Lines & Angles Test

MGMAT Lines & Angles Test (14/15) - 93%

Circles and Cylinders Test

MGMAT Circles and Cylinders Test (13.5/15) - 90%

Geometry

     Quadrilaterals

• Quadrilaterals
  - Parallelogram - Opposite sides and opposite angles are equal
  - Rectangle - All angles are 90°, and opposite sides are equal
  - Rhombus - All sides are equal. Opposite angles are equal
  - Square - All angles are 90°. All sides are equal
  - Trapezoid - One pair of opposite sides is parallel
• Sum of Interior Angles of a Polygon = (n - 2) × 180, where n is the number of sides 
• Perimeter is the sum of the lengths of all sides
• Area
  - Triangle = (Base × Height) / 2
  The base refers to the bottom side of the triangle. The height always refer to a line that is perpendicular (at a 90° angle) to the base
  - Rectangle = Length × Width
  - Parallelogram = Length x Height
  - Cut more complex shapes into rectangles and right triangles, then find the areas of these individual shapes.
  - See p16 in MGMAT Geometry for less common area formulas
• Surface Area = The SUM of the areas of ALL of the faces
• Volume = Length × Width x Height
  - Remember, when you are fitting 3-dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough. You must know the specific dimensions (l,w,h) of each object to determine whether the objects can fit without leaving gaps.
  
  Triangles & Diagonals
• The sum of the three angles of a triangle equals 180°
• Angels correspond to their opposite sides. The largest angle is opposite the longest side, the smallest angle is opposite the shortest side. If two sides are equal, their opposite angles are also equal.
• Length of sides: (x-y)<(x+y)
• Use Pythagorean Theorem to find the hypotenuse (the side opposite the right angle) of a right triangle: a² + b² = c²
• Common right triangles
  3-4-5                6-8-10
  3² + 4² = 5²      9-12-15
  (9 + 16 = 25)    12-16-20
  5-12-13            10-24-26
  5² + 12² = 13²
  25 + 144 = 169
  8-15-17
  8² + 15² = 17²
  64 + 225 + 289
• An isosceles triangle is one with two equal sides, the two angles opposite these two sides are also equal.
  45° 45° 90°
  leg leg hypotenuse
   x :  x  :  x√2
• An equilateral triangle is one with all three sides (and all three angles) equal.
  Two 30-60-90 triangles make up an equilateral triangle
  30°    60°  90°
  short long hypotenuse
     x  :  x√3 : 2x
• Diagonal of a square: d = s√2, where s is a side of a square
• Main diagonal of a cube: d = s√3, where s is an edge of the cube
• To find the diagonal of a rectangle, you must know either the length and the width or one dimension and the proportion of one to the other
• To find the diagonal of a rectangular solid, you can use the Pythagorean theorem twice or the Deluxe Pythagorean Theorem: d² = x² + y² + z²,where x,y and z are the sides of the rectangular solid and d is the main diagonal.
• Triangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion. Once you find that two triangles have two pairs of equal angles, you know that the triangles are similar, furthermore, if two right triangles have one other angle in common, they are similar triangles.  
• If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a²:b² 
• Be able to see any side of a triangle as the base, not just the side that happens to be drawn horizontally, also be able to draw the height from that base. 
• The area of an equilateral triangle with a side length of S is equal to (S²√3) / 4
• Right triangle/Rectangle DS tips, knowing any two of these will allow to solve for the rest:
  1. Side length 1
  2. Side length 2
  3. Diagonal/Hypotenuse
  4. Perimeter
  5. Area
  
  
• Circles & Cylinders
• A radius is any line segment that connects the center point to a point on the circle
• A chord is any line segment that connects two points on the circle. Any chord that passes through the centre of the circle is called a diameter
• The distance around the circle is termed the circumference: C = 2πr
  A full revolution or a turn of a spinning wheel is equivalent to a wheel going around once, a point on the edge of the wheel travels one circumference in one revolution
• A portion of a distance on a circle is termed an arc. Use the central angle to determine what fraction an arc is of the entire circle (out of a total of 360°)
• The boundaries of a sector are formed by the arc and two radii. Slice of pizza.
• Area of a circle: A = πr² 
• You can find the area of a sector by determining the fraction of the entire area the sector occupies, you can do this by looking at the central angle that defines the sector
• An inscribed angle has its vertex on the circle itself. An inscribed angle is equal to half of the arc it intercepts
• If one of the sides of an inscribed triangle is the diameter of the circle, then the triangle must be a right triangle. Conversely, any right triangle inscribed in a circle must have the diameter of the circle as one of its sides.
• Surface area of a cylinder = 2 circles + rectangle = 2(πr²) + 2πrh
• Volume of a cylinder: πr²h
• If you know the circumference, the radius, the diameter, or the area of a circle, you can use one to find any of the other measurements.
• Sphere
  Surface area: 4πr²
  Volume: 4/3πr³
  
  
• Lines and Angles
• Parallel lines are lines that lie in a plane and that never intersect
• Perpendicular lines are lines that intersect at a 90° angle
• Intersecting lines:
  - the interior angles form a circle, so the sum is 360°
  - angles that combine to form a line sum to 180°
  - angles found opposite each other where two lines intersect are equal. These are called vertical angles.
• An exterior angle of a triangle is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle.
• Parallel lines cut by a transversal:
  - All acute angles (less than 90°) are equal
  - All obtuse angles (more than 90° but less than 180°) are equal
  - Any acute angle is supplementary to any obtuse angle (they sum to 180°) 
• Complementary angles sum to 90°
• Supplementary angles sum to 180°

Geometry Test

1. MGMAT Polygons Test (13/15) - 87%

AWA

Issue
Focus on:
• Clear thesis (main idea - expressed clearly)
• Persuasive examples
• Logical structure (distinct paragraphs)
• Transitional words and structure (while, for example, however, in addition)

Strategy:
1. Read the prompt
    - Summarize the main idea and write it down in the prompt
2. Examples + Thesis = Outline
    - Figure out and type up the examples before you decide to agree or disagree (strong points first)
    - Agree or disagree based on how many and how good your arguments are to support that side
3. Write the intro
    - Summarize the issue (what you wrote down earlier, now just use more sophisticated language) (some.. believe, many.. must decide)
    - State your thesis from the outline
4. Write the body
    - One paragraph per example, three examples is optimal
    - Write with purpose - describe examples with supporting details
    - Tie each example to the thesis (restate it) (This shows the folly of, Clearly, the decision.. is the right choice)
    - Conclusion can be a separate paragraph or at the end of the last body paragraph. Clearly restate your argument by looping back to the thesis. Don't summarize the examples.
    - Use transitional words
5. Proofread
    - Add transitional words
    - Make sure examples are tied to the thesis

Argument
• Argument is always flawed.
• Dedicate more time to outlining than you do on issue essays
• Don't develop a separate argument of your own!
• Statements presented as aevidence can themselves depend on questionable assumptions. Look for logical leaps in every sentence. 

Strategy:
1. Read the argument, identify the conclusion
    - Summary is not so important as it is on issue essays, just give it a read - pay attention to recommendations which are common conclusions
2. Identify assumptions
    - Write down counterexamples or alternative causes (same as prephrasing weakeners for assumptions) - this will be the outline 
3. Write the introduction
    - describe the argument from the prompt, state that it is flawed (the argument rests on a questionable chain of logic; without additional evidence this argument cannot stand up to scrutiny), use the word assumption (the author relies on an assumption), briefly state one or two assumptions
4. Write the body paragraphs 
    - use the outline
    - one paragraph per assumption, describe the assumption, cite alternative causes and/or counter examples
    - show insight - conclude by noting info that might support the argument
5. Proofread 
    - logic - make sure you have identified assumptions, alternative causes/counterexamples
    - structure - paragraph breaks, transition
    - grammar - erros, vary syntax (long sentence followed by a shorter sentece)

Common assumptions:

Generalizations - True in one case, so true in general

Questionable analogies - True in one case, so true in a "similar case"

Past vs Present - Something true in the past is still true today

Correlation vs Causation - Events occur together, so one caused the other

Trends - What has been happening recently will continue

Advanced strategies:

• Don't change ideas halfway
• Strengthens can be positioned at the end of the assumption paragraph or at the end of the essay
• Issue essays - discredit opposing evidence

Overlapping Sets Test

MGMAT Overlapping Sets Test (12/15) - 80%

Overlapping Sets

• For problems involving only two categorizations or decisions use the Double-Set Matrix: a table whose rows correspond to the options for one decision, and whose columns correspond to the options for the other decision. The last row and the last column contain totals, so the bottom right corner contains the total number of everything in the problem.
• Make sure the columns and rows correspond to mutually exclusive options for one decision.
• If no amounts are given, pick smart numbers for total, for problems involving percents pick 100, for problems involving fractions, pick a common denominator for the total.
• Read the problem very carefully to determine whether you need to use algebra to represent unknowns.
• You can extended the Double Set Matrix if a decisions requires you to consider more than two options as long as each set of distinct options is complete and has no overlaps.
• Problems hat involve three overlapping sets should be solved using a Venn Diagram. Remember to work from the inside out.
  1) Fill in the innermost circle, items on all three teams.
  2) Fill in circles for items on two teams, remember to subtract the items on all three teams.
  3) Fill in circles for items only on one team, remember to subtract items on two teams and on all three teams.
  To determine the total, just add all numbers together. 
• The union of two sets is the set of all elements that are found in either of the two sets (or in both of them).
• The intersection of two sets is the set of all of the elements that are found in both sets.
  
  

Statistics Test

MGMAT Statistics Test (12/15) - 80%

Statistics

• Arithmetic mean:
  Average = Sum / # of terms
  Sum = Average x (# of terms)
  To keep track of two average formulas in the same problem, you can set up an RTD style table.
• The average of an evenly spaced set is the middle number (or the arithmetic mean of two middle numbers). All you need to do to find the middle number is to find the arithmetic mean of the first and last terms.
• Weighted averages:
  Weighted average = (weight/sum of weights) x (data point) + (weight/sum of weights) x (data point) + ..
  or
  Weighted average = (weight)(data point) + (weight)(data point) / sum of weights
• Having just the ratios of the weights will allow you to find a weighted average. Simply write the ratio as a fraction, and use the numberator and the denominator as weights. If you know the weighted average you know the ratio of weights.
• If you know the two sub-group averages and you know the overall weighted average, then you can solve for the relative weightings of the two sub-groups. 
• If you are finding a weighted average of rates (whose units are fractions), then the "weights" correspond to the units appearing in the denominator of the rate.
• Median is the unique middle value of a set containg an odd number of values arranged in increasing (or decreasing) order, or the arithmetic mean of the two middle values of a set containing an even number of values arranged in increasing (or decreasing) order.
  You may be able to determine a specific value for the median of a set even if one or more unknowns are present.
• Solve problems involving both the arithmetic mean and the median by writing expressions for both.
• You may be required to construct and manipulate a completely abstract set, you can use alphabetical order to make it a little more concrete, or you can place the variables on an abstract number line in order to visualize their relationships, or you can create a column chart.
• Standard deviation indicates how far from the average (mean) the data points typically fall. The more spread out the numbers, the larger the SD.
  If you see a problem focusing on changes in the SD, ask yourself whether the changes move the data closer to the mean, farther from the mean, or neither. If the problem requires comparisons, ask yourself which set is more spread out from its mean.
  The term "variance" is the square of the SD.
• The mode of a set of numbers is the number that occurs most often.
• The range of a set of numbers is the difference between the largest number and the smallest number.
  
  

Sentence Correction

• Subject and verb must agree in number
  - find the subject and the verb through filler
  - verb might come before subject, ask what?
  - identify each subject and verb in the underlined part and pair them
• Unusual subjects
  - compound subjects - and means plural (two things)
  - collective nouns (i.e. group, team) are singular
  - verbals that function as nouns (gerunds aka -ing words or infinitives) are singular
• Fragments
  - looks like a complete sentence, but doesn't express a complete thought
  - check to see that the sentence has a main verb
• Run-on sentence
  - combines two sentences that could stand on their own without proper punctuation, must be separated by a semicolon or a comma and a fanboys word:
  For
  And
  Nor
  But
  Or
  Yet
  So
  - Make sure there are full sentences on both sides of these conjunctions
• Pronouns
  - have to refer to a specific antecedent
  - must agree in number with the antecedent
  - both is plural; each is singular
  - watch out for answer choices that introduce new pronouns
• Modifiers
  - When a sentence starts with a modifier, the object that it is modifying should be the first word after the comma
  - Modifiers should be close to what they modify
• Verb tense indicates the time of verb's action
  - Auxiliary verbs express more complex tenses:
    Past perfect (had) - event in the past preceding another event in the past, (X had before Y was)
    Present perfect (has) - started in the past continuing to present (doesn't need to continue to into the future)
  - Actions in the same time frame should be expressed in the same tense
  - Keywords: as, until, before, since
• Parallel construction:
  - Lists
  - Correlative conjunction: not only.. but also, neither.. nor.. 
  - Comparisons and contrasts - unlike, just as...so, like, as...as, than.., compared with.., in contrast..,
  - "like..., word directly after the comma" 
  - Make sure items are logically comparable

Checklist

• Phrases and clauses
  Find subject and main verb of sentence
  Do subject and verb agree in number?
  Is the sentence a fragment?
  Do phrases/clauses create wordiness?
• Lists 
  Is each item parallel?
• Verbals
  Do verbals violate parallelism?
  Any misplaced/dangling modifiers?
  Are verbals wordy/awkward?
• Pronouns
  Match pronouns to antecedents
  Does each pronoun have one ant.?
  Do pronouns and ant. agree? 
  Is pronoun unnecessary and wordy?
• Comparisons
  Identify items that are compared
  Are items logically able to be compared?
  Are compared items parallel?
  Are comparison idioms correct?
• Quantity Words
  Is correct word used (e.g. fewer/less)? Any redundancy in quantity words?
• “Tell” Words“
  Had” / “if” – tense errors?
  “Being” / “having been” – wordiness?
  Passive verbs – awkward style?
  “There are” / “there is” – wordiness?

Coordinate Plane Test

MGMAT Coordinate Plane Test (13/15) - 87%
One careless mistake.

Coordinate Plane

• Slope of a line is defined as "rise over run" - how much it rises vertically divided by how much it runs horizontally.
  rise / run = y₁-y₂ / x₁-x₂ 
• A point where a line intersects a coordinate axis is called an intercept.
  - The x-intercept is the point on the line at which y = 0
  - The y-intercept is the point on the line at which x = 0
  Plug in 0 for x or y to find the intercepts.
• All lines can be written as equations in the form of y = mx + b, where m represents the slope of the line and b represents the y-intercept of the line.
• Finding the equation of a line:
  1) Find the slope of the line by calculating the rise over run.
      Remember that y-intercept is a point as well
  2) Plug the slope in for m in the slope-intercept equation.
  3) Solve for b, the y-intercept, by plugging the coordinates of one point into the equation. Either point's coordinates will work.
  4) Write the equation in the form of y = mx + b
• Determining which quadrants a given line passes through can be done in two ways:
  1) First, rewriting the equation in form of y = mx + b and then sketching the line
  2) Finding two points on the line by setting x and y equal to zero in the original equation. 
• The perpendicular bisector has the negative reciprocal slope of the line segment it bisects.
  1) Find the slope of the line it bisects
  2) Find the negative reciprocal by flipping the fraction and changing the sign (product must be -1)
  3) Find the midpoint of the line, by finding the midpoints of the x and y coordinates separately.
  4) To find the value of b of the perpendicular bisector substitute the coordinates of the midpoint for x and y. 
• Parallel lines have equal slopes. 
• If two lines in a plane intersect in a single point, the coordinates of that point solve the equations of both lines. To find the find the intersection point of two equations, turn them into slope-intercept form and set them equal to each other.
• When faced with an inequality in the coordinate plane:
  1) Draw the line either by converting to slope-intercept form or using the x and y intercepts
  2) Plug in a point on one side of the line. If this point makes the inequality true, that point is in the solution set. If not, the solution set is on the other side of the line.
• Distance between two points
  Square root of (x₁-x₂)²+(y₁-y₂)²
  
  Need to finish this.
  

Probability

Probability = Number of desired or successful outcomes / Total number of possible outcomes

• All the outcomes must be equally likely - you can use a counting tree
• To determine the probability that independent events X and Y will both occur, multiply the two probabilities together.
• To determine the probability that mutually exclusive events X or Y will occur, add the two probabilities together.
  - if the events cannot occur together, add the probabilities of individual events
  - if the events can occur together, use the formula: P(A or B) = P(A) + P(B) - P(A and B)
• Probability of success + Probability of failure = 1
• If success contains multiple possibilities - especially if the wording contains phrases such as "at least" and "at most" - then consider finding the probability that success does not happen.
• Be careful of situations in which the outcome of the first event affects the probability of the second event, i.e. picking things out of a box.
• When you use a probability tree, multiply down the branches and add across the results.
  

Reading Comprehension

• Read for the big picture, don't dwell on the details, translate ideas and terms into simpler words, ask "why" for each new idea, create a Passage Structure
• Use PS and keywords in the passage to find:
  Main Idea: What? The point or idea that the author expresses in the passage.
  Attitude: How? The feeling or opinion the author expresses about the main idea.
  Purpose:Why? The author's reason for writing
• Pay attention to soft and extreme wordings: some vs all
• Keywords: must, if, then, like
• Read the passage first, don't look at the question before
• Main idea or title should capture the whole passage, not just one paragraph
  
• Global questions
  - Use MAPs
  - Prephrase an answer
  - Look for a match
• Detail question
  - Use MAPs to locate the detail
  - Paraphrase and scan answer choices for the paraphrasing of the passage language
  - Read the complete idea to provide context
• Inference question
  - Think must be true (and are concretely supported by the passage)
  - Eliminate answers one by one using the passage to invalidate
  - (Don't prephrase)
• Distinguish between the author's voice and others
• Common wrong answer choices - out of scope, too broad, distorted details, extreme. 

Combinatorics Test

MGMAT Combinatorics Test (11/15) - 73%
This is tough!

Combinatorics

• Fundamental Counting Principle: If you must make a number of separate decisions, then multiply the number of ways to make each individual decision to find the number of ways to make all the decisions.
• For problems in which certain choices are restricted and/or affect other choices, choose the most restricted options first.
• The number of ways of putting n distinct objects in order, if there are no restrictions, is n!
• When you have repeated items, divide the "total factorial" by each "repeat factorial" to count the different arrangements. 
• To count possible groups, divide the total factorial by two factorials: one for the chosen group and one for those not chosen.
• If you are required to choose two or more sets of items from separate pools, count the arrangements separately - perhaps using a different anagram grid each time. Then multiply the number of possibilities for each step.
• If the problem has unusual constraints, try counting the arrangements without constraints first. Then subtract the forbidden arrangements.
  - for problems in which items or people must be next to each other, pretend that the items "stuck together" are actually one large item.
  - keep in mind that stuck together items can be XY and YX so multiply the result by two.
• A combination is a selection of items from a larger pool, the order of items does not matter.
• A permutation is also a selection of items from a larger pool, the order of items matters.
• If switching the elements in a chosen set creates a different set, it is a permutation. If not, it is a combination.
  

Ratios Test

MGMAT Ratios Test (13/15) - 87%
Two careless errors.

Ratios

• Expresses a part-part relationship or a part-whole relationship.
• If there are only two parts in the whole, you can derive a part-whole ratio from a part-part ratio and vice versa.
• If two quantities have a constant ratio, they are in direct proportion to each other.
• Always write units on either ratio itself or the variables you create, or both.
• Simple ratio problems:
  1) Set up a proportion
  2) Cross multiply to solve
  Note that you can cancel factors either vertically within a fraction or horizontally across an equals sign, but never diagonally across an equals sign.
• Unknown Multiplier
  M/W which is 3/4 becomes 3x/4x which cancel out
  M + W becomes 3x+4x
  Can be used once per problem, you can never have two unknown multipliers in a problem.
  Should be used when neither quantity in the ratio is already equal to a number or a variable expression.
• You can multiply each ratio to make a common term in order to combine ratios.
  
  

Rates & Work Test

MGMAT Rates & Work Test (14/15) - 93%

Rates & Work

Rate x Time = Distance
Rate x Time = Work

• All the units in your RTD chart must match up with one another.
• Always express distance rates as "distance over time" (i.e. 50 km per hour) or work rates as jobs per time unit.
  
• When you have more than one traveler on trip, make a row in your RTD chart for each traveler or trip.
• Sample translations/problems (MGMAT WT p35-38)
  - moving toward each other (add the rates)
  - chasing and catching up (subtract the rates)
  - chasing and falling behind (subtract the rates)
  - faster means lower time
  
• The numbers in the same row of an RTD table will always multiply across. However, the specifics of the problem determine which columns will add up into a total row.
• Use variables to stand for either Rate or Time, rather than Distance.
• Average rate: find the total combined time for the trips and the total combined distance for the trips. You can actually pick any number for the distance. 
• If two or more agents are performing simultaneous work, add the work rates. Only exception is when one agent's work undoes the other agent's work, in this case subtract the rates.
• Use population chart for problems where populations double or triple in size over constant intervals of time.
  

Critical Reasoning

Evidence + Assumption = Conclusion

Evidence - explicit stated support for conclusion
Assumption(s) - unstated support for conclusion
Conclusion - main point

Assumption Strategy:
1) Identify the conclusion
• sounds most important and general
• sounds like an opinion
• preceded by conclusion keywords (therefore, thus, so)
2) Identify the evidence
• sounds like factual info
• sounds like it is contributing to believability of something
• keywords (because, claimed, as)
• is always true (no need to question it)
3) Identify the assumption
• fills the gap between the evidence and the conclusion, a general rule that ties the two
• must be true, if the author didn't believe it, the conclusion would be invalid

Red flags in answer choices:
• extreme statements (keywords: only, never, always)
• opposite
• irrelevant (often irrelevant comparison)
• out of scope (introduces new information)

Stem based approach:
1) Identify the question type
2) Untangle stimulus (read the stimulus)
3) Predict an answer (before looking at the answer choices)
4) Match the prediction with an answer choice

Strengthen / Weaken:
Manipulate the assumption. Correct answer doesn't need to fully prove or disprove the conclusion, just strengthen or weaken it.
Denying an assumption invalidates the conclusion.
Strengthening argument must cause the conclusion to make sense.

Causation:
If X & Y happen at the same time and it is assumed that X causes Y, it could be argued that:
- the reverse is true, Y causes X
- something else caused Y, Z causes Y
- it is just a coincidence

Logical opposite of X is not X

Inference:
What must be true based on what was said in the statement.

Inference Strategy:
1) Logically link each sentence
2) Paraphrase the argument
3) Go through answer choices one by one (difficult to predict an answer)

Degrees of certainty:
- Must be true
- Must be false
- Could be true
Think of what is being asked! All true except - means that can be either 'must be false' or 'could be true'


Some - Most
- Some - at least one
- Most - more than half
Cannot be determine which is greater
Most + Some = Some

Common Wrong AC:
- extreme
- could be true (new info that wasn't mentioned in the stimulus)
- opposite

Formal Logic
X -> Y
not Y -> not X

Algebraic Translations Test

Algebraic Translations Test (11/15) - 73%

Algebraic Translations

• Translating English into algebra:
  1) Assign variables
      - Try to minimize the number of variables
      - Make use of a relationship given in the problem.
  2) Write equations
  3) Solve algebraically
  4) Evaluate the algebraic solution in the context of the problem - make sure you answer the question asked
• You can check your translations with easy numbers.
• Write an unknown percent as a variable divided by 100
• Translate bulk discounts and similar relationships carefully
• When a problem involves several quantities and multiple relationships, it is often a good idea to make a chart or a table to organize the information, i.e. age problems:
  - Put people in rows and times in columns.
  - Use variables to indicate the age of each person now, fill other columns by adding or subtracting time from the now column.
  - Write equations that relate the individuals' ages together
• In a typical Price-Quantity problem, you have two relationships:
  - the quantities sum to a total
  - the monetary values sum to a total
• Look out for hidden constraints (i.e. whole number, positive)
  - Think about what is being measured or counted and whether a hidden constraint applies.
  - To solve algebra problems that have integer constraints, test possible values systematically in a table.
  - When all quantities are positive in a problem, certain algebraic manipulations are safe to perform: 1) dropping negative solutions of equations; 2) dropping negative possibilities with inequalities (see MGMAT WT p21)
  

Subject - Verb Agreement

• Every sentence must have a subject and a (working) verb (a verb that can run a sentence by itself).
• Subject and verb must make sense together.
• Subject and verb must agree in number.
• A noun in a prepositional phrase cannot be the subject of the sentence.
• Compound plural subjects are formed with the word and, additive phrases do not form compound subjects. 
• Phrases such as or, either... or, neither nor - must agree to in number to the nearest noun. Either and neither in a sentence alone (without or or nor) are singular.
• Collective nouns are almost always considered singular.
• Confusing subjects are more often singular than plural. 

This list might not be final.

Subject - Verb Agreement

MGAMT Subject - Verb Agreement Test (9/15) - 60%

Additional VICs Test

MGMAT Additional VICs Test (8/11) - 73%
Last two were RTD problems which I haven't studied yet so I didn't every really try on those.

Advanced Inequalities

MGMAT Advanced Inequalities Test (9/15) - 60%

Grammar, Meaning, Concision Test

MGMAT Grammar, Meaning, Concision Test (10/15) - 67%

Grammar, Meaning, Concision

    Evaluate problems in this order
    1) Grammar

    2) Meaning
    3) Concision

• If a word has more than one meaning, is the author using that word correctly, to indicate the right meaning?
  - Guide 8 p20 for examples
  - Pay attention to helping verbs (such as may, will, must and should) - only change them if the original sentence is obviously nonsensical.
• Changing the position of a single word can alter the meaning of an entire sentence.
  - Look out especially for short words (such as only and all)
  - Pay attention to overall word order.
• Make sure that words that are connected, such as subjects and verbs or pronouns and antecedents, always make sense together.
• Concise is better.
• Don't use a phrase where a single word will do.
• If two words in a sentence mean the same thing, check the sentence for redundancy. Only one of the words might be necessary.
  - Pay attention to expressions of time.
  

Advanced Formulas & Functions

MGMAT Advanced Formulas & Functions Test (9/14) - 64%

Sentence Correction Strategy

• Strategy for SC:
  1. Write down "A B C D E" on your paper.
  2. Read the sentence, nothing any obvious errors as you read.
  3. Scan the answer choices vertically - do not read them - looking for differences that split the answer choices.
  4. Choose a split for which you know the grammatical rule and which side of the split is correct.
  5. Cross out the answer choices that include the incorrect side of the split.
  6. Compare the answer choices by re-splitting.
  7. Continue to split remaining choices until you have one answer left. 
• Make sure you read the entire sentence, as often important words are placed far from the underlined portion.
• Double check that your answer works in the context of the entire sentence.
  

Advanced Equations

MGMAT Advanced Equations Test (12/17) - 71%

Data Sufficiency for Equations, Inequalities and VICs

Things one must know:

• Solved through algebraic manipulations. Sometimes you will need to manipulate the original equation, other times you will need to manipulate the statements, sometimes both. Remember MADS.
• As you rephrase, always keep in mind the variable or variable combo you are trying to isolate.
• Sometimes, even when the variables have multiple potential values, the answer to the question stays the same.
• Frequently, the combos are "hidden": they are not asked about directly, in these cases, you must rephrase the question to find the hidden combo.
• Be on the lookout for hidden meaning in certain statements as you rephrase a question. Sometimes a particular piece of information has an intuitive interpretation that you will not see by simply plodding through the algebra.
  

VICs

MGMAT VIC Test (14/15) - 93%

VICs (Variable Expressions in the Answer Choices)

Things one must know:

• Four most common types:
  1) Word Translations
  2) Algebra
  3) Percent
  4) Geometry

Direct Algebra Strategy
Direct algebraic translation/manipulation as needed until the solution is obtained. Might become too difficult or dangerous as problems become more challenging, however, can be very fast.

•  Break the problem down into manageable parts.
  

Pick Numbers and Calculate a Target
When you don't see a way to do it algebraically.
1) Pick numbers for all or most of the unknowns in the problem
2) Use those numbers to calculate the answer to the problem - the "Target"
3) Plug the numbers you have picked into each answer choice to see which answer choice yields the same value as your Target.

You may accidentally pick numbers that result in two or more answer choices yielding the Target value. Can be slow because it requires a lot of computation.

• Never pick the numbers 1 or 0, for percent problems, also avoid 100, but usually should pick multiples of 10 that are easy to work with.
  Make sure all the numbers you pick are different.
  Pick small numbers.
  Try to pick prime numbers.
  Avoid picking numbers that appear as a coefficient in several answer choices.
• You should ideally test every answer choices, even if you have already found one that equals your Target value. Stop calculating once you realize that an answer choice cannot equal your Target value.
• When variables are related to each other through an equation, you cannot pick a value for each variable. Pick a value for all but one of the variables and then solve for the value of the remaining variable.
  

The Hybrid Method
Pick numbers to help you think through the problem. However, rather than plug these numbers into the answer choices, use the numbers to think through the computations, and therefore the matching algebra, step by step.

• Break the problem down into manageable parts.

Misc

• Always draw a diagram for geometry VIC problems.
• Create intermediate variables to represent key unknowns when solving a VIC problem, just remember that your solution cannot contain these intermediate variables.
• If you are trying to figure out the algebraic manipulation but you get stuck, you should immediately switch to a number picking strategy
• If the variables in a VIC problem are already defined as numbers, find the equation that relates the numbers. This will be the same equation that relates the variables.
  

Inequalities

MGMAT Inequalities Test (11/15) - 73%

Inequalities

Things one must know:

• When you multiply or divide an inequality by a negative number, flip the inequality sign.
• You cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for.
  Set up two cases - the positive and the negative case - and solve each one separately to arrive at two possible scenarios.
• When taking reciprocals flip the sign unless the two sides have different signs. If you don't know the signs of both sides, you cannot take reciprocals. 
• You can only square inequalities when both sides have the same sign.
  If both sides are known to be positive, then do not flip the inequality sign when you square.
  If both sides are known to be negative, then flip the inequality sign when you square.
  If the sides are unclear or different, then you cannot square.
  
• Combining inequalities:
  1) Solve
  2) Simply so that all the inequality symbols point in the same direction
  3) Line up the common variables
  4) Combine by taking the more limiting upper and lower extremes
  
• You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality.
• You can add inequalities together as long as the inequality signs are pointing in the same direction. You can multiply inequalities together only if both sides of inequalities are positive. You can never subtract or divide two inequalities. 
• Use extreme values for problems that involve multiple inequalities and the potential range of values for variables in the problem
• You should know how to solve problems involving both equations and inequalities by: (1) plugging the inequality into the equation using extreme values, and (2) plugging the equation into the inequality using standard algebra.
• Extreme value arithmetic operations (MGMAT 3rd book, p92)
  Flip the extreme value when subtracting, dividing and multiplying a negative number by an extreme value. Multiply two extreme values if you know that both are positive.
• Use extreme values to solve minimization and maximization problems - focus on the largest and smallest possible values for each of the variables. Set up a table with extreme values for all the variables. 
  Set terms with even exponents equal to zero when trying to solve optimization problems.
  
• When you see inequalities with zero on one side of the inequality, you should consider using positive/negative analysis to help solve the problem.
  Common inequality statements:
  xy > 0 implies that x and y are both positive or both negative
  xy < 0 implies that x and y have different signs
  x² - x < 0 or x² < x or 0 < x < 1
• Inequalities and absolute value
  | x + b | = c means the center point of the graph is -b and that x must be exactly c units away from -b.
  | x + b | < c means the center point of the graph is -b and that x must be less than c units away from -b.
  Solve algebraically:
  1) First scenario, simply remove the absolute value brackets and solve
  2) Second scenario, reverse the signs of the terms inside absolute value brackets, remove the brackets, and solve again.
  3) Combine the scenarios into one range of values
• Inequalities involving even exponents require you to consider two scenarios.
  √x² = |x| likewise consider that x can be negative or positive (i.e. positive x < 2 and negative x > -2
• If you have an inequality with zero on one side, see if you can factor the either side to create two factors that can be used for positive/negative analysis.
• Be wary of answer choices A or B on problems involving inequalities and variables. A seemingly in consequential statement, such as "x is positive" or "a is negative" if often crucial on inequality problems.
  
  

Functions

MGMAT Functions Test (12/15) - 80%

After almost a week of studying - every day very little - I'm done with this chapter.

Functions

Things one must know:

• A function rule describes a series of operations to perform on a variable.
• The "domain" of a function indicates the possible inputs, or the x-coordinate. The "range" of a function indicates the possible "outputs", or the y-coordinate.
• Input the value in place of the variable to determine the value of the function. When substituting a variable expression into function, keep the expression inside parentheses.
• The key to solving compound function problems is to work from the inside out - i.e. use the result from the inner function as the new input variable for the outer function. 
• Changing the order of the compound functions changes the answer, to find a value of x for which f(g(x)) = g(f(x)), use variable substitution.
• Finding an unknown constant:
  1) Use the value of the input variable and the corresponding output value of the function to solve for the unknown constant.
  2) Rewrite the function, replacing the constant with its numerical value.
  3) Solve the function for the new input variable.
• Direct proportionality means that the two quantities always change by the same factor and in the same direction: y = kx, where x is the input value, y is the output value and k is the proportionality constant. Can also be written as y / x = k. Set up ratios for the before case and the after case, and then set the ratios equal to each other to solve.
• Inverse proportionality means that the two quantities change by reciprocal factors: y = k / x, where x is the input value, y is the output value and k is the proportionality constant. Can also be written as xy = k. This time set up products to solve the problem. 
• Linear growth - growth at a constant rate, same constant added each period: y = mx + b, where slope m is the constant rate at which the quantity grows, the y-intercept b is the value of the quantity at time zero and the variable x stands for time. Initial time is 0.
• Exponential growth - a quantity is multiplied by the same constant each period of time: y(t) = y₀·k^t, where y₀ represents value at time zero, k represents the constant multiplier (commonly in percentage form), and t represents time. Maybe it is y = k(R^t)
  
• Symmetry, the property that two seemingly different inputs to the function always yield the same output. Solve algebraically or it might be easier to pick a number.
• Pick numbers for advanced function problems that test whether certain functions follow certain properties of mathematics.
• Optimization
  1) Linear functions have straight line graphs. The extremes (max and min) of linear functions occur at the boundaries: at the smallest possible x and at the largest possible x, as given in the problem.
  2) Quadratic functions form parabolas. A parabola has either a peak (a max value) or a valley (a min value) that you need to find. The key is to make the squared expression equal to 0. Whatever value for x makes the squared expression equal to zero is the value of x that minimizes or maximizes the function. The sign of the squared term determines whether the extreme point is a minimum (positive sign) or a maximum (negative sign).
  

Formulas

MGMAT Formulas Test (12/15) - 80%

Formulas

Things one must know:

• Plug-In formulas: simply write the equation down, plug in the numbers carefully, and solve for the required unknown. Be sure to write the formula as a part of an equation.
• Strange Symbol formulas: The symbol is irrelevant. All that is important is that you carefully follow each step in the procedure that the symbol indicates. Watch out for symbols that invert the order of an operation.
• Formula problems that involve unspecified amounts should be solved by picking smart numbers for the unspecified variables.
• Sequences are defined by function rules: each term is a function of its place in the sequence. 
• You must be given the rule in order to find a particular number in a sequence. 
• Recursive formulas - each item of a sequence is defined in terms of the value of the previous items in the sequence. You need to be given the recursive rule and also the value of one of the items in the sequence to solve for the values of a recursive sequence. The subscript n - 1 means "the previous term", and the subscript n means "this term.
  
• Linear sequence, the difference between successive terms is always the same.
  Direct formula: Sn = kn + x, where k is the constant difference between successive terms and x is some other constant.
  Recursive formula: Sn = Sn₋₁ + k and S₁ = k + x
  Compute the difference between two terms to find k, you can then solve for x.
  
• Exponential sequence, each term is equal to the previous term times a constant k, the ratio between successive terms is always the same.
  Direct formula: Sn = x(kⁿ), where x and k are real numbers.
  Recursive formula: Sn = kSn₋₁ and S₁ = xk
  You can find k by combing equations, then solve for x.
  
• If a problem seems to require too much computation, look for a pattern to find an easier way to solve it.
  

Quadratic Equations

MGMAT Quadratic Equations Test (12/15) - 80%

Quadratic Equations

Things one must know:

• Quadratic equations are equations with one unknown and two defining components:
  1) a variable term raised to the second power
  2) a variable term raised to the first power
• Factoring quadratic equations:
  1) Move all the terms to the left side of the equation, combine them, and put them in the form ax² + bx + c (where a, b, and c are integers). The right side should equal 0.
  2) Factor the equation: find two integers whose sum equals b and whose product equals c.
  3) Rewrite the equation in the form (x + ?)(x + ?), where the question marks represent the two integers you solved for in the previous step.
  4) Since this product equals 0, one or both of the factors must be 0. Set each factor independently to 0 and solve for x.  The two solutions for x have the opposite signs of the integers we found in step three.
• If you have a quadratic expression equal to 0, and you can factor an x out of the expression, then x = 0 is a solution of the equation. Be careful not to just divide both sides by x, this eliminates the solution x = 0. You are only allowed to divide by a variable (or any expression) if you are absolutely sure that the variable or expression does not equal zero.
• FOIL - First, Outer, Inner, Last
• Some quadratics have only one solution, these are called perfect square quadratics.
• When zero appears in the denominator of an expression, then that expression is undefined. Making the the denominator of the fraction equal 0 would NOT make the entire expression equal zero.
• Three common quadratic expressions:
  #1 x² - y² = (x + y)(x - y)
  #2 x² + 2xy + y² = (x + y)(x + y) = (x + y)²
  #3 x² - 2xy + y² = (x - y)(x - y) = (x - y)²
• If the other side of the equation is a perfect square quadratic, the problem can be quickly solved by taking the square root of both sides of the equation. Be sure to consider both the positive and the negative square root. 
• You can substitute a variable to create a quadratic.
• Determining the number of solutions a quadratic has: b² - 4ac
  If the result is greater than zero, there will be two solutions. If it is equal to zero, there will be one solution. If it is less than zero, there will be no solutions.
  

Equations with Exponents

MGMAT Equations with Exponents Test (11/15) - 73%

Equations with Exponents

Things one must know:

• Equations that involve variables with even exponents can have two solutions: both a positive and a negative solution.
• If exponential expressions appear on both sides of the equation, rewrite the bases so that either the same base or the same exponent appears on both sides of the exponential equation. You can then eliminate the identical bases and rewrite the exponents as an equation. Be careful if 0, 1, or -1 is the base.
  
• Square both sides of the equation to solve problems that involve variables underneath radical symbols. Be sure to check the solution in the original equation.
  

Basic Equations

MGMAT Basic Equations Test (14/15) - 93%
If I remember correctly, the score I got the first time was pretty horrible.

Basic Equations

Things one must know:

• Solve simultaneous equations by substitution or combination:
  Substitution
  1. Solve the first equation for x
  2. Substitute this solution into the second equation wherever x appears
  3. Solve the second equation for y
  4. Substitute your solution for y into the first equation in order to solve for x
  Combination
  1. Line up the terms of the equations
  2. If you plan to add the equations, multiply one or both of the equations so that the coefficient of a variable in one equation is the opposite of that variable's coefficient in the other equation. If you plan to subtract them, multiply one or both of the equations so that the coefficient of a variable in one equation is the same as that variable's coefficient in the other equation.
  3. Add or subtract the equations to eliminate one of the variables
  4. Solve the resulting equation for the unknown variable
  5. Substitute into one of the the original equations to solve for the second variable.
• Look especially for shortcuts or symmetries in the form of the equations to reduce the number of steps needed to solve the system.
• A master rule for determining whether 2 equations involving 2 variables will be sufficient to solve for the variables:
  1) If both of the equations are linear - that is, if there are no squared terms (such as x² or y²) and no xy terms - the equations will be sufficient unless the two equations are mathematically identical (e.g., x + y = 10 is identical to 2x + 2y = 20)
  2) If there are any non-linear terms in either of the equations (such as x², y², xy or x/y), there will usually be two (or more) different solutions for each of the variables and the equations will not be sufficient.
• Try to manipulate the given equation(s) so that the combo (e.g. x + y) is on one side of the equation.
  Four easy manipulations (MADS):
  M: Multiply or divide the whole equation by a certain number
  A: Add or subtract a number on both sides of the equation
  D: Distribute of factor an expression on one side of the equation
  S: Square or unsquare both sides of the equation
• If after manipulating the given equations in question or statements in DS so that the combo is isolated on one side of the equation, if the other side of an equation from a statement contains a value, the equation is sufficient. If the other side contains a variable expression, that equation is not sufficient.
• Avoid trying to solve for the individual variables in combo questions. 
• Equations involving absolute value:
  1. Isolate the expression within the absolute value brackets
  2. The rule, once we have an equation of the form |x|= a with a > 0, is that x = ± a. Remove the value brackets and solve the equation for two cases:
  CASE 1: x = a (x is positive)
  CASE 2: x = -a (x is negative)
  3. Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal.
• In case of integer constraints, there might be many possible solutions among all numbers, but only one integer solution. Solve for one variable and then test numbers (answer choices)
• You can multiply or divide two complete equations together, because you are doing the same thing to both sides of the equations. Multiply the left sides of the two equations together and also multiply the right sides of the equations together. Set those products equal to each other. You can also divide two equations. Do this when it seems that you can cancel a lot of variables in one move.
  

FDPs

Just finished the MGMAT FDPs book and the OG problems that go with it. I feel I could have done better. I have had so little time to study lately and that's really showing. At this pace there is no way I will be ready for 2010 intake.

Back to basics (cont.)

MGAMT Advanced FDPs Test (18/23) - 78%
Didn't get some of the percent word translation stuff.

Data Sufficiency for FDPs

Things one must know:

• Rephrase questions and statements into equations in order to keep track of what you know and what you need to know. Your ultimate goal in writing equations is to combine them in such a way that you are left with a single equation with only one variable. The variable in the equation should represent the quantity you are asked to find in the original question
• Beware of statements that introduce too many variables, these are usually not sufficient to answer the qeuestion.
  

FDPs

Things one must know:

• Common FDP equivalents (MGMAT FDPs p66)
• Prefer fractions for doing multiplication or division, but prefer decimals and percents for doing addition or subtraction, for estimating numbers or for comparing numbers.
• FDPs and word translations (MGMAT FDPs p68) - X percent is (X/100)
  
  Advanced:
• Repeating decimals: Generally, just do the long division to determine the repeating cycle. However, if the denominator is 9, 99, 999 or another power of 10 minus 1, then the numerator gives you the repeating digits (perhaps with leading zeroes).
• Terminating decimals: When written as a fraction and if, after being fully reduced, the denominator has any prime factors besides 2 or 5, then its decimal will not terminate. If the denominator only has factors of 2 and/or 5, then the decimal will terminate.
• Unknown digits problems:
  1) Look at the answer choices first, to limit your search.
  2) Use other given constraints to rule out additional possibilities.
  3) Focus on the units digit in the product of sum.
  4) Test the remaining answer choices.
• When you work with formulas that act on decimals, avoid shortcuts and follow directions precisely.
• Fractions and Exponents & Roots: The effect of raising a fraction to a power varies depending upon the fraction's value, sign and the exponent. Be ready to generate outcomes with test numbers such as 1/2 or 3/2. Not that taking the square root of a proper fraction raises its value toward 1.
• Use a mixture chart for weighted average problems that involve percents and percent change. You can also use the formula, and leave the percents (keeping the % sign) when you solve.
• You can plug a value that is a fraction, a ratio, a decimal or a percent straight into percent change equations as long as you keep the labels straight.
• Estimating decimal equivalents:
  1) Make the denominator the nearest factor of 100 or another power of 10.
  2) Change the numerator or denominator to make the fraction simplify easily.
  3) Small percent adjustment, increase/decrease the result by the same percent you increased/decreased the nominator/ denominator.
  Try not to change both the numerator and denominator, especially in opposite directions.

Percents

Things one must know:

• A part is some percent of a whole:
  PART / WHOLE = PERCENT / 100
  Set up a proportion like above to solve
• Change is the part, while the original is the whole
• 100% plus or minus a percent change equals the percent of the original quantity that the new quantity represents; ORIGINAL +/- CHANGE = NEW
• You can compute the percent change using the ratio of any two of the following: original, change and new.
• To solve successive percent problems choose real numbers, usually 100 is the easiest, and see what happens
• In a percent problem with unspecified amounts, pick 100 to represent the orginal value
• Simple interest: Principal x Rate x Time
• Set up a mixture chart with the substance labels in rows and 'Original', 'Change' and 'New' in the columns, this way you can keep careful track of the various components and their changes.
  

Back to basics (cont.)

3. MGMAT Percents (14/15) - 93%
4. MGMAT FDPs (13/15) -  87%

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