GMAT Math Cheat (Summary) Sheet

Some important topics for you to consider for GMAT:

• Zero is integer
• 1 is NOT prime number
• Zero is even number (divisible by 2)
• For number 246.31:
• 6 is units, 4 is tens, 2 hundreds digit
• 3 is tenths, 1 is hundredths digit
• Prime numbers:
• 2 is only even prime number
• 0 and 1 are NOT prime numbers
• are always positive
• Absolute value: distance between number and 0. e.g.: |-6| = 6
• |x+1| <= 4 is same as -4 <= x+1 <= 4 (and vice versa)
• Convert -5 <= x <= 3 into absolute, 1) find out length between end points, 3 - (-5) = 8, divide equally: 8/2 = 4, then to convert -5 <= x <= 3 to -4 <= x+1 <= 4 by adding 1 to all sides. therefore |x+1| <= 4
• Odds and Evens:
• If at least 1 factor of a product is even, then product is even
• both odd or both even means...both sum and difference are even, otherwise their sums and differences are odd.
• Consec even #s: 2n + 2, 2n + 4...
• Consec odd #s: 2n + 1, 2n + 3
• A number is divisible by:
• 2 if its even
• 3 if sum of its digits is divisible
• 4 if last two digits are divisible by 4
• 5 if last two digits are 0 or 5
• 6 if it is divisible by 2 and 3

• Fraction: part/whole
• 2 boxes contain 14 shirts, how many in 3 boxes?
• 2 (boxes) / 14 shirts = 3 (boxes) / x shirts
• Decimals in fractions:
• 0.4 = ⅖
• 0.6 = ⅗
• 0.66 = ⅔
• 0.8 = ⅘
• 14.3 x 0.232 = 3.3176 (1 decimal place times 3 decimal places = 4 decimal places)
• Rounding: 9.4 to nearest whole # is 9, 9.5 to nearest whole # is 10
• Ratios:
• part A/ part B, part A + part B = whole
• The ratio of men to women in a room is 3 to 4 (NOTE that this ratio can also be expressed as 5: 4 and as the fraction, 5/4)
• The number of men and the number of women in the room are not necessarily 3 and 4, but they must be some multiple of 3 and 4 (e.g. 6 men / 8 women, 9 men / 12 women, etc). This notion can be expressed with an "unknown multiplier." and expressed as "x" in be 3x: 4x or 3x/4x, where x represents the "unknown multiplier.
• Eg: Women to Men ratio in room is 3 to 4, if total is 28, how many women?
• Whole = part A + part B = 3 + 4 = 7, 28/7 = 4 groups (of 7)
• 4 x 3 women = 12 women / 16 men
• Eg 2: Total 300 vehicles, 30% trucks, 20 % diesel including 15 trucks, what % is non truck non diesal?
• Compound Interest:
• 20 % compounded semi-annually means. Divide interest/# of parts. So, 20% interest/ 2 parts = 10% every six months, After six months it is 10% of 100 and after 1 year it is 10% more of 110
• answer will be a “little” more than simple compound problems
• Percent increases: Year 1: $100, Year 2: $155, Percent increase = Y2-Y1/y1 * 100 = 55/100 *100
• Percent decreases: (similar as increase)...see problem #109 on pg 220 of GMAT book
• Average:
• Average = total sum of the items/total number of the items
• The average of 7 numbers is 5. If two numbers are 11 and 14, what is the average of remaining numbers?
• 5 = x/7, x (total sum) = 35
• 11 + 14 = 25, 25 + y = 35, y = 10
• Since there are 5 numbers left, average = total sum/total #s then 10/5= 2
• Weighted average:
• Applies to many ratio, mixture, percentage problems
• E.g: Mixture A is 0.3% alcohol, mixture B is 0.5% alcohol, in a 10 lb mixture of A and B with 0.35% alcohol, how much is the weight of A?
• Allegation or Scale method (worth learning about this, please google it)
• Median:
• SORT Numbers FIRST before choosing
• if n is odd, the number in middle when sorted
• If n is even, the two middle numbers/2
• Mode:
• Number that occurs the most
• Set of numbers can have more than one node
• Range:
• in a set of numbers 4, 3, 8, 12, 23, 37. Range is 37-3= 34
• Sequences and their Sums:

· Sum of all even from 1 - 99 = sum of first 49 evens = n*(n+1) or 49* (50)

1. (n = 98/2 = 49)

· Sum of all odds from 1 - 99 (inclusive) = sum of first 50 odds = n*n = 50*50

· Arithmetic sequence: a = first term, d = difference between terms, x_n = nth term, then x_n = a + (n-1)d, if x_n = 99 for first even #s, then to find n: 98 = 2 + (n-1)2, 98 = 2 + 2n - 2, n = 98/2 = 49.

· Sum of first 49 sequenced #s (same as all even from 1 - 99, but n = 49 NOT 99!!) = n/2 * (2a + (n-1)d),

· for evens Sum = n (n+1)

· For odds = n*n

· Another way to do this problem:

· Sum of all even numbers from 1 to 199

== 2 + 4 +6+ 8 + ... + 198 == 2 (1 + 2 + 3 + ... 99) = 2 * (99 * 100) / 2 = 9900

· Sum of all odd numbers from 100 to 300

== 101 + 103 + ... + 299

· Equally Spaced sets ==> Mean = (101 + 299)/2 = 200

No. of terms = (299 - 101)/2 + 1 = 100

Sum of the series = 200 * 100 = 20000

· My proffered approach (may not be for you):

· we can simply use AP.

· Sum of all evens from 100 to 301 (inclusive) is: a1 = 100, d = 2, x_n = a1 + (n-1)d = 300, therefore n = 101 Then sub into following formula: formula=(n/2)[2a+(n-1)d]=(101/2)[200+200]=20200

· Geometric sequence

o in which each subsequent term is multiplied by a certain constant. Compound interest in an example of a geometric sequence.

o The standard formula is: a_n = a1*r ^ (n-1) where a_n is the nth term, “r” is rate and a1 is the first term.

o when given the rate of increase and the value of any term, we can find any other term. For example:
Joe originally put $8 in stock in 2000. If stock triples every year in this time (without Joe adding anymore himself), how much money will he have in 2011?
r = 3
a1 = 10

a_11 = ?

• Standard Deviation
• The mean and the median are the same in a normal distribution.
• Distance between mean (average) and each of numbers in the set (measure of how spread out numbers are.)
• Stand dev forumal is not needed but number looks like 2.4 or 3.0
• e.g: 14.6 and 9.3 are both more than two standard devs from average 12 (with stand dev at 1.3)
• it is the square root of the Variance. So now you ask, "What is the Variance?"
• The Variance is defined as:
• The average of the squared differences from the Mean.
• Standard deviation and confidence intervals.
• About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 rule, or the empirical rule, or the 3-sigma rule. 50% of the distribution lies within 0.67448 standard deviations of the mean (source: WikiPedia)

• Exponents:
• Any number to power 0 is 1
• X^2 = 100 means x = 10 OR x = -10 DO NOT FORGET the negative!!!!!!
• Radicals:
• 3sqrt5 = 5 to power ⅓ (not 3 times but small 3 above sqrt sign)
• 3sqrt27 = 27 to power 3 = 3.3.3 = 3
• sqrtXsqrtY = sqrt XY
• sqrt (x/y) = sqrt x/ sqrt y
• sqrt 32 = sqrt 16. sqrt 2 = 4 .sqrt 2
• Remainders:
• for int x and y, qutiont q and remainder r.
• y = xq + r, e.g: For x=8, y = 28, 28 = 8 (3) + 4....3 is quotient and 4 is remainder
• 0 <= r < x
• small int / big int, quotiont is always 0 and remainder is small int
• 5 / 7 , quotiont is 0 and remainder is 5 since 5 = 7 (0) + 5
• If for integers a and b, a/b = 74.32, then what is the remainder?
• Solution: => a = 74.32b OR a = 64b + 0.32b, then 0.32b is remainder where b is an int. Also 0.32b must be an int, find a “b” that makes it true from multiple choice options in a GMAT question.

Algebra:

• Plug in
• write down the variable and value you choose
• then for each option see if equation creates same variable
• Eg: if J can drive distance k miles in 50 mins, how many mins, in terms of k, will it take him to driver 10 miles at the same speed
• if k = 5 miles in 50 mins then, 1 mile = 10 mins
• then 10 miles will take 100 mins
• What option given 100 mins when k is 5?
• eg2: if 80% of x is 50% of y and y is 20% of z. What is x in terms of z?
• Plug in variable of question (avoid 0 and 1):
• Variables in answer choices
• percents (of unknown amount) in answer choices
• fractions or rations (of unspecified amount) in answer choices
• Can also Plug in numbers from answers into question to simplify
• good for problems like: “today Joe is twice as old as Frank, Steve is 2 years younger than Frank. Four years ago if Joe was 4 times as old as Steve , how old is Joe now?
• If answers are sorted and question is straight forward, eg; What is x?
• Start with C, if C is small try bigger, if C is big try smaller A store currently charges the same price for each towel that it sells.
• E.g: If the current price of each bottle were to be increased by $1, 10 fewer of the bottles could be bought for $120 (excluding sales tax). What is the current price of each bottle?
• Perfect for "WHAT IS X" type of question and Answer is given, sub into question and then check given conditions are met.
• Must be/could/cannot be problems: almost always plug in
• if x and y are consec + numbers, which following must be an even int?
• An equation with a variable with even power may have more than one solution but if variable has odd power there must be only one solution.
• Inequalities (<, >): if you multiply or divide both sides of inequality by negative number, direction of inequality symbol changes: -2x > 5, then x < -5/2. also
• if r < 1, then 1/r > 1, if r < 100, 1/r >1/100
• Rate x Time = distance
• Work problem: figure out total time per hour first. If sam completes work in 3 hrs and Mark in 12, how long both take together? Answer: ⅓ + 1/12 = 1/x (total work per hour together). 1/x = 5/12, x = 12/5 (total time taken for all work)
• can also substitute actual work. eg: total work = 12 widgets.
• 0! = 1
• Probability:
• P (ABC) = P(A)*P(B)*P(C)
• P (A OR B) = P(A) + P(B)
• P (not A) = 1 - P(A)
• At least 1 thing will happen: P(at least 1) + P(none) = 1 or another example:
• P (at least 2) + P (0 or 1) = 1, so forth...
• Permutations and combinations:
• Combinations: choose a # of items to fill specific spots each from different source.
• Multiply # of choices in each spot
• Permutations:
• Single source, order matters (keep duplicates)
• n! (multiply # of choices for each spot, but # of choices get smaller)
• Single source, order matters but only for selection (keep duplicates)
• 7 teams, permutations for top 3?
• 7!/(7-3)! OR n!/(n-r)!
• Single source, order does not matter (remote duplicates)
• We only care about unique permutations
• #of unique combinations = n!/(n-r)r!
• 6 horses, how many groups could make up first 3 finishers?
• 6!/(6-3)!3! = 120/6 = 20
• It is interesting to also note how this formula is nice and symmetrical:
• In other words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same number of combinations.

Geometry

· Look for drawn to scale to eliminate crazy answers

· Memorize: pie = 3, sqrt 1 = 1, sqrt 2 = 1.4, sqrt 3 = 1.7

· Triangles: one side of triangle can never be longer than sum of lengths of other two sides or less than their difference. Eg: for 6, 8, third side has to be < 14 and > 2 (can not be equal to either)

o Area of triagle = ½ base x height

o Right triangles: 3:4:5, 6:8:10,

o Right isosceles: side:side: side*sqrt2.

§ e.g: 3:3:3sqrt2

o 30:60:90 triangle: side: 2*side: sqrt3*side

§ e.g: 1:2:sqrt3 OR 3:6:3*sqrt3

o 45:45:90 triangle: side: 1:1:sqrt2, eg: 4:4:4*sqrt2

· Sum of interior angles of any polygon is 180(n-2), where n is number of sides.

· Circles:

o Area: pieR*R

o Circumfurence: 2pieR

· Parallelogram:

o Area: base*height

o Diagonals of parallelogram bisect each other (i.e. divide each other in halves)

· Quadrilateral:

o Area of quadrilater with two sides parellal (trapezoid) = ½ (sum of bases)*height

· Volume of rectangular solid: Area of base * depth (or height)

o (length * width * depth)

· Volume of cylinder: Area of circular base * depth

o pie*R*R * depth

· Surface area: sum of all sides’ areas

· Volume of sphere: 4/3 pie * R*R*R

· Any straight line equation: y = mx + b, where b is y intercept and m is slope of line

o Slope (m) = difference in y coords/difference in x coord

= y1-y2/x2-x1 (remember that numerator is Y coords)

General Strategies

1. Where you have to try every answer choice. HOWEVER, when the only way to get the answer is to try every answer choice, ALWAYS start from E and work up.
More often than not the answer appears near the bottom and the test writers want you to waste time.

2. Don’t miss key words in problems. READ EVERY PROBLEM CAREFULLY! ( first 10-25 secs should be focused on problem alone. I noticed better results after training myself to do this)

3. Questions with variables in the question. Use PLUG IN or algebra (find your preferred approach)

4. Practice as many questions as you can. I did probably over practice 1000 questions in my study time.

5. Track performance on topics you are weak in and work on those.

Important Topics that helped me improve my score (may be different for you):

1. Number properties
2. Sequences (eg: ½ -1/4 + ⅛ - 1/16.... = ?)
3. Ratios (eg. 31/37 on GMAT test 1)
4. Remainders
5. Permutations and Combinations (they show up once in a while but not too much)
6. Some geometry
7. Probability
8. Work/rate problems
9. Medians problems
10. Percent increase/decreases
11. Weighted average questions
12. LCM and GCF (Least common multiple and greatest common factor)

Disclaimer: Every person is different in their needs for GMAT. This review sheet does not guarantee an improved score for anyone. I advise you create your own summary sheet.