GMAT lessons (arithmetic)
What is the sum of consecutive odd integers?
(2X-1) + (2X+1) = 4X. Must be a multiple of 4
The number of consecutive even or odd numbers is?
(L-F)/2 + 1 OR range/2 + 1
How to find number of distinct factors
(P+1)(q+1)(r+1) where p,q,r are the exponents of the prime factorization
What is a multiple of all integers
0
The first 10 non negative integers are
0,1,2,...8,9
How many solutions does a cube root have?
1
What is a factor of all integers?
1
Probability of an event not occuring
1 - probability that the event occurs
millions
1,000,000 or 10^6
The first 10 positive integers are
1,2,3...9,10
How to calculate variance
1. Subtract each score from the mean 2. Square every deviation of the mean 3. Add them all up 4. Divide by how many there is Take the square root of this to calculate standard deviation
√2
1.4
√3
1.7
hundredth digit
10^-2
millionths
10^-6
1/9 is what percent
11%
1/8 is what percent
12.5%
5^3
125
1/7 is what percent
14%
1/6
16 2/3%
13^2
169
14^2
196
How many solutions does a square root have?
2
The second 10 positive integers are?
2,3,4...10,11
√5
2.2
6^3
216
16^2
256
17^2
289
consecutive odd integers
2X-1, 2X+1
consecutive even integers
2X-2, 2X, 2X +2
18^2
324
7^3
343
19^2
361
8^3
512
25^2
625
4^3
64
9^3
729
Intersection (A and B)
A set of all elements that are both in Set A and Set B
prime number
A whole number that has exactly two factors, 1 and itself.
The ratio of males to females is 2 to 3. If the number of males is 10, what is the number of females?
A:B = C:D is equal to AD = BC. So 2:3 = 10:F —> 2F = 30. F =15
Or
Addition
Percent increase or decrease
After - before/ before times 100
If the range is zero
All elements have the same value, standard deviation is 0. Average = median
How to find the nth term of an arithmetic sequence
An = A1 + (n-1)d
How to find the nth term of a geometric sequence
An = A1*R^(n-1)
permutation
An arrangement of objects in which order is important Look for words such as "first" "second" or instanced where they need to pick a president and Vice President
Median
Another type of center for a list of numbers. Order the numbers in least to greatest and find the middle
Transforming fractions to decimals
Count the number or zeros. Example 15/10^7. This has 7 zeros. Write out 0.0000000 and then cross out the last 2 zeros and replace with 15. Answer is 0.0000015. Number of zeros equals the power of 10
The union of A and B (number of elements)
Elements in A + elements in B - the intersection of A and B
If one factor of a number is even then the number is
Even
mutually exclusive
Events that cannot occur at the same time Intersection of two events = 0. Think flipping of one coin, can either be heads or tails
Greatest number of prime factors of a number
Find the prime factorization and count how many unique numbers there are disregarding exponents.
Range
Greatest value - least value
What is the range of the first 10 consecutive even numbers.
L-F/2 + 1 = 10. Range/2 + 1 = 10 Range = 18
How to divide decimals
Make divisor into an integer
|x+y| is always ≤ |x| + |y|
Memorize this. Works with all numbers
Of
Multiply ( write X)
Process to divide fractions
Multiply by reciprocal or later number
If A<B is A^2<B^2
NO, in the case where A is negative and b is positive it's not always true. However if A<B<0 is true, then A^2<B^2 is true
Is 1 a prime number?
No, it does not have indistinct factors.
Standard deviation of 1,2,3 > standard deviation of 2,3,4?
No, parralel movement of data, when you add same amount to each element, does not affect standard deviation. Standard deviation is the separation of numbers. In the example all the elements are 1 away from eachother
Can you simplify AB = 1 to A= 1/B?
No, you cannot divide by variable in case it's equal to 0
geometric average
Nth root of the product. EX: geometric average of 2,9and 12 equal to cube root of (2X9X12) Geometric average a,b = root(ab)
irrational numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.
Is X < X^2< X^3
Only if X>1; if 0<X<1, the opposite is true
Is the percent increase/decrease equal to the percent change?
Only in the case of percent increase. For percent decrease, percent decrease has an opposite sign as percent change. EX: the percent change is -20%. The percent decrease is 20%
Probability of E or F
Probability of E + probability of F - probability of both e and f (intersection) Otherwise known as the union of e and f
Simplify A/BC = AC/B
Reduce by A/B (since it's on both sides of the equation). Left with 1/C = C. C must be 1
Union (A and B)
Set of all elements that are in either A or B
sum of a geometric sequence
Sn=a1(1-r^n)/1-r
arithmetic average
Sum of n divided by n. Also called mean Average of a,b = (a+b)/2
If the average and the least number are equal?
The average is equal to the median
units digit
The digit in the one's place. (For 287, the number 7is in the units or ONES place.)
absolute value
The distance a number is from zero on a number line
Harmonic Mean
The mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers Harmonic average a,b = 2ab/a+b
Standard deviation
The more the data are spread from the mean the higher the standard deviation Square root of the variance
independent events
The outcome of one event does not affect the outcome of the second event Intersection of two events E and F = probability of e multiplied by probablilty of F Think flipping of two coins. One result does not affect the other
Mode
The value that occurs most frequently in a given data set.. note that the mode is not unique, there can be multiple in a data set
If XY<0 and X>Y
Then X HAS TO BE POSTIVE and Y has to be negative. The first part tells you they have opposite signs
Disjoint or mutually exclusive
Two sets that have no common elements
percent
Write 1/100
What is the sum of 2 consecutive integers (odd/even)
X + X+1 = 2X +1 Which must be odd
Remainder formula
X = QN + R Where X = dividend (# being divided) Q = quotient (number yielded when X divided by N) N = divisor (number being divided by) R = remainder
If A<B, is A+ 1< B+ 1?
Yes, you can add a constant to both sides without changing the direction
combination
a grouping of items in which order does not matter Selecting 2 reps Frkm 5 members = n(n-r+1)/r = 5(5-2+1)/2 = 5(4)/2 =10 Formula= (n)(n-1)(n-2)...(n-r+1)/r!
How do you multiply decimals?
count the number of decimal places in the factors because this will determine the number of places in the product. To multiply 1.05 by 5.25, multiply as you would whole numbers: 105⋅525=55125 Then count the number of places in both factors (there are four decimal places). Move four decimal places to the left in the result of 55125, and you get 5.5125, the correct answer.
Even times odd
even
even times even
even
odd + odd
even
A number is divisible by 8
if the number formed by the last three digits is divisible by 8
A number is divisible by 6 if
its divisible by 2 and by 3.
A number is divisible by 4 is...
its last two digits are divisible by 4.
Probability
likelihood that a particular event will occur. Number of cases that an event will occur over all possible events
even + odd =
odd
odd times odd
odd
Integers are
positive and negative whole numbers and zero
A number is divisible by 9 if
the sum of digits is divisible by 9.
A number is divisible by 3 if ...
the sum of its digits is divisible by 3.
Consecutive integers
x, x+1, x+2 BUT should use X-1,X,X+1 when solving problems
Remember this: √(a^2b) =
|a|√b