GMAT - Quant
Is x negative? (1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x.
(1) How can x and x^2 be greater than x^3? When 0<x<1 or when x is negative So Insuff (2) How can x^2 and x^3 be greater than x? When x > 1 and when x is negative Insuff (1) + (2) -- satisfactory when x is negative and why x is positive, parts don't overlap could test if x = 0 or 1 0>x>1 x>1
A certain salesman's yearly income is determined by a base salary plus a commission on the sales he makes during the year. Was the salesman's commission larger than his base salary last year? (1) If the amount of the commission had been 30 percent higher, the salesman's total income (salary plus commission) would have been 10 percent higher last year. (2) The absolute difference between the amount of the salesman's base salary and the amount of the commission was equal to 50 percent of the salesman's base salary last year.
(1) The salesman's total income is defined as s + c. This allows us to construct the following equation: 1.1( s + c) = s + 1.3 c This gives us: 1.1 s + 1.1 c = s + 1.3 c 0.1 s = 0.2 c so s > c (2) INSUFFICIENT: Either s - c = .5 s or c - s = .5 s
If a and b are positive integers, what is the units digit of b ? (1) b is 25% greater than a. (2) If b is decreased by 50%, the result is not an integer.
(1) b = (1.25)*a or b = 5/4*a... so if b is an integer, b/c you are multiplying by 5 as the numerator, your units digit will be 0 or 5 (2) b is odd (1) + (2) -- have to be 5
If x, y, and z are integers greater than 1, what is the value of x + y + z? "(1) xyz = 70 "(2) x/yz=7/10"
(1) prime factorization of 70 - 2 * 5 * 7 (only numbers that can be it if x,y,z > 1 (2) NS can have x = 7 and y and z = 2 and 5 or x = 14 and y and z = 4 10 A
Median properties?
- Odd # of values = unique middle value in the middle of set - Even # of value = average of middle two values (not a value in the set unless value IN THE MIDDLE ARE EQUAL)
Difference between -2^3 and -2^-3
-2^3 = -8 and -2^-3 = 1/8
Average of an ODD # of consecutive integers will be Average of an EVEN # of consecutive integers wil lbe
... an integer ... never be an integer
Add exponents when... Add bases when...
... you have the same base and you are multpliying (3^3 * 3^2) = 3^5 ... you have the same exponents (2^3 * 3^3) = 6^3
Is 1 prime? How many even primes are there?
1 is not prime 2 is the only even prime
1) What is the relationship of mean and median in evenly spaced sets properties (2,4,6,8) or (4, 8, 12, 16, 20)? 2) Does this apply to consecutive?
1) Arithmetic mean and median are equal to one another for any evenly spaced sets 2) Evenly spaced sets are consecutive so yes (mean = median)
If n = 10^10 and n^n = 10^d, what is the value of d?
10 ^ (10 * (10 ^ 10)) = 10 ^ (10 ^ 11)
Jim and Renee will play one game of Rock, Paper, Scissors. In this game, each will select and show a hand sign for one of the three items. Rock beats Scissors, Scissors beat Paper, and Paper beats Rock. Assuming that both Jim and Renee have an equal chance of choosing any one of the hand signs, what is the probability that Jim will win?
3 for Jim * 3 for Renee = 3 total outcomes Jim wins 1/3 of time Tie 1/3 of time Renee wins 1/3 of time
Ratio of radii with two circles where once circle's Area is 4 times greater than the other? What is probability of landing something in a smaller circle?
4pi*r^2 = pi*R^2 4r^2 = R^2 2r = R 2:1 R : r Probability of landing something in small circle is 1/4
If ratio is currently 3:4 for history to reference and proportion of history to reference is doubled, what is new proportion The vegetarians attended the party at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians.
6:4 not 3:8 --> double first number Rate for non-veg is 4:3 --> half the rate would be 2:3
Which of the following integers is NOT a divisor of x if x = (21)(3^7) - (112)?
7(3^8 - 16) 7(3^4-4) (3^4+4) 7(3^2-2)*(3^2+2)(3^4+4) -- see perfect squares or ^ even number... should look to factor
The Italian size of a suit jacket is N centimeters, where N is the linear distance between the armpits. The American size of a suit jacket is 2.5P centimeters, where 2.5P is twice the linear distance between the armpits. If, for a particular jacket, N = P + 10, which of the following is closest to the value of N? 36 47 58 65 72
CAT 4 #22 N = 1.25 P So 1.25P = P + 10 .25P = 10 P = 40 FORGOT THAT I'M SOLVING FOR N!! N = 40 + 10 = 50 CLOSEST VALUE - don't have to have exact value so closest is 47 WITH WORK, MAKE AN EDUCATED GUESS, NOT JUST SOME RANDOM NUMBER
In a certain sequence, each term, starting with the 3rd term, is found by multiplying the previous two terms. What is the difference between the 6th and 3rd terms in the sequence? (1) The 1st term is equal to 8 times the 2nd term. (2) The 4th term is equal to 1.
Can use smart numbers or solve algebraically (1) 8x, x, 8x^2, 8x^3, 8^3x^5, 8^4x^8 (2) assume third number is equal to n so n, 1, n, n --> difference is 0 Can also just asume third number is a random # like 4; so 4, 1, 4, 4 The 1 is the game changer here
If x is a positive number, is x an even integer? (1) 3x is an even integer. (2) 5x is an even integer.
Careful: the problem does not specify that x is an integer. The problem states only that x is a positive number, not necessarily a positive integer, so we must consider non-integer cases of x that satisfy the statements in the problem. Answer: C
It takes the high-speed train x hours to travel the z miles from Town A to Town B at a constant rate, while it takes the regular train y hours to travel the same distance at a constant rate. If the high-speed train leaves Town A for Town B at the same time that the regular train leaves Town B for Town A, how many more miles will the high-speed train have traveled than the regular train when the two trains pass each other?
For trains to cross one another, they will have to travel the whole z miles Add together the two rates So you can add the rates of high-speed and regular to figure out the ours it takes to travel z miles, using R x T = W tables
When you see that the number of parents who volunteered for refreshments was 1.5 times the number of parents who neither volunteered for refreshments or supervision
Need to test logically if it makes sense Equation sounds like x = 1.5y, but in reality
If a number is a perfect square, how many distinct factors does it have? If a number is a perfect square, sum of distinct factors is?
Odd number Odd number (1, 2, 4); (1, 3, 9), (1, 2, 4, 8, 16)
Which of the following expressions CANNOT have a negative value? |a + b| - |a - b| |a + b| - |a| |2a + b| - |a + b| a^2 + b^2 - 2|ab| |a^3 + b^3| - a - b
One possible way: if the answer consists of one expression minus another expression; so if there is a SUBTRACTION SIGN then the equation after that needs to be larger Key is to minimize left side, while maximizing right side a) if a = 2 and b = -3, a-b will be larger b) if a = 4 and b = - 2, a will be larger c) if a = 1 and b = -3 e) if a = 1/2 and b = 1/3 On d) this is a perfect square... can factor into (|a| - |b|)^2... everything inside gets squared
Reiko drove from point A to point B at a constant speed, and then returned to A along the same route at a different constant speed. Did Reiko travel from A to B at a speed greater than 40 miles per hour? (1) Reiko's average speed for the entire round trip, excluding the time spent at point B, was 80 miles per hour. (2) It took Reiko 20 more minutes to drive from A to B than to make the return trip.
Should draw out; if yu are given overall mph for whole trip, then you can almost assume how much time that trip took -- in this scenario the 40mph must be greater or else it's taking more time to drive one-way then the whole trip (1) Suff y the above with real numbers: say the trip is 80 miles long in each direction, so that the round-trip distance is 160 miles. According to this statement, Reiko took (160 miles / 80 miles/hour) = 2 hours to drive the entire round trip. Reiko could not have driven from B to A in zero time, so it must have taken her less than 2 hours to drive from A to B. Therefore, her speed on the trip from A to B must have been (80 miles)/(LESS than 2 hours) = (40 miles)/(LESS than 1 hour) = GREATER than 40 miles per hour. (2) N.S.
If you see a smaller triangle within a triangle, what are things you know and what do you need to figure out the parts?
They share a common angle Need to know if bases are parallel --> if they are parallel, then you have another pair of similar angles which means you have similar triangles
A new tower has just been built at the Verbico military hospital; the number of beds available for patients at the hospital is now 3 times the number available before the new tower was built.
Tricky to interpret, easy to assume 15 original beds and 45 new beds (WRONG b/c old beds not going away) 15 original beds and then add 30 new beds = 45 total beds 45/15 = 3 times more
The Carson family will purchase three used cars. There are two models of cars available, Model A and Model B, each of which is available in four colors: blue, black, red, and green. How many different combinations of three cars can the Carsons select if all the cars are to be different colors?
Use slot method _____ ____ _____ So you have 8 choices for the first slot, then 6 for the next and 4 for the last one 8*6*4 = 192, BUT you can double count (order doesn't matter -- b/c you can get cars in different slots and it will be the same -- A Blue, A Black, A Red is the same as A Black, A Red, A Blue) so need to divide by 3! 8*6*4 / 3! = 32
Keats Library purchases a number of new books, all in the category of biography; the library does not acquire any other books. With the addition of the new biographies, the biography collection of the library amounts to 37.5% of the new total number of books in the library. If prior to the purchase, only 20% of the books in Keats Library were biographies, by what percent has the number of biographies in the library increased?
Use smart numbers, where x = number of books added (20 + x) / (100 + x) = 3 / 8 x = 28 140% increase from before
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?
Want to consider the restriction separately so in driver seat only 2 options, 4 for the front passenger seat, 3, 2, 1 for the back So overall 48 options and need to subtract by number of arrangements where two sisters sitting next to each other consider girls as a unit -- 4 different ways for them to be adjacent, so question is how many different ways can you fill the rest of the car 2 people driving car * 2 in the front passenger seat so: (2*2) * 4 = 16 = front * back 48 - 16 = 32
Point Q lies in the interior of a particular circle. Is point Q the center of the circle? (1) There exist points A, B, and C, all distinct points on the circle's circumference, such that the distances QA, QB, and QC are identical.
Yes
Point Q lies in the interior of a particular circle. Is point Q the center of the circle? (2) At least two different diameters of the circle contain point Q.
Yes, All diameters of a circle intersect at the circle's center and nowhere else. Therefore, if Q is the point of intersection of two or more of the circle's diameters, then Q must be the center of the circle. The statement is sufficient.
*For every positive even integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is "(A) between 2 and 10 "(B) between 10 and 20 "(C) between 20 and 30 "(D) between 30 and 40 "(E) greater than 40"
h(n) = 2*4*6*8... tec. can also be written h(n) = 2(1)2(2)2(3)2(4)2(5)...2(50) If any given prime is a factor of h(100), then that prime cannot be a factor of h(100) + 1, because adding 1 isn't enough. For example, if h(100) has a factor of 5, then h(100) + 5 has a factor of 5, but h(100) + 1 does not. And so on. h(100) includes 2(47) and if there was h(100) + 47, then it would be divisible by 47... but if you add 1, then you would have to add on 46 more All the primes up to 50 are covered, so p has to be something greater than 50. If you want to know, 53 is the first prime number after 50, so the smallest possible value for p is 53. But this doesn't mean that p is 53! In fact, it'd be pretty impossible to figure out what p is without a calculator or computer program and in only 2 minutes (not supposed to figure out answer)
For consecutive integers, the product of k integers is always divisible by...
k factorial (k!) -- product of three consecutive integers is divisible by 3!, 4 by 4!
If x is a multiple of 4 and y is a multiple of 6 then, what are the multiples for xy
largest multiple = 24; 2 * 2 * 2 * 3 = no overlap b/c multiplying together
A driver paid n dollars for auto insurance for the year 1997. This annual premium was raised by p percent for the year 1998; for each of the years 1999 and 2000, the premium was decreased by 1/6 from the previous year's figure. If the driver's insurance premium for the year 2000 was again n dollars, what is the value of p?
n * (1+p) * (1-1/6) * (1-1/6) = n n * (1+p) * 25/36 = n 1+p = 36/25 p = 11/25 Decreased b y 1/6 = 1 - 1/6
If a and n are integers and n > 1, is n odd? (1) a^(n-1) > a^n (2) a^n > a^(3n)
n>1 is crucial information and ASSUMING a and n are integers is important (why it's different from 50 where you can test 0<x<1) (1) How can a^ (n-1) > a^n? because of n, n-1 (one must be odd and one must be even) Tricky part is that if a is negative, one number will be positive and the other negative If a is negative, when n is odd, statement is ALWAYS true (if n is positive, then statement is false) (2) How can a^n > a^(3n)? This statement indicates that, when a is raised to a smaller power (n), the number becomes larger. What kind of numbers would make this true? a must be negative Therefore, if n is even, then statement is false; if n is ODD, then statement is always TRUE b/c n = odd and 3 * n = O * O = O Answer: B DONT MIX UP -2^3 = -8 and -2^-3 = -1/8
If the numerator is odd, then is answer E or O? If the numerator is even, is answer E or O?
numerator odd --> answer is odd numerator even --> answer can be either
Rectangle inscribed in a circle -- what is radius circle?
radius of circle equal to half the diagonal diameter equal to full diagonal, so can use pythagorean theorem to solve
Area of equaliteral triangle with side s
s^2 *sqrt (3) / 4
sqrt(a) * sqrt (b) = ? sqrt (5a) * sqrt (7a) sqrt(7+ sqrt(29))*sqrt (7-sqrt(29)) sqrt 16 / sqrt 4
sqrt (ab) sqrt (35a^2) sqrt((7+sqrt(29))*(7-sqrt(29)) - bring under one sqrt Can combine = sqrt (16/4) = 2
If x^2 < x and x is written as a terminating decimal, does x have a nonzero hundredths digit? (1) 10x is not an integer. (2) 100x is an integer.
x>x^2 --> x is 0<x<1 (1) 10 * .01 = .1 --> YES 10* .305 --> NO -- KEY, MISSED THIS PART N.S (2) 100 * .15 = 15 --> YES 100 * .10 = 10 --> NO (1 + 2) eliminates 10 * .305 as a potential b/c 100 * .305 is not an integer (goes against 2); eliminates 100 * .1 as a potential b/c 10 * .1 is an integer (goes against 1)
y = 2z or y = 3z -- what conclusions can we draw
y = 2z --> y is always even y = 3z --> if z is O then y is O, if z is E then y 1is E
x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT x = w x > w x/y is an integer w/z is an integer x/z is an integer
y = 2z is crucial --> means that z can be O or E but y is always E a) 1+2+3+4+5+6 = 6+7+8 --> y = 6 and z = 3 (this combination) b) 1+2+3+4 > 1+2 c) the sum, x, of y (an even integer), is NEVER an integer d) If z is odd, then w/z will be an integer e) Example: y = 4 = 2(2) = 2z where x = 1+2+3+4 = 10 / 2 = 5
If the original price of an item in a retail store is marked up by m percent and the resulting price is then discounted by d percent, where m and d are integers between 0 and 100, is the item's final price (after both changes) greater than its original price? (1) m > d
(1) INSUFFICIENT: Considering extreme cases, if m = 99 and d = 1, that's a 99% markup followed by a 1% markdown. If the original price is $100, the price after the markup is $199, and the value of the discount is only $1.99. In this case, the final price is much greater than the original, so this is a YES answer to the question. Can we find a NO answer? Considering the other extreme of d (large values), let m = 99 and d = 90. (We could investigate d up to 98 if necessary, but it's best to stick to numbers that provide easy calculation.) With these values, the price after the markup is $199. The discount is 90% of $199, which is very close to 90% of 200 (= $180). Therefore, the price will be marked down by almost $180, substantially less than the original $100 price. This is a NO answer to the question.
A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week? (1) Last week the number of crates of oranges that the grower shipped was 20 more than twice the number of crates of grapefruit shipped (2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped
(1) O = 20 + 2G (2) 15O + 18G = 38700 (need to put the dollar amount in front of quantity when solving for total dollars earned)
If an ≠ 0 and n is a positive integer, is n odd? (1) a^n + a^(n + 1) < 0 (2) a is an integer.
(1) if a is -2 then n has to be even; if a is - 1/2, n is odd (2) tells us nothing about n (1 + 2) a cannot be - 1/2 so a has to be -2 and therefore n is even
-2^n = ? (-2)^n=? If n = 0
-2^0 = -1 (-2)^0 = 1
Surface area of a cylinder
2*pi*r^2 +2*pi*r*h two circles + rectangle
How do you get from 2^x to 10^y... what comparison?
2^10 = 1024 = ~10^3
Prime factorization for a prefect square contains...?
An even power of primes (2^2*3^2*5^4) Prime factorization containing an odd number of primes is not a perfect square (Can explain cubes in the same way - i.e., all powers will be multiple of 3)
If ab ≠ 0, does a = b? (1) x^a = x^b (2) x = x^2
Can't determine b/c a = b if x is not 0 or 1 but if x = 0 or 1, then a and b can be anything 2nd statement true if x = 0 or 1
How is simple interest calculated?
Final value = P * (1+rt) Total Interest = P*rt
If x is a positive integer greater than 1, is x! + x + 1 a prime number?
If x is odd, then x! + x + 1 is never prime 3! + 3 + 1 - have a 2 within 3! to divide both numbers 7! + 7 + 1 - have 4 and 2 in 7! to divide both numbers But can be odd if 2! + 2 + 1; 4! + 4 + 1
Inscribed angle - ratio to central angle
Inscribed angle * 2 = Central angle
What is greater / less than -1/100? Less than -10?
Less than -1/100: - 1/50, -1/25 (further from 0) Greater than -1/100: -1/101, -1/200, -1/400 (closer to 0) Less than - 10: -11, -13, -30 Graph out number line (-1/100 is closer to 0 than -1/25, so closer)
What is the mean / median of evenly spaced sets?
Mean and median are equal to average of (first + last) / 2 e.g., average of 200 to 400 inclusive is 300
What are the limitations of sqrt x?
No negative x
What does n and n-1 indicate?
One is odd and one is even
What is the relation between SD and variance?
SD = sqrt (variance)
What is the area of a hexagon?
Six equilateral triangles - can find if you know a side length
If you see problem with letters in the answer, what does that indicate?
Smart numbers!
What do we know if the sum of integers is odd?
The mean cannot be even (divide something by odd is odd or decimal)
4^x + 4^-x = 2
To get reciprocals to equal 2, both must be 1, so x = 0
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y? 5 5(x - y) 20x 20y 35x
a) 35x / 5 = 7x and 20y / 5 = 4y when x = 1 and y = 1 b) 35x / 5(x-y) = 7x / (x - y) and 20y / 5(x-y) = 4y / (x-y) x = 3, y = 2 c) 35x / 20x = 7/4 -- NOT a divisor d) 35x / 20y, 20y / 20y = 1 so x = 4 , y = 1 e) 35x / 35x = 1 20y / 35x, y = 7 x = 1
On rate or work problems what do I need to do
keep rate, variable, question straight
x + y = xy, what is x and y?
x = 2, y = 2
4 ^ 3.5
4^3 * 4^.5 = 128
If a point anwhere on the circumference of the circle is drawn to center, what must this be?
Radius
A circle has a triangle inside of it If the probability of landing in an area outside of the triangle but inside the circle is 3/4... what is the ratio of the areas
So ratio of outside of triangle but in circle to triangle is 3 : 1 Ratio of circle to triangle is 4 : 1
When a is raised to a smaller power (n - 1), the number becomes larger. What kind of numbers would make this true? a^(n-1) > a^n
a < 0 or 0<a<1 When raising a negative number to a power, two different things will happen: if the power is odd, the resulting number will be negative and if the power is even, the resulting number will be positive.
|a| + |b| = a + b
a and b must be positive
What are key factors when testing x?
could test if x = 0 or 1 0>x>1 x>1 or -1<x<0 or x<-1
x > x^2
x is 0>x>1
x>x^3
x is 0>x>1 x<-1 NOT -1<x<0 -1/2 is not greater than -1/8
Diagonal of cube with side x Diagonal of square
x*sqrt(3) x*sqrt(2)
What is the procedure for finding SD?
The procedure for finding the standard deviation for a set is as follows: 1) Find the difference between each term in the set and the mean of the set. 2) Average the squared "differences." 3) Take the square root of that average.
If x is a prime number, what is the value of x? (2) The average of any x consecutive integers is an integer.
X must be odd Then there are an odd number of members in a consecutive set, the mean/median will be a member of the set and thus an integer (e.g. 5,6,7,8,9; mean/median = 7). In contrast when there are an even number of members in the set, the mean/median will NOT be a member of the set and thus NOT an integer (e.g. 5,6,7,8; mean/median = 6.5).
If i have the mean, median and SD for a set of numbers and I added 5 to each, what changes
mean and median
sqrt (xy) = xy, what is value of x + y?
xy = (xy)^2 xy = 0 or 1 x + y = 2 or 0 so if x = 2 then y = 1/2 or if x = -3 then y = -1/3
If Q is a set of consecutive integers, what is the standard deviation of Q? (1) Set Q contains 21 terms. (2) The median of set Q is 20.
(1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x: [ x - 10, x - 9, x - 8, x - 7, x - 6, x - 5, x - 4, x - 3, x - 2, x - 1, x, x + 1, x + 2, x + 3, x + 4, x + 5, x + 6, x + 7, x + 8, x + 9, x + 10] (2) NOT SUFFICIENT: Since the set is consecutive, we know that the median is equal to the mean. Thus, we know that the mean is 20. However, we do not know how big the set is so we cannot identify the difference between each term and the mean.
Shortest distance from a vertex of the cube to the sphere would be
(Diagonal - Diameter) / 2 or diagonal /2 - radius
If a cone is inscribed in a hempishere, what is ratio of radius to height of cone?
1:1
4^17 - 2^28 -- what is the greatest prime factor?
2^(2 * 17) - 2^28 2^28*(2^6-1) 2^28(2^3+1)(2^3-1) 2^28(9*7) 2^28(3*3*7) PRIME FACTOR -- SO PRIME FACTORIZATION (if number isn't there, then you're ok)
Is x > 10^10 ? (1) x > 2^34 (2) x = 2^35
2^10 = 1024 = ~10^3 2^30 = (10^3*10^3*10^3)= 10^9 2^4 = 16 2^34 = 2^30*2^4 = 10^10 * 1.6 so (1) is sufficient If (1) is true, then (2) is true
Something that is 2x larger is a what % increase 3x? 5x?
2x = 100% increase 3x = 200% incrase 5x = 400% increase % INCREASE IS DIFF THAN HOW MANY TIMES LARGER
How do you mathematically write out summation of consecutive integers
3 consec = n + n+1 + n+2 = 3n+3 5 consec = n +n+1 + n+2 + n+3 + n+4 = 5n + 10
A retail item is offered at a discount of p percent (where p > 10), with a 5% state sales tax assessed on the discounted purchase price. If the state sales tax were not assessed, what percent discount from the item's original retail price, in terms of p, would result in the same final price?
A * (1 - .1) * (1 + .05) = A * (1 - d)
Inscribed triangle inside a circle with diameter as a side
Angle facing diameter is the right triangle
If n is a non-negative integer such that 12^n is a divisor of 3,176,793, what is the value of n^12 - 12^n ? What are the divisibility rules for E and O numbers
Because 12 is an even number, it cannot divide an odd number (3176793), except if n = 0 O must be divide by O to get an answer with no remainder (but even can be divided be O or E)
List P contains m numbers; list Q contains n numbers. If the two lists are combined to produce list R, containing m + n numbers, is the median of list R greater than the median of list P ? What kind of possible sets can you have?
Before looking at the statements, it is important to remember that nothing in the question assumes that the numbers in either list must all be different. For instance, if list P contains 3 numbers, {1, 2, 3} is one possible set, as is {1, 1, 1} (DOES NOT MENTION DISTINCT NUMBERS).
Area of rhombus
Diagonal 1 * Diagonal 2 / 2
Key features of rhombus:
Diagonals bisect angle Diagonals are perpendicular Diagonals bisect each other evenly
What is diameter of a sphere inscribed in a cube?
Diameter = side of cube The sides of a sphere inscribed in a cube will touch the sides of the cube. Therefore, a sphere inscribed in a cube will have a radius equal to half the length of the side of that cube.
How many total factors for 252?
Do prime factorization: 2^2*3^2*7 # of factors = (2+1) * (2+1) * (1+1) Rule is (exponent of base +1)*(exponent of base+1)*etc.
E + O = ? O + O = ? E + E = ? E - O = ? O - O = ? E - E = ? E * O = ? O * O = ? E * E = ?
E + O = O O + O = E E + E = E E - O = O O - O = E E - E = E E * O = E O * O = O E * E = E
1 / (a+b) < 1/a + 1/b
If a and b are positive, then true Need to test a or b as > 1 as well as 0<(a or b - decimal) <1 If negative, then generally won't be true
ab^x if a is even and b is odd, what is x?
If a is even, the solution will be even b/c there is an even number, it doesn't matter what power an odd is raised to (it will still be odd)
If a cylinder is half full, what does this mean?
If it's upright or if it's on its side, then it will still be half full And when it's on its side, the height will be the radius of the circle
Sum of n consecutive numbers is even?
If n is even, n is a multiple of 4 O+E+O+E = E O+E+O+E+O+E+O+E=E n is odd, first number is odd n+(n+1)+(n+2) = O + E + O = E
What is less than 1/100? What is greater?
Less than: 1/101, 1/200, 2/500 Greater than: 1/50, 1/3
If the square root of p^2 is an integer greater than 1, which of the following must be true? I. p^2 has an odd number of positive factors II. p^2 can be expressed as the product of an even number of positive prime factors III. p has an even number of positive factors
Means that p^2 is a perfect square (I) Yes, perfect square has odd number of positive factors (not prime) (II) Yes b/c you need to have two of a prime factor to get a square (3*3 * 2*2) = 6^2 (III) Not always, true if p is not a perfect square, but if p is a perfect square (9^2), then p has an odd number of positive factors
Sum of consecutive integers: For any consecutive integer w/ an odd # of items, the sum of all the integers is always...? And for even consecutive integers?
Multiple of the number of items NEVER a multiple of the items
A list contains n distinct integers. Are all n integers consecutive? (1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed. (2) The positive difference between any two numbers in the list is always less than n.
See it's an INTEGER (1) Yes (2,3,4,5); mean = 3 or 4 with first or last number removed; (2) Yes (2,3,4,5) (n = 4, difference of 3); (4,5,6,7,8), n = 5, difference of 4)
If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the value of x? (1) z is prime (2) x is prime
Simplify to get 5^2 * z = 3 * x ^ y (1) SUFFICIENT: 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. Statement (1) says that z is prime, SO Z CANNOT HAVE ANOTHER FACTOR BESIDES THE 3 Since z = 3, the left side of the equation is 75, so xy = 25. The only integers greater than 1 that satisfy this equation are x = 5 and y = 2, so statement (1) is sufficient. Put differently, the expression xy must provide the two fives that we have on the left side of the equation. The only way to get two fives if x and y are integers greater than 1 is if x = 5 and y = 2. (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.
How are diagrams drawn on the quant section?
The diagram is drawn to scale (problem solving questions are drawn to scale unless otherwise stated, data sufficiency questions are NOT)
What is the median value of the set R, if for every term in the set, Rn = Rn-1 + 3? (1) The first term of set R is 15. (2) The mean of set R is 36.
The set Rn = Rn-1 + 3 describes an evenly spaced set: each value is three more than the previous. For any evenly spaced set, the mean of the set is always equal to the median. A set of consecutive integers is an example of an evenly spaced set. (30,33,36,39,42); (24,27,30,33,36,39,42,45,48)
Directly proportional to m^3 and inversely proportional to q^2 -- what is the equation? What is inversely proportional?
m^3 / q^2 m^3 is directly proportional 1 / q^2 or 1 / m^3 is inversely proportional
Sum of n consecutive numbers is odd?
n is ODD and the first number must be Even n+(n+1)+(n+2) = O + E + O = E = E + O + E = O n is EVEn and it's either 2 + 4x (6, 10, 14) n+(n+1) O + E = O n+(n+1)+(n+2)+(n+3)+(n+4)+(n+5) = O+E +O+E+O+E = O
What is an isosceles right triangle
s:s:s*sqrt(2)
