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1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' DozenTough and Tricky Questions: 1. THE SUM OF EVEN INTEGERS: The sum of the even numbers between 1 and k is 79*80, where k is an odd number, then k=?(A) 79(B) 80(C) 81(D) 157(E) 159The solution given in the file was 79, which is not correct. Also the problem was solved with AP formula thus was long and used the theory rarely tested in GMAT. Here is my solution and my notes about AP. Some may be useful:The number of terms in this set would be: n=(k-1)/2 (as k is odd)Last term: k-1Average would be first term+last term/2=(2+k-1)/2=(k+1)/2Also average: sum/number of terms=79*80/((k-1)/2)=158*80/(k-1)(k+1)/2=158*80/(k-1) --> (k-1)(k+1)=158*160 --> k=159Answer E.MY NOTES ABOUT AP:ARITHMETIC PROGRESSIONSequence a1, a2,…an, so that a(n)=a(n-1)+d (constant)nth term an = a1 + d ( n – 1 )Sn=n*(a1+an)/2 or Sn=n*(2a1+d(n-1))/2 Special cases:I. 1+2+…+n=n(1+n)/2 (Sum of n first integers)II. 1+3+5+… (n times)=n^2 (Sum of n first odd numbers). nth term=2n-1III. 2+4+6+… (n times)=n(n+1) (Sum of n first even numbers) nth term=2nSOLUTION WITH THE AP FORMULA:Sequence of even numbersFirst term a=2, common difference d=2 since even numberSum to first n numbers of AP:Sn=n*(a1+an)/2=n(2*2+2(n-1))/2=n(n+1)=79*80n=79 (odd)Number of terms n=(k-1)/2=79 k=159ORSum of n even numbers n(n+1)=79*80n=79k=2n+1=159 2. THE PRICE OF BUSHEL: The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of cents per day while the price of wheat is decreasing at a constant rate of cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?(A) $4.50(B) $5.10(C) $5.30(D) $5.50(E) $5.60Note that we are not asked in how many days prices will cost the same.Let be the # of days when these two bushels will have the same price.First let's simplify the formula given for the rate of decrease of the price of wheat: , this means that the price of wheat decreases by cents per day, in days it'll decrease by cents;As price of corn increases cents per day, in days it'll will increase by cents;Set the equation: , solve for --> ;The cost of a bushel of corn in days (the # of days when these two bushels will have the same price) will be or $5.6.Answer: E.3. LEAP YEAR:How many randomly assembled people are needed to have a better than 50% probability that at least 1 of them was born in a leap year?A. 1B. 2C. 3D. 4E. 5Probability of a randomly selected person NOT to be born in a leap year=3/4Among 2 people, probability that none of them was born in a leap = 3/4*3/4=9/16. The probability at least one born in leap = 1- 9/16=7/16<1/2So, we are looking for such n (# of people), when 1-(3/4)^n>1/2n=3 --> 1-27/64=37/64>1/2Thus min 3 people are needed.4. ADDITION PROBLEM:AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?(A) 1(B) 3(C) 7(D) 9(E) Cannot be determinedAB and CD are two digit integers, their sum can give us only one three digit integer of a kind of AAA it's 111.So, A=1. 1B+CD=111C can not be less than 9, because no to digit integer with first digit 1 (mean that it's<20) can be added to two digit integer less than 90 to have the sum 111 (if CD<90 meaning C<9 CD+1B<111).C=9Answer: D.5. RACE:A and B ran, at their respective constant rates, a race of 480 m. In the first heat, A gives B a head start of 48 m and beats him by 1/10th of a minute. In the second heat, A gives B a head start of 144 m and is beaten by 1/30th of a minute. What is B’s speed in m/s?(A) 12(B) 14(C) 16(D) 18(E) 20Let x be the speed of B.Write the equation:(480-48)/x (time of B for first heat) - 6 (seconds, time B lost to A first heat) = TIME OF A (in both heats A runs with constant rate, so the time for first and second heats are the same)=(480-144)/x (time of B for second heat) + 2 (seconds, time B won to A second heat)(480-48)/x-6=(480-144)/x+2x=12Answer: A.6. PROBABILITY OF DRAWING:A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?(A) 2/27(B) 1/9(C) 1/3(D) 4/27(E) 2/9This is with replacement case (and was solved incorrectly by some of you):We are multiplying by 2 as there are two possible wining scenarios RW and WR.Answer: D.7. THE DISTANCE BETWEEN THE CIRCLE AND THE LINE:What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?A) 1.4B) sqrt (2)C) 1.7D) sqrt (3)E) 2.0This is tough:First note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).Now we can do this by finding the equation of a line perpendicular to given line (we know it should cross origin and cross given line, so we can write the formula of it), then find the croos point of these lines and then the distance between the origin and this point. But it's very lengthy way.There is another, shorter one. Though I've never seen any GMAT question requiring the formula used in it.We know the formula to calculate the distance between two points and : BUT there is a formula to calculate the distance between the point (in our case origin) and the line:DISTANCE BETWEEN THE LINE AND POINT:Line: , point DISTANCE BETWEEN THE LINE AND ORIGIN:As origin is -->So in our case it would be: So the shortest distance would be: Answer: A.OR ANOTHER APPROACH:In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:If the circle is centered at the origin (0, 0), then the equation simplifies to:So, the circle represented by the equation is centered at the origin and has the radius of .Then note that min distance from the circle to the line would be: length of perpendicular from the origin to the line (as the circle is centered at the origin) - the radius of a circle (which is 1).So we should find the length of perpendicular, or the height of the right triangle formed by the X and Y axis and the line .The legs would be the value of x for y=0 (x intercept) --> y=0, x=4 --> .and the value of y for x=0 (y intercept) --> x=0, y=-3 --> .So we have the right triangle with legs 4 and 3 and hypotenuse 5. What is the height of this triangle (perpendicular from right angle to the hypotenuse)? As perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle: --> --> .Answer: A.8. THE AVERAGE TEMPERATURE:The average of temperatures at noontime from Monday to Friday is 50; the lowest one is 45, what is the possible maximum range of the temperatures?A. 20B. 25C. 40D. 45E. 75Average=50, Sum of temperatures=50*5=250As the min temperature is 45, max would be 250-4*45=70 --> The range=70(max)-45(min)=25Answer: B.9. PROBABILITY OF INTEGER BEING DIVISIBLE BY 8:If n is an integer from 1 to 96 (inclusive), what is the probability for n*(n+1)*(n+2) being divisible by 8?A. 25%B 50%C 62.5%D. 72.5%E. 75%N=n*(n+1)*(n+2)N is divisible by 8 in two cases:When n is even:No of even numbers (between 1 and 96)=48ANDWhen n+1 is divisible by 8. -->n=8p-1 --> 8p-1<=96 --> p=12.3 --> 12 such nembersTotal=48+12=60Probability=60/96=0.62Answer: C10. SUM OF INTEGERS:If the sum of five consecutive positive integers is A, then the sum of the next five consecutive integers in terms of A is:A. A+1 inquiry B. A+5C A+25D 2AE. 5ASum=A, next 5 consecutive will gain additional 5*5=25, so sum of the next five consecutive integers in terms of A is: A+25Answer: C. Hard Questions:A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there? (A) 28 (B) 32 (C) 48 (D) 60 (E) 120 As most of the combination problems this one can be solved in more than 1 way:Sisters sit separately:1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case: 2*12=24Or2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case=8.Total=24+8=32.Another way: Total number of arrangements-arrangements with sisters sitting together=2*4*3!-2*2(sisters together)*2*2*1(arrangement of others)=48-16=32Answer: B.2. What is the probability that a 3-digit positive integer picked at random will have one or more "7" in its digits?(A) 271/900(B) 27/100 (C) 7/25 (D) 1/9 (E) 1/10Total 3 digit numbers 900, 3 digit number with no 7 =8*9*9=648, P(at least one 7)=1-P(no 7)=1-648/900=252/900=7/25Answer: C.3. A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere? (A) (B) (C) (D) (E) Shortest distance=(diagonal of cube-diameter of sphere)/2= Answer: D.4. A contractor estimated that his 10-man crew could complete the construction in 110 days if there was no rain. (Assume the crew does not work on any rainy day and rain is the only factor that can deter the crew from working). However, on the 61-st day, after 5 days of rain, he hired 6 more people and finished the project early. If the job was done in 100 days, how many days after day 60 had rain? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8This one was solved incorrectly:Days to finish the job for 10 people 110 days.On the 61-st day, after 5 days of rain --> 5 days was rain, 55 days they worked, thus completed 1/2 of the job, 1/2 is left (55 days of work for 10 people). Then 6 more people was hired --> speed of construction increased by 1.6, days needed to finish 55/1.6=34.375, BUT after they were hired job was done in 100-60=40 days --> so 5 days rained. They needed MORE than 34 days to finish the job, so if it rained for 6 days they wouldn't be able to finish the job in 100(40) days.Answer: B.5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t? (A) 2 (B) 4 (C) 8 (D) 20 (E) 45 divided by yields the remainder of can always be expressed as: (which is the same as ), where is the quotient and is the remainder.Given that , so according to the above , which means that must be a multiple of 3. Only option E offers answer which is a multiple of 3Answer: E.6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?(A) 3510(B) 2620(C) 1404(D) 700(E) 635Committee can have either: 2 men and 4 women OR 3 men and 3 women (to meet the condition of at least 2 men and 3 women).Ways to chose 6 members committee without restriction (two men refuse to server together): Ways to chose 6 members committee with two particular men serve together: 700-65 = 635 Answer: E.7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ? I. x^2<2x<1/x II. x^2<1/x<2x III. 2x<x^2<1/x (A) None (B) I only (C) III only (D) I and II only (E) I II and III First note that we are asked "which of the following COULD be the correct ordering" not MUST be.Basically we should determine relationship between , and in three areas: .When --> is the greatest and no option is offering this, so we know that x<2.If --> is greatest then comes and no option is offering this.So, we are left with :In this case is least value, so we are left with:I. --> can ? Can , the expression can be negative or positive for . (You can check it either algebraically or by picking numbers)II. --> can ? The same here , the expression can be negative or positive for . (You can check it either algebraically or by picking numbers)Answer: D.8. In the xy plane, Line k has a positive slope and x-intercept 4. If the area of the triangle formed by line k and the two axes is 12, What is the y-intercept of line K ?(A) 3(B) 6 (C) -3 (D) -6 (E) -4Positive slope, positive (4) x-intercept --> negative y-intercept. --> 1/2*4*|y|=12 --> |y|=6. --> y=-6Answer: D9. Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y? (A) 135 (B) 120 (C) 115 (D) 105 (E) 90 20%X=X&Y=15 --> X=75 --> Only X=75-15=6025%Y=X&Y=15 --> Y=60 --> Only Y=60-15=45Only X or Y=60+45=105Answer: D.10. What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive.(A) 420 (B) 840 (C) 1260 (D) 2520 (E) 5040 The integer should be divisible by: 2, 3, 4(=2^2), 5, 6(=2*3), and 7. LCM=2^2*3*5*7=420Answer: A. Set of Hard questions:1. ABCDE is a regular pentagon with F at its center. How many different triangles can be formed by joining 3 of the points A,B,C,D,E and F?(A) 10(B) 15(C) 20(D) 25(E) 30Answer: C.2. The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is prime, then f(p) = (A) P-1 (B) P-2 (C) (P+1)/2 (D) (P-1)/2 (E) 2Answer: A.3. How many numbers that are not divisible by 6 divide evenly into 264,600? (A) 9 (B) 36 (C) 51 (D) 63 (E) 72Answer: D.4.A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale? (A) 20 (B) 36 (C) 48 (D) 60 (E) 84Answer: C.5. Mrs. Smith has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Smith been given if there are 120 ways that she could distribute the vouchers?(A) 13(B) 14(C) 15(D) 16(E) more than 16Answer: C.6. This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year? (A) 1/(r+2)(B) 1/(2r+2)(C) 1/(3r+2) (D) 1/(r+3) (E) 1/(2r+3)Answer: E.7. Before being simplified, the instructions for computing income tax in Country Rwere to add 2 percent of one's annual income to the average(arithmetic mean)of 100units of Country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in Country R's currency, for a person in that country whose annual income is I? (A) 50+I/200(B) 50+3I/100(C) 50+I/40 (D) 100+I/50 (E) 100+3I/100 Answer: C.8. How many positive integers less than 10,000 are such that the product of their digits is 210?(A) 24 (B) 30 (C) 48 (D) 54 (E) 72Answer: D.9. Find the number of selections that can be made taking 4 letters from the word"ENTRANCE". (A) 70 (B) 36 (C) 35 (D) 72 (E) 32Answer:B.Find in the above word, the number of arrangements using the 4 letters. Answer:606.10. How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates (x,y) satisfying 1≤x≤3 and 1≤y≤3? (A) 72 (B) 76 (C) 78 (D) 80 (E) 84 Answer: B. Standard Deviation Questions:Please note the following: A. I was assured MANY TIMS, by various GMAT tutors, that GMAT won't ask you to actually calculate SD, but rather to understand the concept of it. Though KNOWING how it's calculated helps in understanding the concept.B. During the real GMAT it's highly unlikely to get more than one ot two question on SD (as on combinatorics), actually you may see none, so do not spend too much of your preparation time on it, it's better to concentrate on issues you'll definitely face on G-day.Many questions below are easy, some are tough, but anyway they are good to master in solving SD problems. I'll post OA after some discussions. Please provide your way of thinking along with the answer. Thanks.A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is close to the standard deviation for the original 5 numbers? (A) -1 and 9 (B) 4 and 4 (C) 3 and 5 (D) 2 and 6 (E) 0 and 8 Answer: D.2. A certain list of 100 data has an average of 6 and standard deviation of d where d is positive. Which of the following pairs of data, when added to the list must result in a list of 102 data with the standard deviation less than d?(A) 0 and 6(B) 0 and 12(C) 0 and 0(D) -6 and 0(E) 6 and 6Answer: E.3. For a certain examination, a score of 58 was 2 standard deviations below the mean, and a score of 98 was 3 standard deviations above the mean. What was the mean score for the examination? (A) 74 (B) 76 (C) 78 (D) 80 (E) 82Answer: A.4. Which of the following distribution of numbers has the greatest standard deviation? (A) {-3, 1, 2} (B) {-2, -1, 1, 2} (C) {3, 5, 7} (D) {-1, 2, 3, 4} (E) {0, 2, 4} Answer: A.5. Which of the following has the same standard deviation as {s,r,t}?I. {r-2, s-2, t-2}II. {0, s-t, s-r}III. {|r|, |s|, |t|}(A) I only (B) II only (C) III only (D) I and II only (E) I and III only Answer: D.6. A certain characteristic in a large population has a distribution that is symmetric about the mean m. If 68% of the distribution lies one standard deviation d of the mean, what percent of the distribution is less than m+d? (A) 16% (B) 32% (C) 48% (D) 84% (E) 92%Answer: D.7. Which of the following data sets has the third largest standard deviation? (A) {1, 2, 3, 4, 5} (B) {2, 3, 3, 3, 4} (C) {2, 2, 2, 4, 5} (D) {0, 2, 3, 4, 6} (E) {-1, 1, 3, 5, 7} Answer: A.8. The table below represents three sets of numbers with their respective medians, means and standard deviations. The third set, Set [A+B], denotes the set that is formed by combining Set A and Set B. Median Mean StandardDeviation Set A: X, Y, Z. Set B: L, M, N. Set [A + B]: Q, R, S. If X – Y > 0 and L – M = 0, then which of the following must be true? I. Z > N II. R > M III. Q > R (A) I only (B) II only (C) III only (D) I and II only (E) NoneAnswer: E9. E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?(A) 3 (B) 4 (C) 5 (D) 6 (E) 7Answer: B10. If a certain sample of data has a mean of 20.0 and a standard deviation of 3.0, which of the following values is more than 2.5 standard deviations from the mean? (A) 12.0 (B) 13.5 (C) 17.0 (D) 23.5 (E) 26.5 Answer: A.11. Arithmetic mean and standard deviation of a certain normal distribution are 13.5 and 1.5. What value is exactly 2 standard deviations less than the mean? (A) 10.5(B) 11(C) 11.5(D) 12(E) 12.5Answer: A. Exponents and roots problems are very common on the GMAT. So, it's extremely important to know how to manipulate them, how to factor out, take roots, multiply, divide, etc. Below are 11 problems to test your skills. Please post your thought process/solutions along with the answers. I'll post OA's with detailed solutions tomorrow. Good luck.Exponents and Roots Questions: 1. What is the value of ?A. B. C. D. 50E. 60Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292162. What is the units digit of ?A. 0B. 2C. 4D. 6E. 8Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292193. If and what is the value of ?A. 14/5B. 5C. 28/5D. 13E. 14Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292214. What is the value of ?A. 5^6B. 5^7C. 5^8D. 5^9E. 5^10Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292225. If and is a multiple of , where is a non-negative integer, then what is the value of ?A. -26B. -25C. -1D. 0E. 1Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292236. If then which of the following must be true?A. B. x>-2C. x^2<4D. x^3<-8E. x^4>32Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292247. If , then which of the following must be true: A. x<6 B. 6<x<8C. 8<x<10D. 10<x<12E. x>12Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292278. If is a positive number and equals to , where the given expression extends to an infinite number of roots, then what is the value of x?A. B. 3C. D. E. 6Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p10292289. If is a positive integer then the value of is closest to which of the following?A. B. C. D. E. Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p102922910. Given that and , then what is the value of ?A. 5B. 10C. 15D. 20E. Can not be determinedSolution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p102923111. If , and then what is the value of ?A. 2B. 2^(11)C. 2^(32)D. 2^(37)E. 2^(64)Solution: tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029232Tough and Tricky Exponents and roots questions: Answers by Bunuel:What is the value of ?A. B. C. D. 50E. 60Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:Must know fro the GMAT: (while ). So we get: . Note that sum of the first and the third terms simplifies to , so we have --> . Also must know for the GMAT: , thus .Recall that we should un-square this value to get the right the answer: .Answer: C.2. What is the units digit of ?A. 0B. 2C. 4D. 6E. 8Must know for the GMAT:I. The units digit of is the same as that of , which means that the units digit of is that same as that of and the units digit of is that same as that of .II. If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:and not , which on the other hand equals to .So:;.Thus, and .III. The units digit of integers in positive integer power repeats in specific pattern (cyclicity): The units digit of 7 and 3 in positive integer power repeats in patterns of 4:1. 7^1=7 (last digit is 7)2. 7^2=9 (last digit is 9)3. 7^3=3 (last digit is 3)4. 7^4=1 (last digit is 1)5. 7^5=7 (last digit is 7 again!)...1. 3^1=3 (last digit is 3)2. 3^2=9 (last digit is 9)3. 3^3=27 (last digit is 7)4. 3^4=81 (last digit is 1)5. 3^5=243 (last digit is 3 again!)...Thus th units digit of will be 1 (4th in pattern, as 12 is a multiple of cyclicty number 4) and the units digit of will be 3 (first in pattern, as 9=4*2+1).So, we have that the units digit of is 1 and the units digit of is 3. Also notice that the second number is much larger then the first one, thus their difference will be negative, something like 11-13=-2, which gives the final answer that the units digit of is 2.Answer B.3. If and what is the value of ?A. 14/5B. 5C. 28/5D. 13E. 14First thing one should notice here is that and must be some irrational numbers (4,900 has other primes then 5 in its prime factorization and 25 doesn't have 2 as a prime at all), so we should manipulate with given expressions rather than to solve for x and y.--> --> Answer: E.4. What is the value of ?A. 5^6B. 5^7C. 5^8D. 5^9E. 5^10This question can be solved in several ways:Traditional approach: Note that we have the sum of geometric progression in brackets with first term equal to 5 and common ratio also equal to 5. The sum of the first terms of geometric progression is given by: , where is the first term, # of terms and is a common ratio .So in our case: .30 sec approach based on answer choices:We have the sum of 6 terms. Now, if all terms were equal to the largest term 4*5^5 we would have: , so the actual sum must be less than 5^7, thus the answer must be A: 5^6.Answer: A.5. If and is a multiple of , where is a non-negative integer, then what is the value of ?A. -26B. -25C. -1D. 0E. 1so is an odd number. The only way it to be a multiple of (even number in integer power) is when , in this case and 1 is a factor of every integer. Thus --> . Must know for the GMAT: , for - any nonzero number to the power of 0 is 1. Important note: the case of 0^0 is not tested on the GMAT.Answer: C.6. If then which of the following must be true?A. B. x>-2C. x^2<4D. x^3<-8E. x^4>32Must know for the GMAT: Even roots from negative number is undefined on the GMAT (as GMAT is dealing only with Real Numbers): , for example .Odd roots have the same sign as the base of the root. For example, and .Back to the original question:As then must be a little bit less than -2 --> . Thus , so option D must be true.As for the other options:A. , is not true.B. , thus x>-2 is also not true.C. , thus x^2<4 is also not true.E. , (2^4=16, so anyway -2.1^4 can not be more than 32) thus x^4>32 is also not true.Answer: D.7. If , then which of the following must be true: A. x<6 B. 6<x<8C. 8<x<10D. 10<x<12E. x>12Here is a little trick: any positive integer root from a number more than 1 will be more than 1. For example: .Now, (as 3^2=9) and (2^3=8). Thus Answer: E.8. If is a positive number and equals to , where the given expression extends to an infinite number of roots, then what is the value of x?A. B. 3C. D. E. 6Given: and --> , as the expression under the square root extends infinitely then expression in brackets would equal to itself and we can safely replace it with and rewrite the given expression as . Square both sides: --> --> or , but since then: .Answer: B.9. If is a positive integer then the value of is closest to which of the following?A. B. C. D. E. Note that we need approximate value of the given expression. Now, is much larger number than . Hence will be very close to itself, basically is negligible in this case. The same way will be very close to itself.Thus .You can check this algebraically as well: . Again, -1, both in denominator and nominator is negligible value and we'll get the same expression as above: Answer: D.10. Given that and , then what is the value of ?A. 5B. 10C. 15D. 20E. Can not be determinedRearranging both expressions we'll get: and . Denote as and as . So we have that and . Now, and as then --> . Thus we get two equations with two unknowns: and --> solving for --> --> . Solving for --> Finally, .Answer: B.11. If , and then what is the value of ?A. 2B. 2^(11)C. 2^(32)D. 2^(37)E. 2^(64)--> (note that is not a valid solution as given that ). Second step: --> .OR second step: --> --> since then .Answer: D.12 Easy Pieces or not:After posting some 700+ questions, I've decided to post the problems which are not that hard. Though each question below has a trap or trick so be careful when solving. I'll post OA's with detailed solutions after some discussion. Good luck.1. There are 5 pairs of white, 3 pairs of black and 2 pairs of grey socks in a drawer. If four socks are picked at random what is the probability of getting two socks of the same color?A. 1/5B. 2/5C. 3/4D. 4/5E. 1Solution: 12-easy-pieces-or-not-126366.html#p10339192. If x is an integer and 9<x^2<99, then what is the value of maximum possible value of x minus minimum possible value of x?A. 5B. 6C. 7D. 18E. 20Solution: 12-easy-pieces-or-not-126366.html#p10339213. Fanny and Alexander are 360 miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and 65 mph respectively, how far apart will they be exactly 1.5 hours before they meet?A. 25 milesB. 65 milesC. 70 milesD. 90 milesE. 135 milesSolution: 12-easy-pieces-or-not-126366.html#p10339244. If -3<x<5 and -7<y<9, which of the following represent the range of all possible values of y-x?A. -4<y-x<4B. -2<y-x<4C. -12<y-x<4D. -12<y-x<12E. 4<y-x<12Solution: 12-easy-pieces-or-not-126366.html#p10339255. The angles in a triangle are x, 3x, and 5x degrees. If a, b and c are the lengths of the sides opposite to angles x, 3x, and 5x respectively, then which of the following must be true?I. c>a+bII. c^2>a^2+b^2III. c/a/b=10/6/2 A. I onlyB. II onlyC. III onlyD. I and III onlyE. II and III onlySolution: 12-easy-pieces-or-not-126366.html#p10339306. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same color and the first and the last marbles are of different colors. How many different arrangements are possible?A. 30 B. 60C. 120D. 240E. 480Solution: 12-easy-pieces-or-not-126366.html#p10339327. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the total ratio of negative numbers to non-negative numbers be 2 to 1?A. 11/14B. 13/18C. 4/7D. 3/7E. 3/14Solution: 12-easy-pieces-or-not-126366.html#p10339338. There are 15 black chips and 5 white chips in a jar. What is the least number of chips we should pick to guarantee that we have 2 chips of the same color?A. 3B. 5C. 6D. 16E. 19Solution: 12-easy-pieces-or-not-126366.html#p10339359. Julie is putting M marbles in a row in a repeating pattern: blue, white, red, green, black, yellow, pink. If the row begins with blue marble and ends with red marble, then which of the following could be the value of M?A. 22B. 30C. 38D. 46E. 54Solution: 12-easy-pieces-or-not-126366.html#p103393610. If is an integer and , then what is the value of n?A. 1 B. 2 C. 3 D. 4 E. 5Solution: 12-easy-pieces-or-not-126366.html#p103393811. The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p?A. 97B. 151C. 209D. 211E. 219Solution: 12-easy-pieces-or-not-126366-20.html#p103393912. If and , what is the least value of possible?A. -1/100B. -1/50C. -1/36D. -1/18E. -1/6Solution: 12-easy-pieces-or-not-126366-20.html#p103394912 Easy Pieces or not – Solutions by Bunuel: There are 5 pairs of white, 3 pairs of black and 2 pairs of grey socks in a drawer. If four socks are picked at random what is the probability of getting two socks of the same color?A. 1/5B. 2/5C. 3/4D. 4/5E. 1No formula is need to answer this one. The trick here is that we have only 3 different color socks but we pick 4 socks, which ensures that in ANY case we'll have at least one pair of the same color (if 3 socks we pick are of the different color, then the 4th sock must match with either of previously picked one). P=1.Answer: E. _________________2. If x is an integer and 9<x^2<99, then what is the value of maximum possible value of x minus minimum possible value of x?A. 5B. 6C. 7D. 18E. 20Also tricky. Notice that can take positive, as well as negative values to satisfy , hence can be: -9, -8, -7, -6, -4, 4, 5, 6, 7, 8, or 9. We asked to find the value of , ans since and then .Answer: D.3. Fanny and Alexander are 360 miles apart and are traveling in a straight line toward each other at a constant rate of 25 mph and 65 mph respectively, how far apart will they be exactly 1.5 hours before they meet?A. 25 milesB. 65 milesC. 70 milesD. 90 milesE. 135 milesMake it simple! The question is: how far apart will they be exactly 1.5 hours before they meet? As Fanny and Alexander's combined rate is 25+65 mph then 1.5 hours before they meet they'll be (25+65)*1.5=135 miles apart.Answer: E. _________________4. If -3<x<5 and -7<y<9, which of the following represent the range of all possible values of y-x?A. -4<y-x<4B. -2<y-x<4C. -12<y-x<4D. -12<y-x<12E. 4<y-x<12To get max value of y-x take max value of y and min value of x: 9-(-3)=12;To get min value of y-x take min value of y and max value of x: -7-(5)=-12; Hence, the range of all possible values of y-x is -12<y-x<12.Answer: D.5. The angles in a triangle are x, 3x, and 5x degrees. If a, b and c are the lengths of the sides opposite to angles x, 3x, and 5x respectively, then which of the following must be true?I. c>a+bII. c^2>a^2+b^2III. c/a/b=10/6/2 A. I onlyB. II onlyC. III onlyD. I and III onlyE. II and III onlyAccording to the relationship of the sides of a triangle: the length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. Thus I and III can never be true: one side (c) can not be larger than the sum of the other two sides (a and b). Note that III is basically the same as I: if c=10, a=6 and b=2 then c>a+b, which can never be true. Thus even not considering the angles, we can say that only answer choice B (II only) is left.Answer: B.Now, if interested why II is true: as the angles in a triangle are x, 3x, and 5x degrees then x+3x+5x=180 --> x=20, 3x=60, and 5x=100. Next, if angle opposite c were 90 degrees, then according to Pythagoras theorem c^2=a^+b^2, but since the angel opposite c is more than 90 degrees (100) then c is larger, hence c^2>a^+b^2.6. Anna has 10 marbles: 5 red, 2 blue, 2 green and 1 yellow. She wants to arrange all of them in a row so that no two adjacent marbles are of the same color and the first and the last marbles are of different colours. How many different arrangements are possible?A. 30 B. 60C. 120D. 240E. 480Seems tough and complicated but if we read the stem carefully we find that the only way both conditions to be met for 5 red marbles, which are half of total marbles, they can be arranged only in two ways: R*R*R*R*R* or *R*R*R*R*R. Here comes the next good news, in these cases BOTH conditions are met for all other marbles as well: no two adjacent marbles will be of the same color and the first and the last marbles will be of different colors.Now, it's easy: 2 blue, 2 green and 1 yellow can be arranged in 5 empty slots in 5!/(2!*2!)=30 ways (permutation of 5 letters BBGGY out of which 2 B's and 2 G' are identical). Finally as there are two cases (R*R*R*R*R* and *R*R*R*R*R. ) then total # of arrangement is 30*2=60.Answer: B. ______________7. After 2/9 of the numbers in a data set A were observed, it turned out that 3/4 of those numbers were non-negative. What fraction of the remaining numbers in set A must be negative so that the total ratio of negative numbers to non-negative numbers be 2 to 1?A. 11/14B. 13/18C. 4/7D. 3/7E. 3/14If choose variable for set A there will be too many fractions to manipulate with, so pick some smart #: let set A contain 18 numbers."2/9 of the numbers in a data set A were observed" --> 4 observed and 18-4=14 numbers left to observe;"3/4 of those numbers were non-negative" --> 3 non-negative and 1 negative;Ratio of negative numbers to non-negative numbers to be 2 to 1 there should be total of 18*2/3=12 negative numbers, so in not yet observed part there should be 12-1=11 negative numbers. Thus 11/14 of the remaining numbers in set A must be negative.Answer: A.8. There are 15 black chips and 5 white chips in a jar. What is the least number of chips we should pick to guarantee that we have 2 chips of the same color?A. 3B. 5C. 6D. 16E. 19Worst case scenario would be if the first two chips we pick will be of the different colors. But the next chip must match with either of two, so 3 is the answer.Answer: A.9. Julie is putting M marbles in a row in a repeating pattern: blue, white, red, green, black, yellow, pink. If the row begins with blue marble and ends with red marble, then which of the following could be the value of M?A. 22B. 30C. 38D. 46E. 54There are total of 7 different color marbles in a pattern. Now, as the row begins with blue marble and ends with red marble (so ends with 3rd marble in a pattern) then M=7k+3. The only answer choice which is multiple of 7 plus 3 is 38=35+3.Answer: C.10. If is an integer and , then what is the value of n?A. 1 B. 2 C. 3 D. 4 E. 5Also no need for algebraic manipulation. 1/10^(n+1) is 10 times less than 1/10^n, and both when expressed as decimals are of a type 0.001 (some number of zeros before 1) --> so the given expression to hold true we should have: 0.001<0.00737<0.01, which means that n=2 (1/10^n=0.01 --> n=2).Answer: B.11. The numbers {1, 3, 6, 7, 7, 7} are used to form three 2-digit numbers. If the sum of these three numbers is a prime number p, what is the largest possible value of p?A. 97B. 151C. 209D. 211E. 219What is the largest possible sum of these three numbers that we can form? Maximize the first digit: 76+73+71=220=even, so not a prime. Let's try next largest sum, switch digits in 76 and we'll get: 67+73+71=211. Question is it a prime number? If you notice 210=2*3*5*7=the product of the first four primes. So, 210+1=211 must be a prime. For example: 2+1=3=prime, 2*3+1=7=prime, 2*3*5+1=31=prime.Answer: D.12. If and , what is the least value of possible?A. -1/100B. -1/50C. -1/36D. -1/18E. -1/6To get the least value of , which obviously will be negative, try to maximize absolute value of , as more is the absolute value of a negative number "more" negative it is (the smallest it is).To maximize pick largest absolute values possible for and : . Notice that: -1/18<-1/36<-1/50<-1/100, so -1/100 is the largest number and -1/18 is the smallest number (we cannot obtain -1/6 from x^2*y or else it would be the correct answer).Answer: D.BaKers Dozen: A password on Mr. Wallace's briefcase consists of 5 digits. What is the probability that the password contains exactly three digit 6?A. 860/90,000B. 810/100,000C. 858/100,000 D. 860/100,000E. 1530/100,000Solution: baker-s-dozen-128782-20.html#p10575022. If , then y is NOT divisible by which of the following?A. 6^4B. 62^2C. 65^2D. 15^4E. 52^4Solution: baker-s-dozen-128782-20.html#p10575033. For the past k days the average (arithmetic mean) cupcakes per day that Liv baked was 55. Today Bibi joined and together with Liv they baked 100 cupcakes, which raises the average to 60 cupcakes per day. What is the value of k?A. 6B. 8C. 9D. 10E. 12Solution: baker-s-dozen-128782-20.html#p10575044. What is the smallest positive integer such that is the square of a positive integer?A. 14B. 36C. 144D. 196E. 441Solution: baker-s-dozen-128782-20.html#p10575055. There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble to remain in the jar?A. 460B. 490C. 493D. 455E. 445Solution: baker-s-dozen-128782-20.html#p10575076. A pool has two water pumps A and B and one drain C. Pump A alone can fill the whole pool in x hours, and pump B alone can fill the whole pool in y hours. The drain can empty the whole pool in z hours, where z>x. With pumps A and B both running and the drain C unstopped till the pool is filled, which of the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?A. B. C. D. E. Solution: baker-s-dozen-128782-20.html#p10575087. Metropolis Corporation has 4 shareholders: Fritz, Luis, Alfred and Werner. Number of shares that Fritz owns is 2/3 rd of number of the shares of the other three shareholders, number of the shares that Luis owns is 3/7 th of number of the shares of the other three shareholders and number of the shares that Alfred owns is 4/11 th of number of the shares of the other three shareholders. If dividends of $3,600,000 were distributed among the 4 shareholders, how much of this amount did Werner receive?A. $60,000B. $90,000C. $100,000D. $120,000E. $180,000Solution: baker-s-dozen-128782-20.html#p10575098. A set A consists of 7 consecutive odd integers. If the sum of 5 largest integers of set A is -185 what is the sum of the 5 smallest integers of set A?A. -165B. -175C. -195D. -205E. -215Solution: baker-s-dozen-128782-20.html#p10575129. If x and y are negative numbers, what is the value of ?A. 1+yB. 1-yC. -1-yD. y-1E. x-ySolution: baker-s-dozen-128782-20.html#p105751410. If x^2<81 and y^2<25, what is the largest prime number that can be equal to x-2y?A. 7B. 11C. 13D. 17E. 19Solution: baker-s-dozen-128782-20.html#p105751511. In an infinite sequence 1, 3, 9, 27, ... each term after the first is three times the previous term. What is the difference between the sum of 13th and 15th terms and the sum of 12th and 14th terms of the sequence?A. 10*3^11B. 20*3^11C. 10*3^12D. 40*3^11E. 20*3^12Solution: baker-s-dozen-128782-40.html#p105751712. x, y and z are positive integers such that when x is divided by y the remainder is 3 and when y is divided by z the remainder is 8. What is the smallest possible value of x+y+z?A. 12B. 20C. 24D. 29E. 33Solution: baker-s-dozen-128782-40.html#p105751913. If , what is the product of the tens and the units digits of ?A. 0B. 6C. 7D. 12E. 14Solution: baker-s-dozen-128782-40.html#p1057520Bakers Dozen Solution by Bunuel: A password on Mr. Wallace's briefcase consists of 5 digits. What is the probability that the password contains exactly three digit 6?A. 860/90,000B. 810/100,000C. 858/100,000 D. 860/100,000E. 1530/100,000Total # of 5 digit codes is 10^5, notice that it's not 9*10^4, since in a code we can have zero as the first digit.# of passwords with three digit 6 is : each out of two other digits (not 6) has 9 choices, thus we have 9*9 and is ways to choose which 3 digits will be 6's out of 5 digits we have.Answer: B.2. If , then y is NOT divisible by which of the following?A. 6^4B. 62^2C. 65^2D. 15^4E. 52^4.Now, if you analyze each option you'll see that only is not a factor of , since the power of 13 in it is higher than the power of 13 in .Answer: E.3. For the past k days the average (arithmetic mean) cupcakes per day that Liv baked was 55. Today Bibi joined and together with Liv they baked 100 cupcakes, which raises the average to 60 cupcakes per day. What is the value of k?A. 6B. 8C. 9D. 10E. 12Total cupcakes for k days was 55k, which means that total cupcakes for k+1 days was 55k+100. The new average is (55k+100)/(k+1)=60 --> 55k+100=60k+60 --> k=8Answer: B.4. What is the smallest positive integer such that is the square of a positive integer?A. 14B. 36C. 144D. 196E. 441, so in order to be a square of an integer must complete the powers of 2 and 7 to even number, so the least value of must equal to 2*7=14, which makes the leas value of equal to 14^2=196.Answer: D.5. There are 7 red and 5 blue marbles in a jar. In how many ways 8 marbles can be selected from the jar so that at least one red marble and at least one blue marble to remain in the jar?A. 460B. 490C. 493D. 455E. 445Total ways to select 8 marbles out of 7+5=12 is ;Ways to select 8 marbles so that zero red marbles is left in the jar is ;Ways to select 8 marbles so that zero blue marbles is left in the jar is ;Hence ways to select 8 marbles so that at least one red marble and at least one blue marble to remain the jar is .Answer: D.6. A pool has two water pumps A and B and one drain C. Pump A alone can fill the whole pool in x hours, and pump B alone can fill the whole pool in y hours. The drain can empty the whole pool in z hours, where z>x. With pumps A and B both running and the drain C unstopped till the pool is filled, which of the following represents the amount of water in terms of the fraction of the pool which pump A pumped into the pool?A. B. C. D. E. With pumps A and B both running and the drain unstopped the pool will be filled in a rate pool/hour. So, the pool will be filled in hours (time is reciprocal of rate).In hours A will pump amount of the water into the pool.Answer: B.7. Metropolis Corporation has 4 shareholders: Fritz, Luis, Alfred and Werner. Number of shares that Fritz owns is 2/3 rd of number of the shares of the other three shareholders, number of the shares that Luis owns is 3/7 th of number of the shares of the other three shareholders and number of the shares that Alfred owns is 4/11 th of number of the shares of the other three shareholders. If dividends of $3,600,000 were distributed among the 4 shareholders, how much of this amount did Werner receive?A. $60,000B. $90,000C. $100,000D. $120,000E. $180,000Fritz owns is rd of the shares of the other three shareholders --> Fritz owns th of all shares;Luis owns is th of the shares of the other three shareholders --> Luis owns th of all shares;Alfred owns is th of the shares of the other three shareholders --> Alfred owns th of all shares;Together those three own th of all shares, which means that Werner owns . Hence from $3,600,000 Werner gets .Answer: D. _________________8. A set A consists of 7 consecutive odd integers. If the sum of 5 largest integers of set A is -185 what is the sum of the 5 smallest integers of set A?A. -165B. -175C. -195D. -205E. -215Say 7 consecutive odd integers are: , , , , , , .Question: Given: --> --> --> Answer: D.9. If x and y are negative numbers, what is the value of ?A. 1+yB. 1-yC. -1-yD. y-1E. x-yNote that . Next, since and then and . So, Answer: D.10. If x^2<81 and y^2<25, what is the largest prime number that can be equal to x-2y?A. 7B. 11C. 13D. 17E. 19Notice that we are not told that and are integers.means that and means that . Now, since the largest value of is almost 9 and the largest value of is almost 10 (for example if ), then the largest value of is almost 9+10=19, so the actual value is less than 19, which means that the largest prime that can be equal to is 17. For example: and .Answer: D.11. In an infinite sequence 1, 3, 9, 27, ... each term after the first is three times the previous term. What is the difference between the sum of 13th and 15th terms and the sum of 12th and 14th terms of the sequence?A. 10*3^11B. 20*3^11C. 10*3^12D. 40*3^11E. 20*3^12You don't need to know geometric progression formula to solve this question. All you need is to find the pattern:;;;;...;Answer: B.12. x, y and z are positive integers such that when x is divided by y the remainder is 3 and when y is divided by z the remainder is 8. What is the smallest possible value of x+y+z?A. 12B. 20C. 24D. 29E. 33Given , where is a quotient, an integer . Which means that the least value of is when , in that case . This basically means that is less than . For example 3 divided by 4 yields remainder of 3. Thus we have that:is divided by the remainder is 3 --> minimum value of is 3;is divided by the remainder is 8 --> minimum value of is 8 and minimum value of is one more than 8, so 9 (8 divided by 9 yields the remainder of 8);So, the smallest possible value of is 3+8+9=20.Answer: B.13. If , what is the product of the tens and the units digits of ?A. 0B. 6C. 7D. 12E. 14Apply : .Next, . Now, since has 2 and 5 as its multiples, then it will have 0 as the units digit, so will have two zeros in the end, which means that will have 00-38=62 as the last digits: 6*2=12.Answer: D. ................
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