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Combinations: “4 choose 2”
Combinations: “5 choose 3 or 2”
Anything applied to one side of an equation must be applied to the other side
Factoring, multiplying, & reducing: apply to all added or subtracted terms
Watch out for parentheses and addition or subtraction within fractions.
Distributing exponents & radicals
apply only to multiplied or divided terms
total interest = (interest rate)(time)
Look for repeated terms among equations and apply transitive property.
You can stack equations and add/subtract
Simplifying Funky numbers or decimals
1) Prime factor and/or use scientific notation.
2) Stay away from large calculations - just write out the math; Numbers will often reduce in the end.
3) Look to rephrase decimals or percents into fractions.
Fractions
1) Alone: reduce by canceling from all added/subtracted terms in numerator/denominator.
* Simplify multi-level fractions.
2) In expressions: get common denominator by multiplying all added/subtracted terms by “1”.
3) In equations: eliminate them (multiply all added/subtracted terms by common denominator -- BOTH sides of eqn)
4) Multiplication: reduce terms before multiplying across.
Common factors
1) In expressions: factor from each added/subtracted term**
2) In equations: cancel from each added/subtracted term**
** (watch out for parentheses).
Repeated variables
Combine by adding or subtracting before factoring, multiplying, and/or dividing.
Ex) add example ;)
Exponents & Radicals, Part 1
1) Prime factor everything!!
2) Radicals: pull out perfect squares; eliminate radicals in denominator.
* Look to rephrase radicals into exponents.
* Radical in equation: isolate radical then square both sides (eliminate radical).
3) Negative exponents: rephrase into reciprocals and/or fractions into negative exponents.
Exponents & Radicals, Part 2
4) Common bases: combine terms (apply “one-step down” rules).
* Adding/subtracting common exponents: factor terms (no “one step down”).
5) Adding/subtracting exponents: look to break out into multiple bases
Exponents & Radicals, Part 3
6) No common bases: look to create common exponents and use distribution rules.
* Distribute exponents/radicals only to multiplied/divided terms.
7) Subtraction of terms with even exponents: expand into (x + y)(x – y).
8) x^{2} in an equation: set equal to zero and factor. *Rephrase classic quadratics.
9) Square root of each side yields positive & negative result (squaring both sides = positive).
Inequalities
1) When multiplying/dividing both sides by negative, switch direction of the sign.
2) DS: look to pick numbers or multiply/divide both sides by variable whose sign is unknown.
3) “xy < 0” or “xy greater than zero”: think in terms of positive/negative rules.
4) Multiple inequalities: Stack inequalities in same direction and add (can’t subtract).
5) Absolute value w/ inequality: create 2 inequalities, switch sign direction for negative option.
1) With a grin, read over everything (including answer choices) before writing anything.
2) Given any apparent trigger, immediately do the right thing (ignore what you don’t understand)
3) Apply specific approach: pick numbers, use answer choices, create equations, define elements,
look to assign variables to unknown values , simplify, and/or use logic.
When there are variables in answer choices and/or no actual numbers
Picking Numbers:
Fractions, percents, or ratios in the question/answers AND no actual numbers
1) Pick a workable number for an unknown value (sometimes pick 2 sets).
Fractions: pick common multiple for an unknown whole (following “of”)
Percents: pick 10 or 100 for an unknown whole (immediately following “of”)
Ratios: pick common multiple for largest term OR unify terms in fractions.
2) Use the picked number to define everything
3) Use the numbers from step 2) to answer the Q
Picking Numbers:
Must/CANNOT/Could be true
Pick acceptable numbers (when necessary).
* Pick a) -1, -1⁄2, 0, 1⁄2, 1, 2
b) the same numbers (for x and y)
and c) extreme numbers.
*You must have an intuitive understanding of the logic of the question.
Picking Numbers:
Variables in the answers
1) Pick basic and different numbers for each variable (do not pick 0 or 1).
2) Use the picked numbers to solve the problem by generating a specific answer.
3) Plug the picked numbers into each choice, looking for the same specific answer.
Picking Numbers:
Fractions and/or percents w/ no actual numbers AND variables(s) in choices
1) Pick a common multiple for an unknown whole.
2) Use the picked number to define everything you can figure out, incl. what remains.
3) Identify both the resulting value of the variable and the specific answer to the question.
4) Plug the number for the variable into each choice, looking for the specific answer.
Variables as percents and in answer choices
Try to pick easy & absurd numbers (like 0,10, 50 or 100), but watch for repeats
A tricky word problem with basic numbers in the answers (esp. “least/greatest”) Backwards Method:
1) Set the question equal to answer (B) or (D)
2) Using the number in (B)/(D), write down everything you can figure out.
* What remains???
3) Determine whether (B)/(D) “fits”.
Data Sufficiency: 3-Step Method
(Step 1)
1) With a grin, read over everything (including statements) before writing anything.
* A lack of information means anything goes (x could be fraction, 0, negative, etc.)
* Physical entities (positive integers) can create limitations.
* Watch out for inequalities.
Data Sufficiency: 3-Step Method
(Step 2)
2) Consider rules of multiple equations, picking numbers, specific approach, or just use logic.
Data Sufficiency: 3-Step Method
(Step 3)
3) Rephrase and/or simplify anything and everything.
* Look to create multi-variable equations, inequalities, and/or expressions.
* Place the question mark appropriately!!
* Look to rephrase statements into the form of the question or given information.
* Look to plug simplified statements into the question or given information.
rules of multiple equations (#1)
1) “x” linearly independent equations are sufficient to solve for “x” variables
* Non-independent equations: always simplify, looking for insufficiency: (2 equations that are the same)
* Non-linear equations: check for sufficiency (could be single value) Ex) x^{2 }or xy is not linear (x or y could have 2 values).
* Physical entities/geometry: non-linear is sufficient (only one value)
rules of multiple equations (#2-4)
2) Compound expression [Ex: (x – y), ] in the question: look to rephrase the
statements into the same form as that expression.
3) Addition of 2 variables w/ weird coefficients; all terms positive integers: sufficient??
* Pick numbers or see “Algebra” for solving.
4) Equation in question stem: look to plug into the question (rare).
Data Sufficiency: 3 Steps for Picking Numbers (Step 1)
(inequalities, absolute value, prime, digits, statistics)
1) Pick number(s) for variable(s) consistent with the statement and the given information.
* Pick -1, -1⁄2, 0, 1⁄2, 1, 2, the same numbers (for x and y) and extreme numbers.
* Try to pick numbers consistent with both statements (when feasible).
Data Sufficiency: 3 Steps for Picking Numbers (Steps 2 and 3)
(inequalities, absolute value, prime, digits, statistics)
2) Plug those numbers into original question and take note of the answer (yes/no/value).
3) Repeat step 1) using different numbers, trying to get a different answer found in step 2).
* Combining statements: use prior numbers and look for commonalities.
NOTE: No numbers and question asks for fraction, ratio, or percent: pick #s for “whole”.
Fractions in word problems
* Actual numbers: set total = x and define each term (especially what remains).
* No numbers: set total = x OR pick number(s).
* “What fraction of x is y?”, “x is what fraction of y?”: “of...” = denominator.
* Look to break out complex fractions:
x+y/z = x/z+y/z
x^{2} with an equation: set = 0, factor, solve for x
Memorize the three classic quadratics:
1) (x+y)^{2} =(x+y)(x+y) = x^{2 }+ 2xy + y^{2 }
2) (x–y)^{2} =(x–y)(x–y)=x^{2} –2xy + y^{2}
3) x^{2k} – y^{2k} = (x^{k}+ y^{k})(x^{k} – y^{k})
x^{2} – 1 = x^{2} – 12 = (x – 1)(x + 1)
Even/odd; positive/negative; “xy < 0”:
think in terms of rules, try NOT to pick numbers.
2k = even
2n – 1 = odd
0 = even integer (not + or –)
x^{even} > 0.
Multiples/factors, divisible by, integer
* Prime factor and reduce, but don’t eliminate fraction (usually). * Variable in numerator: variable is multiple of denominator. * Variable in denominator: variable is factor of numerator.
pick numbers, plug in answers, or rephrase as 2 equations (positive/negative)
Remainder
pick numbers, plug in answers, use remainder = decimal x divisor
M/F = I
Consecutive numbers: Sums
Sums: recall formulas or rephrase as x + (x + 1)
Sum = Average*Number of terms
Average = (first + last/2)
Number of terms = (first - last +1)
Products: think about prime factors.
Consecutive multiples of x
Consecutive multiples of x: factor out the x
1) Total is provided: add elements using common variable (3x + 5x + 7x...)
2) No total: rephrase as fractions, looking to unify or combine algebraically; pick numbers?
1) Statement: “x is y percent of z”: x = y/100(z)
2) Question: “x is what percent of y?” OR “what percent of y is x?” = x/y (sometimes x 100).
3) Word problem: use the grid (label the “part and percent”), look to pick numbers.
Part = Percent*Whole
1) Use the grid, looking to assign variables & relative values, maybe pick numbers (rare).
2) One element (D, R, or T) is often the same.; look for totals or catching up/overtaking.
1) Same work for each element (filling a tank, producing 500 units): apply 1/A + 1/B = 1/Total
2) Different work (total work differs from individual): apply grid (infer rates from stem).
Overlapping Sets
1) Mentally confirm the overlap (“can have/be both...”) to identify the 2 groups.
2) Apply summation grid, looking to assign variables and relative values.
3) DS: rephrase statements in context of the grid.
Counting, grouping, ordering, or probability of multiple events
1) Apply slots method, looking to use permutation or combination formulas.
2) Order matters when switching 2 elements would create another possibility.
3) "At least" questions: look to subtract from 1.
Geometry
1) Problem-solving drawn to scale, so look to guesstimate.
2) Data Sufficiency (not drawn to scale): look to write down geometric equations.
3) Apply rules of triangles and look for radii in circles.
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