Learning from Mistakes
If x, y, and z are integers greater than 1, and (3^27)(5^10)(z) = (5^8)(9^14)(x^y), then what is the value of x? (1) y is prime (2) x is prime
(3^27)(5^10)(z) = (5^8)(9^14)(x^y) Simplify the 914 (3^27)(5^10)(z) = (5^8)(3^28)(x^y) Divide both sides by common terms 5^8, 3^27 (5^2)(z) = 3x^y (1) INSUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Similarly, x must have at least one factor of 5. Statement (1) says that y is prime, which does not tell us how many fives are contained in x and z. (2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5 to balance out the 5^2 on the left side. Since statement (2) says that x is prime, x cannot have any other factors, so x = 5. Therefore statement (2) is sufficient.
When a is divided by b, the quotient is c and the remainder is d. Which of the following expressions is equal to d?
(a/b) = c + (d/b)
In the figure above, SQRE is a square and AB = AC. Is the area of triangle ABC greater than the area of square SQRE? Square SQRE with Triangle ABC inside, BR and CE are the right and left bits of the triangle beyond the square base (1) The length of RE is less than twice the length of BR. (2) AS = AQ
Area formulas - (1/2)bh for triangle and bh for square Figure out the base of triangle Triangle base = ER + CE + BR = b + CE + BR Height of triangle is same as base To rephrase the question, it is helpful to see what connections exist between the triangle and the square and furthermore how it is possible to relate their areas. Draw a vertical line from point A straight down to the base of the triangle (and label the bottom point of this line Z). This new line, AZ (of length x), is the height of triangle ABC and has the same length as any side of the square. Label the diagram with the rest of the given information: AB = AC, implying that triangle ABC is isosceles and that angle B must equal angle C. The area of the square can be expressed as x2 and the area of the triangle as (label all lengths equal to AZ as x)Since BC is comprised of ER + RB + CE, and ER is also a side of the square or x, you can rewrite BC asx + RB + CE. Thus the question becomes "Is ?"If RB and CE together sum to the length of x, then the left side of the inequality will be identical to the right:For the left side of the inequality to be greater than the right, however, RB and CE must sum to more than x, thus the final rephrase can be stated as "Is RB + CE > x ?" (1) NOT SUFFICIENT: The statement can be translated as ER < 2RB x < 2RB Again replacing ER with the variable xTo know if RB + CE > x, you would need to know something about CE as well. Can you assume that CE = RB? From the given, triangle ABC is isosceles and thus symmetrical. However, there is no guarantee that is symmetrically placed in the square. If the triangle is symmetrically placed, RB = CE and both are greater than and would thus sum to greater than x (scenario II below). Scenario III below would also satisfy the question, but scenario I would not necessarily.(2) NOT SUFFICIENT: This statement establishes the symmetric placement of triangle ABC within the square SQRE. On its own it does not relate RB and CE to x.(1) AND (2) TOGETHER SUFFICIENT: Given the symmetric placement of triangle ABC within the square SQRE from statement (2) along with the information from statement (1), scenario II above is established and RB + CE > x. The correct answer is (C).
Six mobsters have arrived at the theater for the premiere of the film "Goodbuddies." One of the mobsters, Frankie, is an informer, and he's afraid that another member of his crew, Joey, is on to him. Frankie, wanting to keep Joey in his sights, insists upon standing behind Joey in line at the concession stand, though not necessarily right behind him. How many ways can the six arrange themselves in line such that Frankie's requirement is satisfied?
Combinatorics Ignoring Frankie's requirement for a moment, observe that the six mobsters can be arranged 6! or 6 x 5 x 4 x 3 x 2 x 1 = 720 different ways in the concession stand line. In each of those 720 arrangements, Frankie must be either ahead of or behind Joey. Logically, since the combinations favor neither Frankie nor Joey, each would be behind the other in precisely half of the arrangements. Therefore, in order to satisfy Frankie's requirement, the six mobsters could be arranged in 720/2 = 360 different ways.
If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?
Consecutive sets and using avg formula to get sum and derive largest prime factor First, let us simplify the problem by rephrasing the question. Since any even number must be divisible by 2, any even multiple of 15 must be divisible by 2 and by 15, or in other words, must be divisible by 30. As a result, finding the sum of even multiples of 15 is equivalent to finding the sum of multiples of 30. By observation, the first multiple of 30 greater than 295 will be equal to 300 and the last multiple of 30 smaller than 615 will be equal to 600. Thus, since there are no multiples of 30 between 295 and 299 and between 601 and 615, finding the sum of all multiples of 30 between 295 and 615, inclusive, is equivalent to finding the sum of all multiples of 30 between 300 and 600, inclusive. Therefore, we can rephrase the question: "What is the greatest prime factor of the sum of all multiples of 30 between 300 and 600, inclusive?" The sum of a set = (the mean of the set) × (the number of terms in the set) Since 300 is the 10th multiple of 30, and 600 is the 20th multiple of 30, we need to count all multiples of 30 between the 10th and the 20th multiples of 30, inclusive. There are 11 terms in the set: 20th - 10th + 1 = 10 + 1 = 11The mean of the set = (the first term + the last term) divided by 2: (300 + 600) / 2 = 450 k = the sum of this set = 450 × 11 Note, that since we need to find the greatest prime factor of k, we do not need to compute the actual value of k, but can simply break the product of 450 and 11 into its prime factors: k = 450 × 11 = 2 × 3 × 3 × 5 × 5 × 11 Therefore, the largest prime factor of k is 11.
When a cylindrical tank is filled with water at a rate of 22 cubic meters per hour, the level of water in the tank rises at a rate of 0.7 meters per hour. Which of the following best approximates the radius of the tank in meters?
Cylinder Volume = (pi)(r^2)h Assume 1 hr Volume = 22*1 = 22; height = 0.7*1=0.7 22 = (pi)(r^2)(0.7) r = square root of 10
Joan, Kylie, Lillian, and Miriam all celebrate their birthdays today. Joan is 2 years younger than Kylie, Kylie is 3 years older than Lillian, and Miriam is one year older than Joan. Which of the following could be the combined age of all four women today?
Draw it out and notice the consecutive numbers Since the ages are consecutive integers, they can all be expressed in terms of L: L, L + 1,L + 2, L + 3. The sum of the four ages then would be 4L + 6. Since L must be an integer (it's Lillian's age), the expression 4L + 6 describes a number that is two more than a multiple of 4: 4L + 6 = (4L + 4) + 2 [4L + 4 describes a multiple of 4, since it can be factored into 4(L + 1) or 4 * an integer.] 54 is the only number in the answer choices that is two more than a multiple of 4 (namely, 52).
Which of the following, when multiplied by itself, will yield a fraction greater than 2/3 ? 5/7 2/3 0.7 0.8 0.027/0.03
Multiply by itself and find number OR Note also that you could rephrase this question to "which of the following numbers is the greatest?" Since all numbers are positive, squaring the numbers will not change their order. According to the question only one number can be large enough that, when squared, it is greater than 2/3. Thus looking for the largest answer choice will lead you to the correct answer.
In a group of 68 students, each student is registered for at least one of three classes - History, Math and English. Twenty-five students are registered for History, twenty-five students are registered for Math, and thirty-four students are registered for English. If only three students are registered for all three classes, how many students are registered for exactly two classes?
Overlapping sets - create Venn Diagram History students: a + b + c + 3 = 25 Math students: e + b + d + 3 = 25 English students: f + c + d + 3 = 34 TOTAL students: a + e + f + b + c + d + 3 = 68 The question asks for the total number of students taking exactly 2 classes. This can be represented as b + c + d. If we sum the first 3 equations (History, Math and English) we get: a + e + f + 2 b +2 c +2 d + 9 = 84.
A small company employs 3 men and 5 women. If a team of 4 employees is to be randomly selected to organize the company retreat, what is the probability that the team will have exactly 2 women?
Probability - desired/total outcomes Combinatorics - 2 different sets Total outcomes = 8!/(4!4!) [8 people/select 4 not select remaining 4] = 70 w = the number of possible teams with exactly 2 women This means there must be 2 men on the team so if we calculate the number of ways that 2 out of 5 women can be selected and the number of ways that 2 out of 3 men can be selected, we can multiply the two to get to a team of 2 men and 2 women Women = 5!/(2!3!) = 10 Men = 3!/(2!1!) = 3 w = 10*3 = 30 Probability = 30/70 = 3/7
Q = x3 − x Given that x is a positive integer such that x ≥ 75, which of the following is the remainder when Q is divided by 6?
Q = x3 − x = x(x2 − 1) = x(x + 1)(x − 1) Q = (x − 1)x(x + 1) If x is a positive integer, then (x − 1) is the integer immediately before it, and (x + 1) is the integer immediately after it. Thus, Q is the product of three consecutive integers. We know that, in any group of three consecutive integers, either one is even and two are odd, or one is odd and two are even. Thus, it's assured that at least one is even, and so Q is definitely divisible by 2. We also know that, in any group of three consecutive integers, there must be exactly one factor of three, so Q is definitely divisible by 3. Any number that is simultaneously divisible by 2 and by 3 has to be divisible by 6, so Q must be divisible by 6.
John and Jacob set out together on bicycle traveling at 15 and 12 miles per hour, respectively. After 40 minutes, John stops to fix a flat tire. It takes John one hour to fix the flat tire and Jacob continues to ride during this time. Once John has resumed his ride, at a rate of 15 miles per hour, how many hours will it take him to catch up with Jacob? (consider John's deceleration/acceleration before/after the flat tire to be negligible)
Rate*Time = Distance Divide out the parts and create table: 40 min = 2/3 hr In 2/3 of an hour, John traveled 15 × 2/3 = 10 miles (rt = d) In that same 2/3 of an hour, Jacob traveled 12 × 2/3 = 8 miles John therefore had a two-mile lead when he stopped to fix his tire. It took John 1 hour to fix his tire, during which time Jacob traveled 12 miles. Since John began this 1-hour period 2 miles ahead, at the end of the period he is 12 - 2 = 10 miles behind Jacob. The question now becomes "how long does it take John, traveling 15 miles per hour, to bridge the 10-mile gap between him and Jacob (while Jacob is still traveling at 12 miles per hour)?" We can set up an rt = d chart to solve this. Jacob travels some distance, d, during this time period; John needs to cover that distance plus another 10 miles. The two are traveling for the same length of time (beginning when John starts riding again). John's travel during this "catch-up period" can be represented as 15t = d + 10Jacob's travel during this "catch-up period" can be represented as 12t = d If we solve these two simultaneous equations, we get: 15t = 12t + 10 3t = 10 t = 3 1/3 hours
If a car traveled from Townsend to Smallville at an average speed of 40 mph and then returned to Townsend along the same route later that evening, what was the average speed for the entire trip? (1) The trip from Townsend to Smallville took 50% longer than the trip from Smallville to Townsend. (2) The route between Townsend and Smallville is 165 miles long.
Rate*Time = Distance To determine the average speed for the trip from Townsend to Smallville and back again, we need to know the average speed in each direction. Because the distance in each direction is the same, if we have the average speed in each direction we will be able to find the average speed of the entire trip by taking the total distance and dividing it by the total time. (1) SUFFICIENT: This allows us to figure out the average speed for the return trip. If the return time was 2/3 the outgoing time, the return speed must have been 3/2 that of the outgoing. Whenever the distance is fixed, the ratio of the times will be the inverse of the ratio of the speeds. We can see this by looking at an example. Let's say the distance between the two towns was 120 miles. We can calculate the "going" time as 3 hours. Since, the "going" trip took 50% longer than the "return" trip, the "returning time" is 2 hours. Thus, the average rate for the return trip is Distance/Time or 120/2 = 60 miles per hour. We can use this table to calculate the average speed for the entire trip: take the total distance, 240, and divide by the total time, 5. This results in an average speed of 240/5 = 48 miles per hour. It does not matter that we chose a random distance of 120; we would able to solve using any distance or even using a variable x as the distance, as long as we adhere to the given information that the distance is the same in both directions. The times would adjust accordingly based on the distance we used and the same average speed of 48 would result. (2) INSUFFICIENT: If all we know is the distance from Townsend to Smallville, we will be able to find the time traveled on the way there but we will have no indication of how fast the car traveled on the way back and therefore no way of calculating the average overall speed.
If x and y are non-zero integers, and 9x^4 - 4y^4 = 3x^2 + 2y^2, which of the following could be the value of x^2 in terms of y?
Simplify (algebra's most important rule: (a^2)-(b^2) = (a-b)(a+b) (3x2 + 2y2)(3x2 - 2y2) = (3x2 + 2y2) Note that if x and y are non zero integers, these are not equal to zero and we can divide out the same set (3x^2 - 2y^2) = 13 x^2 = 2y^2 + 1 x^2 = (2y^2+1)/3
if and y are integers and (15&x + 15^(x+1))/(4^y) = (15^y), what is the value of x?
Simplify using exponent rules (3^x)(5^x)(2^4) = (3^y)(5^y)(2^2y) Since both sides of the equation are broken down to the product of prime bases, the respective exponents of like bases must be equal. 2y = 4 so y = 2. x = y so x = 2.
If 2 + 5 a - b/2 = 3 c, what is the value of b? (1) a + c = 13 (2) -12 c = -20 a + 4
Simplify! 4 + 10a - 6c = b Question is what is 10a - 6c? (1) INSUFFICIENT: Knowing the sum of a and c is not enough to determine the value of 10a - 6c. For example, if a = 10 and b = 3, then 10a - 6c = 10(10) - 6(3) = 82. However, if a = 6 and b = 7, then 10a - 6c = 10(6) - 6(7) = 18. (2) SUFFICIENT: Manipulating the equation gives us the following: -12c = -20a + 4 20a - 12c = 4 10a-6c = 2
90 students represent x percent of the boys at Jones Elementary School. If the boys at Jones Elementary make up 40% of the total school population of x students, what is x?
The boys of Jones Elementary make up 40% of the total of x students. Therefore: # of boys = 0.4x x% of the # of boys is 90. Use x/100 as x%: (x/100) × (# of boys) = 90 Substitute for # of boys from the first equation: (x/100) × 0.4x = 90 (0.4x2) / 100 = 90 0.4x2 = 9,000 4x2 = 90,000 x2 = 22,500 x = 150
Is d negative? (1) e + d = -12 (2) e - d < -12
The question asks about the sign of d. (1) INSUFFICIENT: When two numbers sum to a negative value, we have two possibilities: Possibility A: Both values are negative (e.g., e = -4 and d = -8) Possibility B: One value is negative and the other is positive.(e.g., e = -15 and d = 3). (2) INSUFFICIENT: When the difference of two numbers produces a negative value, we have three possibilities: Possibility A: Both values are negative (e.g., e = -20 and d = -3) Possibility B: One value is negative and the other is positive (e.g., e = -20 and d = 3). Possibility C: Both values are positive (e.g., e = 20 and d = 30) (1) AND (2) SUFFICIENT: When d is ADDED to e, the result (-12) is greater than when d is SUBTRACTED from e. This is only possible if d is a positive value. If d were a negative value than adding d to a number would produce a smaller value than subtracting d from that number (since a double negative produces a positive). You can test numbers to see that d must be positive and so we can definitively answer the question using both statements.
If (243^x)(463^y) = n, where x and y are positive integers, what is the units digit of n? (1) x + y = 7 (2) x = 4
Units rule - unit digit x unit digit --> unit digit (3^x)*(3^y) = 3^(x+y) we need to know what is x+y 1) Sufficient 2) Not Sufficient
