Fractions

Things one must know:

• Proper fractions - the numerator is smaller than the denominator
• Improper fractions - the numerator is greater than the denominator
  
• Multiplying two proper fractions yields a smaller number
• Dividing two proper fractions yields a larger number
• Comparing fractions: multiply the numerator of one fraction with the denominator of the other fraction and vice versa
• When simplifying fractions that incorporate sums or differences, you may split up the terms of the numerator, but you may never split the terms of the denominator
• Two numbers are reciprocals of each other if their product equals 1. 
  
• When using benchmark values round some numbers up and others down
• Smart numbers: choose numbers equal to common multiples of the denominators of the fractions in the problem (MGMAT FDP p37)
• Pick smart numbers when no amounts are given in the problem, do not pick smart numbers when any amount or total is given
  
  Rules of positive fractions:
• Numerator goes up, the fraction increases
• Denominator goes up, the fraction decreases
• Adding the same number to both the numerator and the denominator brings the fraction closer to 1.
  - Adding to a proper fraction increases its value
  - Adding to an improper fraction decreases its value
  
• Multiplying or dividing both the numerator and the denominator by the same number does not change the value of the fraction

Digits & Decimals

Things one must know:

• Place value: hundreds digit x 100 + tens digit x 10 + units digits x 1
• Use variables for unknown digits, xy and the reverse yx
• If the right-digit-neighbor is 5 or greater, round up
• Multiplying by powers of ten moves decimal forward specified number of places, dividing moves the decimal back. Negative powers reverse the regular process
• To find the units digit of a product or a sum of integers, only pay attention to the units digits of the numbers.
• Heavy Division Shortcut: move the decimals in the same direction and round to whole numbers
• Powers and roots: rewrite the decimal as the product of an integer and a power of ten, and then apply the exponent
  

Back to basics (cont.)

1. MGMAT Digits & Decimals (12/15) - 80%
2. MGMAT Fractions (13/15) -  87%

Exponents & Roots Advanced

1. MGMAT Exponents & Roots Advanced Test (20/26) - 77%
Couple of careless errors here, overall, I think my grasp of this is pretty good.

O&E/P&N/CI Advanced

1. MGMAT O&E/P&N/CI Advanced Test (9/14) - 64%

Advanced Divisibility & Primes

1. MGMAT Advanced Divisibility & Primes Test (20/26) - 77%

Data Sufficiency

Things one must know:

• Rephrase both the question and the statements if you can by reducing it into its simplest form and focusing how the piece of information relates to the question.
  
• Factoring algebraic expressions is a great way to un-disguise hidden information
• Value questions - find a single number to answer the question.
• Yes/No questions - conclusively answer yes or no
• When testing numbers try to prove the statements not sufficient, be sure to try fractions, negatives and zero unless you are specifically told in the question that the variables have constraints.
• The statements cannot contradict each other - allows you to detect any mistakes you might have made.
  

PEMDAS

Things one must know:

• PEMDAS
• When an expression with multiple terms is subtracted, the subtraction must occur across every term within the expression, each term in the subtracted part must have its sign reversed.
• Pretend that there are parentheses around the numerator and denominator of a fraction. This is easy to forget if you eliminate the fraction bar or add or subtract fractions.
  

Back to basics (cont.)

MGMAT PEMDAS (8/10) - 80%

Back to basics (cont.)

Kaplan MW Exponents Test (17/19) - 89%

Roots

Things one must know:

• Odd roots will have the same sign as the base
• Even roots only have a positive value
• Break numbers into prime factors to find their roots
• Only roots linked by multiplication and division can be simplified
• Properties of roots (MGMAT NP p81)
• Squares and square roots to memorize (MGMAT NP p82)
• Simplifying a root: 1) factor the number under the radical sign into primes; 2) pull out any pair of matching primes from under the radical sign and place one of those pairs outside the root; 3) consolidate the expression
• Use the conjugate to rationalize the denominator of any fraction with a square root plus or minus another term, to find the conjugate, simply change the sign of the square root term.
  

Back to basics (cont.)

1. MGMAT Roots Test (11/15) - 73%
Three of the wrong answers were careless errors, I'm just tired

Back to basics (cont.)

Started going over Kaplan math tests to cement my knowledge.
1. Number Properties Test (16/19) - 84%

Exponents

Things one must know:

• An even exponents hides the sign of the base
• Numbers between 0 and 1 decrease when raised to a higher exponents
• Exponents rules! (MGMAT NP p68)
• Exponential expressions can only be simplified if they are linked by multiplication or division and have either a base or an exponent in common
• Terms added or subtracted can be factored whenever the base is the same or whenever the exponent is the same and bases contain something in common
• An exponent of 0 always yields 1
  
• A base of 0 always yields 0
• A base of 1 always yields 1, -1 yields 1 if the exponent is even and -1 if the exponent is odd
• Always considering factoring or distributing a given expression if it's possible. 
• 4 step simplifying process: 1) simplify or factor an additive or subtractive terms; 2) break every non-prime base down into prime factors; 3) distribute the exponents to every prime factor; 4) combine the exponents for each prime factor and simply.
• Advanced distributed/factored forms (MGMAT NP p164)

Back to basics (cont.)

5. Exponents (13/15) - 87%

Consecutive Integers

Things one must know:

• Sets of consecutive integers are sets of consecutive multiples, a special case of evenly spaced sets, where all the values in the set are multiples of the increment.
• The arithmetic mean and median are equal to each other.
• The mean and median equal the average of first and last terms of the set.
• The sum of elements in the set is equal to the arithmetic mean times the number of items in the set. 
• The sum of a set of an odd number of consecutive integers is divisible by the number of items.
• The sum of a set of an even number of consecutive integers is never divisible by the number of items.
• Counting consecutive integers: Last - First + 1
• Counting consecutive multiples: (Last - First) / Increment + 1
• The product of n consecutive integers is always divisible by !n

Back to basics (cont.)

4. Consecutive Integers (14/15) - 93%

Positives & Negatives

Things one must know:

• Absolute value: how far a number is from zero on a number line.
• Double negative = positive.
• When multiplying or dividing two numbers: if the signs are the same, the answer is positive, if not, the answer is negative. 
• When multiplying or dividing a group of nonzero numbers, the result will be positive if there's an even number of negative numbers, and the result will be negative if there's an odd number of negative numbers.
• Use a table to test positive or negative numbers systematically.
• The absolute value | x - y | can be interpreted as the distance between x  and y. Ie. | x - 3 | < 4 can be rephrased as "the distance between x and 3 is less than 4 units", this means -1 < x < 7. Find the midpoint or the average to set up the inequality.
• Whenever you see inequalities with zero on either side of the inequality, you should consider testing positive/negative case
  
  Complex absolute value equations
  
• Two different variables and no constants in absolute value expressions, it is usually easiest to test positive/negative numbers
• One variable and one or more constants, it is usually easier to solve with an algebraic approach. You need to consider two real cases:
  1) neither expression changes sign
  2) one expression changes sign
  Check the validity of the solutions!
  

Back to basics (cont.)

3. Positives & Negatives (15/15) - 100%
I'm getting the hang of it!

Odds & Evens

Things one must know:

• Add or subtract two odds or two evens, the result will be even.
• Add or subtract an odd with an even, the result will be odd.
• When multiplying, if any of the integers is even, the result will be even.
• When multiplying, if none of the integers is even, the result will be odd.
• Divisibility rules (see MGMAT NP p30)
• Set up a table listing all the possible odd/even scenarios if you are dealing with multiple variables that can be odd or even.
• Odd integers leave a remainder of 1 after division by 2
• Even integers leave a remainder of 0 after division by 2
• An arbitrary even number can be written as 2n, where n is any integer
• An arbitrary odd number can be written as 2n + 1 or 2n - 1, where n is any integer
  

Back to basics (cont.)

2. Odds & Evens (15/15) - 100%
Yay!

Divisibility & Primes

Things you should know:

• The sum, difference and product of two integers are always integers.
• The result of dividing two integers is sometimes an integer.
• Rules of divisibility (see MGMAT NP p14)
• Divisibility without rules:
  1) Find a multiple of the divisor that is close to the dividend
  2) Find the difference of the multiple and the initial number, then check if this difference is divisible by the divisor
  
• A factor is a positive integer that divides evenly into an integer.
• A multiple is formed by multiplying that integer by an integer.
• An integer is always both a factor and a multiple of itself.
• 1 is a factor of every integer.
  
  
• Adding or subtracting multiples of N result in a multiple of N.
• Adding or subtracting a multiple of N to/from a non-multiple of N results in a non-multiple of N.
• Adding two non-multiples of N can result in either a multiple of N or a non-multiple of N.
• When adding or subtracting two integers, neither of which is divisible by 2, the result will always be divisible by 2.
  
  
• First ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
• Determining if a number is prime: test divisibility with odd numbers up to its root.
  
  
• Any integer is divisible by all of its factors and also divisible by all of the factors of its factors.
• An integer n is divisible by all the possible products of its primes.
  
  
• GCF (shared primes): Take the lowest count of each prime factor found across all the integers (smallest count is zero)
  
• LCM (all the primes less the shared primes): Take the highest count of each prime found across all the integers. 
• (GCF of m & n) x (LCM of m & n) = m x n
• GCF of m & n cannot be larger than the difference between m & n
• Consecutive multiples of n have a GCF of n 
• Total prime factors (length): Add the exponents of the prime factors. 
• Total factors - if a prime factors appears to Nth power then there are N+1 possibilities for the occurrence of that factor, multiple the number of ways to make each individual decision to find the number of ways to make all the decisions
  
• All perfect squares have an odd number of total factors and the prime factorization contains only even powers of primes
  
  
• N! must be divisible by all integers from 1 to N
  
  Remainders:
  
• X/N = Q + R/N
• X = Q x N + R
• Remainders range from 0 to (N - 1), there are N possible remainders.
• Remainders can be added or subtracted directly, as long as the excess or negative remainders are corrected by just adding or subtracting the divisor.
• Remainders can be multiplied as long as the excess is corrected at the end.
• The decimal part of a quotient equals the remainder divided by the divisor.
 

Back to basics with MGMAT again

Since I still don't feel quite comfortable with math I've decided to go over MGMAT books again.

1. Divisibility & Primes (13/15) - 87%

Kaplan Math Workbook Word Problems

2. Data Sufficiency Practice Test 1 (11/25) - 44%
A real bloodbath.

Kaplan Math Workbook Data Sufficiency

1. Data Sufficiency Practice Test 1 (15/24) - 63%

I hate DS.

Kaplan Math Workbook Word Problems

1. General Word Problems Test (33/40) - 83%

Kaplan Math Workbook Word Problems

1. Percent, Ratio and Rates Word Problems Test (36/37) - 97%

I'm getting better at this.

Kaplan Math Workbook Word Problems

1. Level 1 Test (24/25) 96%
This is easy. One mistake because I haven't mastered algebra yet.

Kaplan Math Workbook Geometry (cont. 2)

5. Multiple Figures (13/15) - 87%
Did this one yesterday.

6. Solids (7/8) - 87%