GRE Math Meh Problems
Sums of Sequences : Hard What is the sum of all integers from 45 to 155 inclusive?
# integers in sequence sum of arithmetic sequences n ( n + 1 ) sum = -------------- 2 n = 111 inclusive therefore 155 - 45 + 1 = 111 111 ( 45 + 155 ) / 2 = 11, 100
FCP w/ Restriction note: it is faster to use the "complement" to solve these problems than to count the arrangements directly. Count : TOTAL arrangements - # arrangements we DON'T WANT
# w/o restrictions 5 ! = 120 # w/ A & B together (AB) C D E 4! = 24 x 2 for (BA) = 48 120 - 48 = 72
Even + Odd Integers - REVIEW The integers P, Q, and R are all positive odd integers. Which of the following also must be odd?
- an odd number raised to ANY power is odd.
Motion Problems [when distance is the same] Andy drove from Townville to Villagton at an average speed of 40 mph. He then drove the same route back from Villageton to Townville at an average speed of 60 mph. column A: 50 column b: the average speed of Andy's entire trip.
- remember if the distance is the same then the TOTAL distance = 2D - also you can make up a number for D. Use that if nothing else works.
Algebra Problems
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Algebra Problems : Hard
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Counting Problems : Hard
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Counting Problems : Medium
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Counting Problems: Very Hard
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Geometry Things to Know
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Powers & Roots
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Probability : Very Hard
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Word Problems : Hard
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Word Problems : Medium
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Integer Problems
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Mixture Questions : Hard Solution Y is 40 percent sugar by volume, and solution X is 20 percent sugar by volume. How many gallons of solution X must be added to 150 gallons of solution Y to create a solution that is 25 percent sugar by volume?
1. EQ: solute amount [ concentration ] = -------------------- x 100 TOTAL amount of solution 150 x 0.4 = 60 60 + 0.2x 0.25 = ------------ 150 + x x = 450
Working with Percents / Assigning Variables Whenever Art Dealer sells a sculpture, he earns a 20 percent commission on the first $12,000 of the sale price plus 15 percent of the sale price in excess of $12,000. If Art earned a $3,900 commission on the sale of a certain sculpture, what was the sale price?
3,900 = (0.2 x 12,000) + 0.15(x-12,000) A : x = 22,000
LCM Word Problem / Assigning Variables A large aquarium contains only two kinds of fish, guppies and swordtails. If 3/4 of the number of guppies is equal to 2/3of the number of swordtails, then what fraction of fish in this aquarium are guppies?
A: 8/17
Percent Increases and Decreases Cam is 20 percent taller than Bea, and Bea is 20 percent taller than Ann. Column A : C - B height Column B: B - A height
Ann = x Bea = 1.2x Cam = 1.44x Answer: Column A
GCM & LCD Rule : Powers & Roots : Hard
If 724 is the greatest common divisor of positive integers A and B, and 726 is the least common multiple of A and B, then AB=
Generalized OR Rule P ( A or B ) = P(A) + P(B) - P(A and B) note using because: the events are NOT mutually exclusive there is overlap. Among all the students at a certain high school, the probability of picking a left-handed student is (1/4), and the probability of picking a student who is learning Spanish is (2/3). Which of the following could be the probability of picking a student who is either left-handed or learning Spanish or both? Indicate all such numbers.
NOTICE HERE that it is asking for A OR B and also overlap. Since it already asking for A or B, the overlap is 0. The maximum is P (A ) x P ( B ) = 11/12 and all numbers less than 11/12 is an answer.
Simple AND Rule Events A and B are independent. The probability that events A and B both occur is 0.6. column A: probability that event A occurs column b: 0.3
P ( A & B ) = P (A) x P (B) 0.6 = A x 0.3 here A = 2 we can't have a p = 2 THEREFORE P (A) = 0.6 or greater and P (B) = 0.6 or greater
What is the Greatest Common Factor (GCF) of 18x^8y^20 and 24x^12y^15? REVIEW remember: the GCF between 2 algebraic terms is just the one with the lowest power!
THEREFORE GCF here is ... 6x^8y^15
Age Problems Seven years ago Bob was k times as old as Ann. If Ann is now 11 years old, what is Bob's present age in terms of k?
answer: 4k + 7
Parallel Lines + Angles
can make 75 degree assumption because of parallel lines
Case III & II : Exponential Growth : Hard -1 < x < 0 y > 1 column a: x^y column b: y^x
case III : base between -1 & 0 x^y because x is (-) x^y will be negative whenever y is an odd number, positive when even. case II: positive base, negative exponent y^z must be positive because a positive number raised to any power remains positive. answer: the relationship cannot be determined from the information given.
Circle Things to Know
circumference = 2πr area = πr^2 notice : area is in terms of radius , we can also express circumference in terms of radius SO if we know ' r ' , we can find out all its other values. ~ always find r ~ central angle properties measure of the central angle equals the measure of the arc. inscribed angle properties the measure of the inscribed angle is 1/2 the measure of the arc it intercepts. ° inscribed angles & semicircles hint : the test loves this fact any inscribed angle that intercepts a semicircle has to be a right angle. ° if 2 inscribed angles in the same circle intercept the same arc or the same chord, then the 2 inscribed angles are equal. ---> reference photo arcs & sectors set up a part to whole proportion arc length angle ------------ = ----------- 2πr 360 °
Geometry Volume Things to Know
cube v = s ^ 3 SA = 6s ^2 space diagonal s = l , w, h (AB^2) = s^2 + s^2 + s^2 = 3s^2 AB = √3s note: the test LOVES this fact face diagonal pythagoreans theorem cylinder volume: πr^2 x h surface area: 2 ( π x r ^2) + π x d x h think: paper rolled around can, If we unroll this we get a flat rectangle. the top edge of this rectangle the l ) = wrapping around top of circle therefore it is the circumference 2πr lateral area = 2πrh total area = 2πr^2 + 2πrh
large numbers raised to large numbers : powers & exponents column a: 90^800 column b: 8000^400
easy to compare when large bases have the same exponent
Percent Increase + Decrease $ : Hard After her birthday, Fiona had D dollars in gift money. She spent 3/5 of this on an electric skateboard. The next day, she spent 1/3 of what was left on a movie passcard, and then finally put the last $80 in the bank. If she made no other purchases with that money, then what was the value of D?
focus : writing initial equations correctly
Percent Increase + Decrease $ : Hard The sum of the pre-tax costs of Item A and Item B is $300. In Alumba, each item would be charged a flat 7%. In Aplandia, Item A is subject to 5% tax and Item B is subject to 10% tax. If the tax in Aplandia on the purchase of both items is exactly $3 more than it is in Alumba, then what is the pre-tax price of Item A?
focus : writing initial equations correctly
sums while working with means equation PQ: If the average (arithmetic mean) of x, y and 15 is 9, and the average of x, 2y and 2 is 7, then y =
if m is the mean of n numbers, then the sum of the numbers is nm.
changing mean + median
if m is the mean of n numbers, then the sum of the numbers is nm. fr
Integers Things to Know
integers & squaring only the squares of integers have an odd number of positive divisors. squares of PRIME INTEGERS have exactly 3 positive divisors.
Units Digits Question : Powers & Roots : Hard
lastly 9 / 5 = 4
Part to Whole Proportions for Length of Sector Arc --> Area of Sector
length of arc proportion 15π 135 (3) ---- = ---- 2πr 360 (8) r = 20 sector area 3 ------------- = ------- πr^2 = 400π 8 area of sector = 150π
Polygons, Sum of Angles
n sided polygon = (n-2) 180° = total degrees
Triangle Things to Know
pythagorean triplets ~ be aware of multiples! 3, 4, 5 5, 12, 18 8, 15, 17 special right triangles 45, ,45, 90 s, s, s√2 30, 60, 90 s, 2s, s√3 area of triangle a = (1/2) b h area of equilateral triangle (REVIEW PQ) (√3/4) (s)^2 = area equilateral triangle
Recursive Sequences s1, s2, s3, s4, s5, .... In the sequence above, each term after the first term is equal to the preceding term divided by a positive number p, such that p > 1. If s3 = 24 and s5 = 6, which of the following is the value of s8?
s4 = 24 from that we can get the expression for the next term ... s5 = (24/p) / p = 24 / p^2 = 6 SO 24 = 6p^2 2 = p a3 = 24 a4 = 12 a5 = 6 a6 = 3 a7 = 1.5 a8 = 0.75
Motion Problem 2 : Hard While driving from A-ville to B-town, Harriet drove at a constant speed of 115 kilometers per hour. Upon arriving in B-town, Harriet immediately turned and drove back to A-ville at a constant speed of 135 kilometers per hour. If the entire trip took 5 hours, how many minutes did it take Harriet to drive from A-ville to B-town?
set up 2 D = RT , equations
Trapezoid Things to Know
supplementary 2 angles on a leg are supplementary isosceles trapezoids a trapezoid with 2 equal legs diagonals have equal lengths trapezoid area (b1 + b2) A = ------------ x h 2 think: trapezoid has 2 bases (2 parallel sides) so we must find the average of the bases.
Average Speed : Hard Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove the same route back from Villageton to Townville at an average speed of 60 miles per hour. column A : 50 column B : the average speed of Andy's entire trip in mph
think ... 1. need to find total distance & total time total distance ave speed = -------------------- total time 2. find ind. D = RT 3. combine remember to use 2D if going back and forth across same distance. t = d/40 t2 = d/60 LCD : 120
Shrinking & Expanding Gaps : Hard Car X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y, but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm, Car X reaches B, and at that moment, Car Y is still 35 miles away from B. What is the speed of Car X?
think ... 1. shrinking (-) , expanding (+) 2. rate at which expanding / shrinking - find gap time - gap distance 3. find missing variable question is asking for
Geometric Sequence : Power & Roots : Hard In a certain sequence of all positive terms, {a1, a2, a3, ...} each term equals the previous term times a constant factor. If (a1)(a5) = 900, what is the value of a3?
tips 1. write out pattern for first ~ 5 terms defined with unknown variables THEN 2. write out overarching pattern do not try to identify pattern w/o writing out trend first.
Quadratic Equations & Roots
x = - 6