMAT fractions, decimals, and percents note: Track mean, median, range, distribution, and how operations change spread. Having just the ratios of the weights will allow you to find a weighted average.


GMAT fractions, decimals, and percents note: Simply write the ratio as a fraction, and use the numerator and the denominator as weights A mixture of "lean" ground beef (10% fat) and "super-lean" ground beef (4% fat) contains twice as much lean beef as super-lean beef. What is the percentage of fat in the mixture?


GMAT fractions, decimals, and percents note: The ratio of lean beef to super-lean beef is 2: 1, and so we can use 2 as the weight for the lean beef and 1 as the weight for the super-lean beef The percentage of fat in the mixture is [(10%)*2} + (4%)*1]/ (2+ 1) = 24%/3 = 8%.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. MODE, RANGE AND STANDARD DEVIATION Mode is the term that occurs most frequently.


GMAT statistics note: There may be no mode, or more than one mode in a given series RANGE Maximum value – minimum value Range does not change when a constant value is either added to or subtracted from every term in a set Range is multiplied or divided by that constant value which is used to multiply/ divide every term in a set Range = 0 when all values are the same Range is always > = 0 Range of a set has to be equal or greater than difference between any two numbers in the set.


GMAT statistics note: Ge = 0 when all values are the same Range is always > = 0 Range of a set has to be equal or greater than difference between any two numbers in the set STANDARD DEVIATION Formula: √ (term1 – average)2 + (term2 – average)2 + (term3 – average)2 / √number of terms Rules The set in which numbers are farther away from each other than the numbers in some other set will have greater value of standard deviation A = {2,4,6,8,10} B = {3,6,9,12,15} In set B, the numbers are wider away from each other and hence will have greater value of standard deviation Standard deviation does not change when a constant value is either added to or subtracted from every term in a set Standard deviation is multiplied or divided by that constant value which is used to multiply or divide every term in a set Taking the absolute value of the elements of a set will not change the standard deviation if all elements are positive or negative If y = ax + b, and if the standard deviation of x series is ‘S’, then the standard deviation of y series will be a*s The SD of any list is not dependent on the average, but on the deviation of the numbers from the average.


GMAT statistics note: So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD If mean.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Ything about their standard deviation - different averages can have the same SD If mean = maximum value it means that all values are equal and SD is 0 VARIANCE = STANDARD DEVIATION2.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. FRACTIONS,DIGITS AND DECIMALS AND PERCENTAGES.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. FRACTIONS PROPER FRACTIONS Proper fractions fall between 0 and 1.


GMAT fractions, decimals, and percents note: In proper fractions, the numerator is always smaller than the denominator IMPROPER FRACTIONS In improper fractions, the numerator is always greater than the denominator. Improper fractions are therefore always greater than 1.


GMAT fractions, decimals, and percents note: As an integer and a proper fraction COMPARING FRACTIONS The traditional method of comparing fractions entails finding a common denominator and comparing the numerators The shortcut method entails multiplying the numerator of one fraction with the denominator of the other fraction.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.


GMAT fractions, decimals, and percents note: Ominator and comparing the numerators The shortcut method entails multiplying the numerator of one fraction with the denominator of the other fraction FRACTION RULES FOR POSITIVE, PROPER FRACTIONS As the numerator increases, the value of the fraction increases as it approaches 1, assuming that the denominator is held constant As the denominator increases, the value of the fraction decreases as it approaches 1, assuming that the numerator is held constant Increasing both the numerator and denominator by the same value brings the fraction closer to 1, regardless of the original value of the fraction.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT sequences and patterns note: DIGITS – LAST DIGIT Last digit of (xyz)n is the same as that of Zn Determine the cyclicity number of Z Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base Integers ending with 2, 3, 7 and 8 have a cyclicity of 4 2 – 2, 4, 8, 6 3 – 3, 9, 7, 1 7 – 7, 9, 3, 1 8 – 8, 4, 2, 6 - Integers ending with 4 have a cyclicity of 2.


GMAT sequences and patterns note: When n is odd, last digit will be 4 and when n is even, last digit will be 6 - Integers ending with 9 have a cyclicity of 2. What is the last digit of 12739?


GMAT sequences and patterns note: Last digit of 12739 is the same as that of 739. First we need to determine the cyclicity of 7, which is 4.


GMAT sequences and patterns note: Now divide 39 (power) by 4 (cyclicity), remainder is 3. Given that x is a positive integer, what is the units digit of 24(2x+1)* 33 (x+1)* 17(x+2)* 9(2x).


GMAT sequences and patterns note: Given that x is a positive integer, what is the units digit of 24(2x+1)* 33 (x+1)* 17(x+2)* 9(2x) x is a positive integer, so 2x is always even, while 2x + 1 is always odd Thus, 4(2x + 1) = (4)(odd), which always has a units digit of 4 9(2x) = 9(even), which always has a units digit of 1 That leaves us to find the units digit of (3)(x + 1)*(7)(x + 2).


GMAT sequences and patterns note: Rewriting, and dropping all but the units digit at each intermediate step, = (1)(x + 1)(7) = 7, for any value of x So, the units digit of is (4)(7)(1) = 28, then once again drop all but the units digit to get Solve the problem step by step. What is the units digit of 5*5*9*9*4*4*4?


GMAT sequences and patterns note: Multiply the last digit of each of the intermediary products 5*1*4 = 20. Units digit will therefore be 0.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. Multiply the last digit of each of the intermediary products 5*1*4 = 20.


GMAT sequences and patterns note: Units digit will therefore be 0 Last digit of a product Last digit of a power What is the units digit of 1727? 1727 will end in the same units digit as 727 27 / 4 results in a remainder of 3.


GMAT sequences and patterns note: Therefore, the units digit of 1727 will be the same as the units digit of 73, which is 3 If x is a positive integer, what is the units digit of 24(2x+1) 33(x+1) 17(x+2) 9 (2x) Note that x is a positive integer, so 2x is always even, while 2x + 1 is always odd.


GMAT sequences and patterns note: Thus, (4)(2x + 1) = (4)(odd), which always has a units digit of 4 (9)(2x) = (9)(even), which always has a units digit of 1 That leaves us to find the units digit of (3)(x + 1)(7)(x + 2).


GMAT sequences and patterns note: Rewriting, and dropping all but the units digit at each intermediate step, (3)(x + 1)(7)(x + 2) = (3)(x + 1)(7)(x + 1)(7) = (3 × 7)(x + 1)(7) = (21)(x + 1)(7) = (1)(x + 1)(7) = 7, for any value of x. So, the units digit of (4)(2x + 1)(3)(x + 1)(7)(x + 2)(9)(2x) is (4)(7)(1) = 28, then once again drop all but the units digit to get 8.


GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. How much sugar should you add so that the mixture is 60% sugar and 40% water Always remember: the key to these problems is the component that does not change.


GMAT word problem note: In this case, we are adding sugar, while water remains constant. Therefore, we will focus on the water for most of the problem First, determine how much water is in the mixture.


GMAT word problem note: 60% of 50 is 30 ounces Because we are not adding or subtracting any water, we will still have 30 ounces of it after we add more sugar. However, we now want that 30 ounces to represent 40% of the total, rather than 60%.


GMAT word problem note: X = 75 Since the increase in total volume is only made up of additional sugar, and we went from 50 total ounces to 75 total ounces, we must add 25 ounces of sugar.


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread. If mixture X and mixture Y are combined, the new mixture is 15% acid.


GMAT word problem note: What percentage of the new mixture is mixture X? (0.1X + 0.30Y)/ (X+Y) = 0.15 Solving for X and Y, we get 0.05 X = 0.15 Y X/Y = 3 / 1, which means that for every 3 parts of X, there is 1 part of Y The ratio of mixture X to the total ratio is 3/4.


GMAT word problem note: Therefore, mixture X constitutes 75% of the new mixture.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Or every 3 parts of X, there is 1 part of Y The ratio of mixture X to the total ratio is 3/4.


GMAT fractions, decimals, and percents note: Therefore, mixture X constitutes 75% of the new mixture How many liters of a solution that is 15 percent salt must be added to 5 liters of a solution that is 8 percent salt so that the resulting solution is 10 percent salt?


GMAT fractions, decimals, and percents note: The amount of salt in the 15% solution [0.15n] plus the amount of salt in the 8% solution [(0.08)(5)] must be equal to the amount of salt in the 10% mixture [0.10 (n + 5)].


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. If that price represented a 25 percent discount on the original price of the dress, what was the original price of the dress?


GMAT fractions, decimals, and percents note: Solution # 1 If p is the original price of the dress, then 0.75p is the discounted price and 0.75p = $24, or p = $32. The original price of the dress was $32.


GMAT fractions, decimals, and percents note: These two discounts are equal to an overall discount of what percent? Solution # 2 If p is the original price of the item, then 0.8p is the price after the first discount.


GMAT fractions, decimals, and percents note: The price after the second discount is (0.7)(0.8)p = 0.56p. This represents an overall discount of 44 percent (100% – 56%).


GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. The price after the second discount is (0.7)(0.8)p = 0.56p.


GMAT word problem note: This represents an overall discount of 44 percent (100% – 56%) SIMPLE INTEREST SI = Simple Interest, P = Principal, T = Time period in years, R = Rate of interest per annum S. = (PTR)/100 If simple interest for 2 years is 200, then S.I for 4 years is 400, for 1 year it is 100, for 10 years it is 1000.


GMAT word problem note: For any length of time is known, then it can be calculated for any other length of time A man invests $1000 in a bank which pays him 10% p.a. Rate of simple interest for 2 years.


GMAT word problem note: How much does the man get after 2 years? = (1000 ×2 ×10)/100 = 200 Therefore, amount after 2 years = P + S.I.


GMAT word problem note: = 1000 + 200 = 1200 COMPOUND INTEREST A = P × [1 + (R/100)] ^n If interest is compounded monthly, divide the annual rate by 12, and multiply n by 12 If interest is compounded quarterly, divide the annual rate by 4, and multiply n by 4 If interest is compounded semi-annually, divide the annual rate by 2, and multiply n by 2.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. PERCENTAGES – 5 All of the furniture for sale at Al’s Discount Furniture is offered for less than the manufacturer’s suggested retail price.


GMAT fractions, decimals, and percents note: Once a year, Al’s holds a clearance sale. If jamie purchased a certain desk during the sale, did she get a discount of more than 50% of Al’s regular price for the desk?


GMAT fractions, decimals, and percents note: Al’s regular price for the desk is 60%, rounded to the nearest percent, of the MSRP of $2000 The sale price was $601 less than Al’s regular price for the desk.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Ular price for the desk is 60%, rounded to the nearest percent, of the MSRP of $2000 The sale price was $601 less than Al’s regular price for the desk In order to determine the percent discount received by Jamie, we need to know two things: the regular price of the desk and the sale price of the desk.


GMAT fractions, decimals, and percents note: Alternatively, we could calculate the percent discount from the price reduction and either the regular price or the sale price (1) INSUFFICIENT:. This statement tells us the regular price of the desk at Al’s, but provides no information about how much Jamie actually paid for the desk during the annual sale.


GMAT fractions, decimals, and percents note: This statement tells us how much the price of the desk was reduced during the sale, but provides no information about the regular price. On the other hand, if the regular price was $602, then the discount was nearly 100%.


GMAT fractions, decimals, and percents note: At first glance, it seems that the statements together provide enough information.


GMAT statistics note: Percent problems usually relate part, whole, rate, and percent change. $602, then the discount was nearly 100%.


GMAT statistics note: This indicates that the regular price of the desk at Al’s is 60% of the MSRP, plus or minus 0.5% of the MSRP. Rather than clearly stating that the regular price is (0.60)($2000) = $1200, this statement gives a range of values for the regular price: $1200 plus or minus $10 (0.5% of 2000), or between $1190 and $1210.


GMAT statistics note: If the regular price was $1190, then the discount was ($601/$1190) × 100% = 50.5% (you can actually see that this is greater than 50% without calculating). If the regular price was $1210, then the discount was ($601/$1210) × 100% = 49.7% (you can actually see that this is less than 50% without calculating).


GMAT statistics note: The uncertainty about the regular price means that we cannot answer with certainty whether the discount was more than 50% of the regular price.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. COMMON FRACTIONS, DECIMALS AND PERCENTAGE EQUIVALENTS.


GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. RATIO AND PROPORTION A ratio expresses a particular relationship between two or more quantities.


GMAT fractions, decimals, and percents note: The two partners spend time working in the ratio of 1 to 3. For every 1 hour the first partner works, the second partner works 3 hours.


GMAT fractions, decimals, and percents note: Three sisters invest in a certain stock in the ratio of 2 to 3 to 8. For every $2 the first sister invests, the second sister invests $3, and the third sister invests $8 The ratio of men to women in the room is 3 to 4.


GMAT fractions, decimals, and percents note: For every 3 men, there are 4 women Ratios can express a part-part relationship or a part-whole relationship A part-part relationship:. The ratio of men to women in the office is 3:4 A part-whole relationship:.


GMAT fractions, decimals, and percents note: There are 3 men for every 7 employees Notice that if there are only two parts in the whole, you can derive a part-whole ratio from a part-part ratio, and vice versa Remember that ratios only express a relationship between two or more items. They do not provide enough information, on their own, to determine the exact quantity for each item.


GMAT fractions, decimals, and percents note: All we know is that the number of men is less than the number of women. However, ratios are surprising.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. EXPONENTS Simplifying exponential expressions Always try to simplify exponential expressions when they have the same base or the same exponent.


GMAT algebra note: You can only simplify exponential expressions that are linked by multiplication or division. The largest prime factor of 421 + 422 + 423 is 7 *** When both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal.


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. ROOTS A ROOT is the inverse operation of raising a number to an exponent, and answers the question:.


GMAT algebra note: Which number do I multiply by itself n times in order to get a product of b? EVEN ROOTS ONLY HAVE ONE SOLUTION – A POSITIVE VALUE Unlike even exponents, which yield both a positive and a negative solution, even roots have only one solution.


GMAT algebra note: √18 = √50.18 = √900 = 30 √144:16 = √144: √16 = 12/4 = 3 √25+16 = √41  you cannot simplify this one 2√7 + 3√7 = 5√7.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. √18 = √50.18 = √900 = 30 √144:16 = √144: √16 = 12/4 = 3 √25+16 = √41  you cannot simplify this one 2√7 + 3√7 = 5√7 KNOWING COMMON ROOTS GMAT requires you to know all the perfect square roots from 1 to 30 and also the following imperfect rules:.


GMAT algebra note: SUM OF THE ROOTS OF AN EQUATION ax2+bx+c = 0 IS (-B/A).


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. RATE AND WORK Basic motion problems (speed, distance, time) Average rate problems Simultaneous motion problems Work problems Population problems.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. RATE, DISTANCE TIME RATE Rate is expressed as a ratio of distance and time, with two corresponding units.


GMAT word problem note: DISTANCE Distance is expressed using a unit of distance. RATE = DISTANCE / TIME RATE * TIME = DISTANCE Speed and time are inversely proportional to each other.


GMAT word problem note: Therefore, if a man increases his speed by 3/2, then the time taken will become 2/3*original time If three men cover the same distance with speeds in the ratio a: b: c, the times taken by these three will be in the ratio 1/a: 1/b: 1/c respectively.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – SPEED RELATIONS.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – THE KISS/CRASH.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – CATCHING UP/OVERTAKING.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – ROUND TRIP.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. MULTIPLE RATE, DISTANCE TIME PROBLEMS – SAMPLE SITUATIONS.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating. MISCELLANEOUS PROBLEMS BOAT PROBLEMS If the speed of a boat (or man) in still water is X km/hour, and the speed of the stream (or current) is Y km/hour, then:.


GMAT quant note: Speed of boat with the stream (or Downstream or D/S) = (X + Y) km/hour Speed of boat against the stream (or upstream or U/S) = (X – Y) km/hour X = [(X + Y) + (X – Y)] / 2 and Y = [(X + Y) – (X – Y)] / 2 Boat’s speed in still water = [Speed downstream + Speed upstream] / 2 Speed of current = [Speed downstream – Speed upstream] / 2.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. AVERAGE RATE / RELATIVE SPEED 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.


GMAT word problem note: 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 It is not necessary to know the total distance travelled or the total time taken while computing the average speed When two objects travel in the same direction, the time to catch up/overtake will be = lead distance/ difference of speeds (note: in order for A to overtake B, A needs to be traveling at a higher speed than B otherwise A will never catch up) When two objects travel in different directions, time to meet will be = lead distance/ sum of speeds AVERAGE RATE = TOTAL DISTANCE/ TOTAL TIME OR AVERAGE SPEED = (2*SPEED OF A*SPEED OF B) / (SPEED A + B)  only to be used when the distance travelled is the same.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. WORK In distance problems, if the rate (speed) is known, it will normally be given to you as a ready-to-use number.


GMAT word problem note: An perform one surgery in 1.5 hours, the work rate = # of given jobs/ given amount of time i.e., 1/1.5 hours, which is equal to 2/3 surgeries per hour WORKING TOGETHER – ADD WORK RATES If A can finish a job alone in x days and B can finish the same job in y days alone, then One-day work of both A and B is (x + y)/x*y Number of days A and B take to finish the same job working together is: (x*y) / (x + y) Exception: one agent undoes the other’s work, like a pump putting water into a tank and another drawing water out.


GMAT word problem note: Working alone, A finish a job in 2 days and B can finish a job in 3 days. If A and B work together, in how many days will they finish the work?


GMAT word problem note: A’s rate = 1/ 2 B’s rate = 1/ 3 In one day, A and B finish 5 / 6 of the job. Four men working together all day, can finish a piece of work in 11 days.


GMAT word problem note: But two of them having other engagements can work only one half–time and quarter time respectively. How long will it take them to complete the work?


GMAT word problem note: Each man will take 11 x 4 = 44 days to complete the work. If one man works half day/day he will take 44 x 2 = 88 days to finish the work.


GMAT word problem note: Similarly, a man working quarter day/day will take 44 x 4 = 176 days to finish the work.


GMAT word problem note: The second set in which we need to find number of days: 2 men × 3 days × 4 hours = 3 men × N days × 2 hours 2 × 3 × 4 = 3 × N ×2 Therefore, N = 4 days If a man can do a piece of work in N days (or hours or any other unit of time), then the work done by him in one day will be 1/N of the total work If A is twice as good a workman as B, then A will take half the time B takes to finish a piece of work.


GMAT quant note: Translate the wording into variables, relationships, and equations before calculating.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure.


GMAT number properties note: NUMBER PROPERTIES OF INTEGERS IMPORTANT CONCEPTS TO ALWAYS KEEP IN MIND An integer can be either positive or negative, even or odd, a prime or a consonant (0 and 1 are neither prime nor consonant) Never assume a number to be an integer unless EXPLICITLY stated in the problem If n2 is an integer, it is not necessary that n is also an integer.


GMAT number properties note: Impact of arithmetic operations on integers When integers are added to, subtracted from, or multiplied with each other, the result is ALWAYS an integer. An odd integer divided by an even integer ALWAYS results in a non – integer.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. An odd integer divided by an even integer ALWAYS results in a non – integer Divisibility The product of 2 even numbers will always be divisible by 4 The product of x consecutive integers will always be divisible by x and x!.


GMAT number properties note: The smallest number that has all given numbers as a factor GCF:. The largest number which is a factor of all given numbers Product of two numbers = LCM * GCF.