Quant Strategies Flashcards
Absolute value
Complex abs val where there’s multiple, like |4x-7| = |x-8|, what’s x?
”- Only need to solve for 2 scenarios, both are positive, any one is - /+; like x-2 = 2x-3, and x-2 = - (2x-3)
- DON’T FORGET TO PLUG BACK IN TO CHECK”
Backsolving
Follow the flow of the problem, plug in the choices
Careless Typo - don’t write J as 1
Slow down
Careless 9 and a in a problem
Squigly a and straight 9, be careful
Careless 19+7
26
Combo If you see the question asking for 2 variables, ie x/y =?
- Simplify question if you can
- Try to manipulate answer choices to solve the combo, not indiv var”
Combo If DS option is combo, ie x/y and the question asks what xy=?
TEST CASES: Instead of plugging it in, you can just say x could be 4, y could be 2
Combo You don’t need to solve if in DS, you have combo in stem
- if you can manipulate the choices to the same as the problem, or the 2 choices are actually the same - not C
DS Data Sufficiency
AD/BCE, BD/ACE
DS When you have a gnarly fraction, look at answer choices to try to manipulate it there, ie (2t+t-x) / (t-x)
”- Signal: 2t+t must be like that for a reason, don’t just go 3x
- 2t / (t-x) + 1 bc that’s in answers”
DS Don’t forget to test which number?
“1; practice starting with 1 unless it’s not allowed
- Helps you get faster with testing cases”
DS When you have equation in question
Don’t assume that it’s given, write out equation with “?” for each stem
Equations System of equations
”- Figure out what to solve for, x
- Isolate other variables, y (put y on 1 side, and rest on other side), plug into 2nd equation
- Practice both subsitution ^ or add/subtract equations (if it’s already laid out)”
Equations 2 variables in 1 equation
Can’t solve if you have 2 variables
Equations If the question has x and y, asks what y could be and gives you multiple choice, solve for…
Solve for y –> x = y something, that way, you can plug y into the equation
Equations If you have 2 equations, you can; ie xy=2, yz=3, xz=4
Multiple, add, subtract together; ie x^2y^2z^2=24
Equations If you have 3 equations where all variables don’t have coefficient,
Add the 3 together, then you can sub 1 equation with 2 variables at a time
Equations If problem asks for what the smallest possible value is, |37-5y|
”- Signal: 5y is a divisor, so you can think of 0, 5
- Other strategy is to test number”
Exponents Same exponent but different base in a fraction (no addition / subtraction in between)
“Consolidate the fraction and put it all over exponent
- if unsure, check if it works the other way”
Exponents Same base with exponent, if division, you can
If division, you can subtract exponents. If multiplication, you can add
Exponents Exponent to the exponent (2^3)^4
Multiple the exponents: 2^12
Exponents See exponential terms being added (11^3 + 11^2)
Can’t multiple exponents, try factoring out
.Exponents Fraction in exponent (8^(2/3)
“Can’t split fraction by multiplying it out with the same base; not 8^2*8^(1/3)
- Instead, simplify base and see if you can x or / exponents”
Exponents Memorize 4^n, ending unit pattern is
4, 6 | 4, 6
Exponents Memorize 9^n, ending unit pattern is
9, 1 | 9, 1
Exponents Memorize 5^n, ending unit pattern is
Always 5
Exponents Memorize 6^n, ending unit pattern is
Always 6
Exponents Memorize 7^n, ending unit pattern is
7, 9, 3, 1 | 7, 9, 3, 1
Exponents Memorize 3^n, ending unit pattern is
3, 9,7, 1 | 3, 9, 7, 1
Exponents Memorize 8^n, ending unit pattern is
8, 4, 2, 6 | 8, 4, 2, 6
Exponents Memorize 2^n, ending unit pattern is
2, 4, 8, 6 | 2, 4, 8, 6
Exponents Memorize that n^n ends in n, single digit options are
1, 5, or 6
Exponents Memorize that n^n ends in only 2 digits
4 or 9
Exponents -10 * 10^2
-(10)^3, not -10^1
Formulas Sequence of numbers, given value of a6, need to find a100, if pattern is +3
Find gap, 100-6 = 94 * 3 (because it’s just +3)
Formulas when you see complicated formula, and variable is also complicated f(2rt3)=2x^4-x^2
”- Simplify formula without complicated variable first, x^2(2x^2-1)
- Then plug in
‘- Don’t try to simplify with all the complicated variables, easy to mess up”
Fractions Ugly fraction in equation
”- Get rid of denominator first
- Cross multiply”
Fractions 2 fractions divided by each other
Multiple top 1 and bottomest one / middle 2
Fractions If see many terms in a fraction, if see same terms in num and den
Split the numerator to turn 1 of the frxns into 1
Inequalities if you see 2 inequalities, 1>1-ab>0, what to do if you want to -1?
Remember to do the same for each of the 3 (-1 for all 3, multiple -1 for all 3)
Inequalities If you have 2 separate inequalities, x+a<2, y+a<3
You can add them (make sure same direction of the sign), never subtract or divide; a
Inequalities Retailer has less than twice as many dogs as cats
“d<2c
- Use numbers, D=4, C=2, so you know its D=2C
- But less than so D=5 or 3, 3 so it’s D<2C”
Inequalities If asked for what the max product of 2 variables, and you’re given 2 inequalities. Also, in general, what’s the strategy
Test all 4 extremes (don’t forget negatives). Don’t forget 1000, -1000
Inequalities If you know that x is negative and you have to square x<3
“Flip ineq: x^2>9
- Doesn’t work if x can be positive”
Inequalities If you know that x is pos, can you square x>3 (both left and right sides are pos)
Yes, “Don’t flip ineq: x^2>9
- Doesn’t work if x can be negative”
Inequalities There are dogs and cats, there are more than 5 dogs
D>5, not D=5+C
Negative - Adding negatives
Pay attention, double check, slow down
Negative - Subtracting 3-4
Pay attention, double check, slow down
Negative - Equations with negatives
Do 1 thing per line, don’t try to save time
Negative - Negative in answer
Don’t forget to choose the answer with negative!
Negative If there is no parenthesis, ie -2^4
the negative does not distribute with an exponent
-16, not 16
Proportionality If you see 2 sets of 2 variables in ratio form, think –
”- Direct: y=kx, so you can do y1/x1 = y2 / x2
- Inverse: y=k/x so y1x1 = y2x2”
PS If you see answer choices, if it’ll be hard to determine right equations (ugly quadratics / fractions), or formulas are messy
”- Backsolve from answer choices
- Start with B or D (if can’t tell to go up or down, do B, then do D and see how much closer you are)”
PS If get fraction, but need whole numbers in answer choice when backsolving
Look out for even choices
Quadratic If you see variable in denom and different powers,
Don’t move constant to other side
Keep them on same side, because you’re going to quadratic equation; may work for linear equations
Quadratic If you see all answer choices with rt 5, you know you can
divide out by that number
Quadratics Typically if you see an equation with ^2 or greater
Watch out for 2 solutions, often x=0 is 1 of them
Quadratics Memorize x^2-y^2
(x-y)(x+y)
Quadratics Memorize x^2 - 2xy + y^2
(x-y)^2
Quadratics Memorize x^2 + 2xy + y^2
(x+y)^2
Quadratics If you see 2xy
Simplify by moving it to the side with x^2 and y^2
Quadratics If question asks for x^2+x2y+y^2
Simplify it to (x+y)^2, always simplify to the possibilities
Quadratics To figure out if there is no solution, 1 or 2 solutions, look at
“Discriminant (b^2 - 4ac) from ax^2 + bx + c = 0, so if it’s +4ac then c is negative
- if +, 2 solutions
- if = 0, 1 solution
- if -, 0 solution”
Roots If there’s a x or / inside the root,
Can break it into 2 roots
Can’t for +/- i.e. sq rt (16+25)
Roots (sq rt of 4) ^ 5
”- Don’t get confused with sq rt, first simplify to 2^5
- Or, it’s 4 ^(5/2)”
Roots Memorize sq rt 2
1.4
Roots Memorize sq rt 3
1.7
Roots Memorize sq rt 5
2.25
Roots Memorize sq rt 169
13
Roots Memorize sq rt 196
14
Roots Memorize sq rt 225
15
Roots Memorize sq rt 256
16
Roots Memorize sq rt 625
25
Roots 10^3
1000
Roots Any time you simplify a sq rt, you must remember
+/- the simplified answer, 2 solutions
Roots When you have gnarly roots, try
“Squaring it instead of calc the root
- See if it’s close to a square, like 49, 64, 81, etc.”
Smart numbers How to pick smart numbers?
”- Don’t pick 0, 1
- Pick odd and even numbers
- Don’t pick #s already in the problem”
Smart numbers When to pick smart numbers?
”- When there are no real numbers, just variables/%/frxn; if there is any real #, try plugging in choices
- Variables in answer”
Mental math Divide by 5
Divide other number by 2, multiply by 10
Subtraction When there’s a carryover
Write out the carryover instead of doing it in your head
Test cases Often with DS in inequalities
“Test 0, 1, frxn, odd/even, neg frxn, neg odd/even (unless constraint)
- 0, 1, 1/2, -1/2, 2 or 3, -2 or-3”, really big and really small numbers (extremes)
Variables Combination of variables (x+y = ?)
”- Almost always don’t need to solve for x, y separately
- Figure out x+y together”
Word problem See difficult word problem without any real numbers just variables/%/frxn, and variable in answer
”- Use smart numbers
- Be careful- plug in the correct smart number”
Word problem If the questions asks for a ratio
Don’t multiply out everything, try crossing out things
Word problem If you see constraints in the problem, like y<1
Eliminate all answers where y isn’t <1
Word problem Be careful of units, but don’t need to change all units if the units cross out; ie mph / sec = mph / sec (direct proportional)
Can leave mph without converting to sec
Word problem - ask yourself
Can I backsolve, instead of algebra
Inequalities: If more than twice as many dogs as cats
D>2C
When equation is y=x+1, that means y will always be xxx than x
y will always be 1 greater than x
Inequalities: if question asks: is a>b, stems have a+2b = 1
Test cases
OR plug in , 1-2b>b, insuff bc is b<1/3? don’t know
Testing cases: if you have answer choices and constraints in problem, you can
- simplify answer choices, 51 only has 1, 51 or 3, 17 so if you know you can square the numbers < 100, 51 doesn’t work.
- start with max answer choice if ask for max
Guessing randomly strategies
- choose answer that looks most similar to other answers, least similar than numbers in problem (narrow down most similar 3, than to similar 2, then see if there’s another reason
- SIMILAR TO CHOICES, DIFFERENT TO PROBLEM
To figure out if you should go for it or guess and move on: If you recognize how to solve it / know strategy
If you get to 2 min and still far from finish line
Do it- but if you’re still far from the finish line, get out
If you’re confused - move on, don’t even try
If minute away, don’t rush or give up at the very end
How confidence you’ll get to answer VS how quickly you can do it
If you see roman numeral problem, and you find 1 of them is wrong,
cross out all choices with that choice
Can you square rt (x-y) = A?
can square it and (x-y) is positive
If you have decimals,
convert to fraction (often can cross out denom)
When you square a pos fraction, it gets
smaller than the original (closer to 0)
When you square a neg fraction, it gets
bigger than the original (closer to 0)
0.9999999 translates to
1 - 0.00000001 which becomes 1 - 10^-8
Think 0.1 = 10^-1
- 1 translates to …
0. 1^n translates to
10^-1
10^-n
Step 1 - read the problem AND….
look at answer choices, you can eliminate things as you go through
.375
3/8
.167
1/6
.625
5/8
.8333
5/6
GCF
Largest divisor
Distinct factors
Don’t forget 1, and any repeats
x^2 = 1, x, x (3 factors)
Trapezoid area
A= 1/2 * (b1+b2) *h
Area of piece of pie in circle if given angle s
A of pie / A of circle = s / 360
Central angle vs Inscribed angle in circle
2x vs x
Special rt triangles
x, x, xrt2 –> 45, 45, 90
x, xrt3, 2x –> 30, 60, 90
Pythag triplets
3,4,5
6,8,10
5,12,13
8,15,17
Ratio of angles in triangle vs ratio of sides
a deg > b deg > c deg
opposite a deg > opposite b deg > opposite c deg
Surface area of a cube
6x^2
Slopes
y coordinates / x coordinates
Vertical has no slope
Horiz has 0 slope
Perpendicular slope
negative reciprocal
2–> -1/2
Area of a sector
Fraction of the total area of the circle
Fraction = angle / 360
V of Cylinder
pi *r^2 *h
Total sum of angles of a polygon
(n-2)*180
What makes an impossible triangle
diff of 2 other sides < side < sum of 2 other sides
Exterior angle (outside the triangle) =
Sum of 2 interior angles on the opposite side of triangle
Similar triangles mean
2 angles are the same
Area of rhombus
d1*d2 /2
Largest area if given the perimeter for any quadrilateral
square
Smallest perimeter if given the area or a diagonal for a quadrilateral
If given area, square
if given diagonal, really skinny rectangle (diagonal ~ long side)
Area of equilateral triangle
[s^2 rt 3] / 4
similar triangles have a:b sides, area ratio will be
a^2:b^2
diagonal of a cube
s rt 3
diagonal of a rectangular solid
sq rt of x^2 + y^2 + z^2 –> super pythag th
Max perimeter of a quadrilateral when they give you an area is 48
infinite
(10^-infinity) * 48 x 10^infinity
Max area of triangle
Right triangle
If you can’t figure out how to get a length of a triangle
Try finding the A, then changing where you draw the b and h to calc one of the sides
If y=x is the perpendicular bisector to a line, to find the other point….
Flip the point given (x,y) –> (y,x), watch out for signs (visualize it)
x^2 + bx+c, if they’re asking if the products of the roots are +/- and the stem says c<0
c = rs so rs<0, you know that roots are - when multipled
3 phrases of inequalities 1+2x < x+5 < 2x + 6
you can split them into 2
OR try to get rid of the outer x’s so you’re left with 1 in the middle (do the same for each clause)
2^4 - 5^2, how do you solve?
Simplify; (2^2 - 5)(2^2+5)
Smallest side of a triangle and largest side
difference of 2 other sides < SIDE < sum of 2 other sides
When checking choices, look out for
Integers - if question asks for one, eliminate all without int
Parallelogram has —- angles
2 pairs of equal angles
When you have quadratic for geometry problem
negative doesn’t matter bc can’t have negative length
When given a geometry problem of the size of something without telling you the sides
Think extremes - big volume, but can have the tiniest height, vs skinniest width but highest height
When you lay a cylinder flat, the height is
the diameter or 2*radius
Isoc triangle in circle must be in
center of circle (r and r `are its sides)
Assume similar triangles if you suspect it!!
xx
Weighted averages trick
Identify it is one bc it gives you ratios in a group
ALWAYS BETWEEN the 2 ratios and closer to the heavier one
Check answer by
Check constraints, makes logical sense.
For DS, check constraints in question AND STEM
REREAD PROBLEM
play devil’s advocate
box was 1/2 taller than it was before –>
1.5x
Split the pot and given % and # type question. How to tackle?
- Plug in choices
2. Figure out the remaining % which equals the only actual number given
1st step in quant - practice
Glancing at choices
If far apart can estimate
As you go through, start eliminating - if something is supposed to be small, cross out all big, +/-, fraction vs integer, odd vs even
Estimate or benchmark
Use round numbers, get in the range
When testing cases
Identify what they’re testing or trying to trick you, test those
30 is what % of 50
Shortcut is
30/50 = x%
Faster calc with odd percent like 7%
Do 10%/2 to get 5%
Then 1%+1%
Compounding interest
Trick - new value vs old value
Compound quarterly if given annual rate
A=P(1+r/n)^nt
– n is if quarterly, but you can first annualize the rate
Divide annual rate by 4 for quarterly
May be easier to just go year by year if calc is too hard. 100 becomes 102 becomes 104.04
Also can try $1 if asks you to compare to original
For 2nd year, don’t need to add end of 1st yr + end of 2nd yr, 2nd year already accounts for end of 1st yr
New value isn’t too different than original, takes a lot of years to accumulate
Watch out when given a time series - original amount is x, in 5 years
Original amount is Y0, don’t put Y1, Always write Y0
In addition or substraction, if % increase or decrease is the same, then total
Ex: Gross (-2%)- Deduct (-2%) = profit (??%)
will also change by same % increase or decr
-2% also
If % changes differ, can’t subtract add %s
Smart numbers trick - find lowest, how?
Think about how low you need, if there’s a fraction, go as low to get integer, but don’t go higher. Need to think and pause about which number is lowest
For ratios, if you have 3 variables, you just need
2 ratios of the 3 var to solve
Tens and units digit, see if there’s pattern using
10x+y (ex, 10x+y vs 10y-x, you know the common factor is 9)
Given fraction, figure out if terminating decimal
Don’t forget numerator, may cross out something in denom
3, 7, 11 etc. prime #s in denom after fraction is reduced
If given A is dir prop to B, B dir prop to C, A is
dir prop to C.
If Q asks for A if given C, you just need A to C ratio or you need both A to B and B to C
15*6
90
Ratios, sum of the ratios must be what compared to the total
Divisible by total amount
2:3:5:6, sum is 16, so the answer cannot be total 192 which isn’t divs by 16
Sum of consec integer, you can figure out range
Take smallest and largest number * #of terms in range
If you know there’s at least a pair consec numbers in a set, you know there’s duplicates and
Subtract 1 from any number yields same number of distinct values
Slope greater than 1
Steeper than y=x line
In circle, if point is on the circle and creates an angle to the other side of the circle, the length of arc is
2* angle
Inscribed angle
Prime numbers <30
2,3,5,7
11,13,17,19
23,29
When you see change in %, you can either
Estimate
OR benchmark - look at choices and start eliminating if it’s clearly not 50% or higher
when you have SD (d) and mean (m), the area below m+d
50% + 1/2 of 1 SD – don’t forget 50%, no need to add up that small tail
Hidden FDP problems, when they talk about arranging number of people in groups of or tables of x, look out for
look out for factors, common divisors
When the Q asks for big enough area to cover 2 spots, think
size and distance apart
which fractions can you take the first 1/# and multiply it for subsequent ones
1/9 –> try 5/9=.115
1/11 –> try 5/11=/095
1/6 –> try 5/6 = .167*5=.833
1/7 –> try 5/7 = .14 *5=.7
Be ok with switching smart numbers
or even variables to solve for (total instead of the variable they ask)
33^43 ends in what digit. TRICK
Divide by 4, not 3
Word problems - when you see work, wage, earnings think
work produced or earnings or $ PAID OR LABOR COST = wage * # hrs
Rates - when 2 people walk towards each other, you can —- rates. Typically ask to figure out the delta in difference
add rates. Use the total distance / sum of rates = time. With time, you can apply indiv rate to get dist of ea
Rates - when 2 people walk away from ea other, you can – rates
add rates
Rates - when 2 people walk in the same direction, you can — rates
subtract rates
when you have a slower and faster rate, the average is
NEVER THE simple avg
**ALWAYS closer to the slower rate (smaller #) (estimate if needed) – doesn’t apply to 2 working together
To get average rate of 2 workers
To combine 2 workers and get their rate together
Add distance and time –>
Add rates –>
Rate problem steps
Read through problem, get sense for what is being asked (avg rate?)
Think what is needed
Simplify what’s given and derive the real Q
When you see estimate within 0.5, think
+/- 0.5, so test the extremes
Overlapping sets - WATCH OUT FOR some % or number OF…. VERB
where, who, ….can change denominator
Look for OF…..VERB
Overlapping sets - minimum can be
0, not 1
simultaneously and indep working together means you can add rates?
add rates
count the book ends of a range vs not, how to count how many #s in between (ie. number of int between 1-11 not incl 1 and 10)
try with smaller range
if have evenly spaced set, mean is what to median
mean = median
Count integers in a set (divis by 7 btwn 100-150)
Last - first / incr + 1
Find last and first divis by 7 - 105, 147
(147-105)/7+1
Sum of consec integers
Avg * # of terms bc avg = sum/# of terms
Trick to finding sum of 1st 3 int in a set of 6 when given last 3 int sum is 624
n+3+n+4+n+5
-(n+n+1+n+2)
=9 so take 624-9
(only works for even number set, all n’s cross out)
terms
(High # - Low #) / interval + 1
Interval is 2 for even or odd
** avg is add, # of terms is subtract
Simplified Q for set of consec integers –>
What is the value of any of the numbers?
Weighted avg - given the 2 scores (80, 100) and weighting (40%, 60%) , find weighted avg score
- Find diff of scores = 20
- Multiply by HEAVIER weight = 20*.6 = 12
- Add to LIGHTER side 80+12
Could flip 2 and 3, just take OPPOSITE ONES
weighted avg - given 2 scores (80,100) and weighted avg of 92. Find weighting of the 100
- Denom is diff of scores so 20
- Num is diff of the longer dist so 92-80=12
- 12/20 is heavier weighting (since you took longer)
If weighted avg is closer to 1st number
1st number is weighted heavier, 1st number is teetered down
weighted avg - if you get the 2 distances from the weight avg, to get the ratio, ex: 10—12—–16, ratio fo the 10 to 16 is
just flip the 2 distances
4:2 not 4:6
if question asks for what value of 100x+200y, given x+y=1 and choices are 10, 150, 250
WEIGHTED AVG PROBLEM IN DISGUISE
x+y=1 is hint!
Can’t be 10 or 250 - has to be btwn 100 - 250
When you label average and median, use
Me -for mean and Md for median
If # of terms * 1 of the numbers = # of terms *avg, then avg =
median
variance=
SD^2
What components needed for SD
difference between mean and each int, # of int
or variance
Number prop - strategy
test cases
sum of an odd number of consecutive integers is divisible by (i.e. 2, 3, 4)
the number of items
-divis by 3
sum of an even number of consecutive integers is divisible by ie 2,3,4,5
NOT the number of items (avg of even set is not an integer so sum isn’t divis)
avg is 3.5 so sum is 13, not divis by 4
number divis by 8 if
last 3 digits are
or if you can divide by 2 3 times
number divis by 9 if
sum of digits divis by 9, just like 3
12 is equal to 3n where n is an integer (really saying)
12 is divis by 3
if N is a divisor of x and y, then N is a divisor of what?
x+y or x-y (adding/ subtr multiples will yield another multiple)
if a is a factor of b and b is a factor of c, then
a is a factor of c
if 72 is divis by 12, then all factors of 12 are divis by 72
if sum or product of 2 prime numbers is even, you know 1 of them must be
2
97/7=
NOT 13!!! IT’S 13.SOMETHING
2! is
3! is
2
6
4!
24
5!
120
6!
720
When there are repeats in the combinatorics,
for ex 2 bronze and 3 silver
divide by repeats
total! /2!3!
when there are 2 groups (tricky bc may say choose from 7 colors, pick 2)
1st do the 1: T!/(1!6!)
Then 2nd T!/(2!5!)
THEN MULTIPLY THE 2 TO GET TOTAL
when you see AND (2 events happening) in comb / prob
multiply
when you see OR in comb / prob
add
when you see at least 1, trick is to
1 - P(none)
when you see at most 1, trick is to
P(none) + P(1)???
LCM calc: 100, 140
Find all prime factors, for each unique factor, find the largest multiple
100: 2^25^2
140: 2^25*7
LCM: 2^2 * 5^2 * 7
perfect squares contains only odd or even power of primes?
even (everything is doubled)
144 = 2^4*3^2
If you see 3x+1 is prime, the +1 signals x should most likely be
even
check your work / calc!! when
- if you finish quickly
- if there’s simple calc (longer calc less likely to get choices if you screw up)
- if you’re doing something “new”, may have to come up w strategy on the fly so need to play devil’s advocate
- -> Slow down, don’t worry about speed
- -> Question every step’s logic of problem (think about trap) - NOT REDO ACTUALLY CALC (10 #s divis by 3, then does 6 really have 2 3’s)
* *THINK ABOUT WHAT’S CAUSING PROBLEM / CARELESSNESS
Use test trick - in PS, if you want to test multiple numbers, know that
there is only 1 choice, so the cases you test should always yield the same choice (don’t need to test multiple cases)
DS - never guess E unless you know it’s E, esp harder DS Qs
much lower % chance it’s E
x! is a multiple of
any number between 1 through x
10! + 7 is a mult of
7 bc both have 7
when you see rt x + rt y, think of
x + 2 rt (xy) + y
Avg of a set of numbers is
Avg of 1st and last terms or (1st + Last) /2
** avg is add, # of terms is subtract
2^3a * 2^-3b
2^ (3a - 3b)
Overlapping matrix - how do you combine the 2 columns
ADD NOT MULTIPLY
Geometry tricks
- make sure you solve for right thing - area, perim (reread problem)
- Look to Q for clues if stuck - cubic meters –> volume
- test extremes (max area, can triangle be very fat or skinny)
- Use the silent polygon or bigger shape
When you need to work with right triangles, always think
sides (3-4-5, 6-8-10, etc.)
AND
angles (45-45-90, 30-60-90, etc.)
Geometry shapes in PS are drawn to scale
so visualize and estimate!
DS are not
12000/ 25
easier to say 1000/25 = 40 and 12*40 = 480
Rate word problem tricks
- visualize
- simplify Q, so if given R and ask for T, simplify and say what’s D?
- like overlapping set, fill in as much of grid as possible, more suff
if given a few factors (1, 2, 3, 4 ) for ex: 24, you can figure out the other factors by
multiply the other factors that you know
so 6, 8, 12, 24
if given median = average and all but 1 numbers in a set, you can
figure out that missing number
overlapping sets - careless mistakes
descriptive label (instead of L, do LB) reread each label in Q
When you want to minimize 1, you
maximize all the other
Geometry trick - triangle
if your gut says similar tirangles, just assume
LOOK FOR RIGHT angle - COULD BE AT THE TOP , RIGHT OR LEFT
Given slope is -3 and line passes through point (-2, r), it hits the origin if r is
6 bc you can get the slope with the other point (0,0) and (-2, r) so slope -3 = r / -2
what percent of x is y
P/100 * x = y or P is y/x*100
rate problem trick - if given person A starts before person B
figure out how much perosn A did first
If you see stems that look similar, giving you same info, guess
A or B, likely a trick that your’e not seeing
UNLESS it’s giving you the same exact info when you simplify - D or E
if question starts using ratios and asks what the total number is, think
add the ratios and find common multiple
ie 2:3 so total must be multiple of 5
and always simplify so 4:6 is 2:3
I have 20% fewer candies than Susan means
I have 80% of Susan
Ron’s DS yes/no steps
Use 2 line device - be explicit with 2 options Top: question, x>2 Bottom: x <=2 Row: A, B, C
Ron’s WP steps
- answer intuitively
- build Excel (no variables)
- plug in info (no variables), draw allll relationships
- solve - now use var. Backsolve, pick #s, est, brute
when you add a width to a frame, don’t forget to
add 2w to the length and width of frame
Ron’s DS 2 line device trick
you can assume that the top row is checked, just need to prove bottom row
20% more in her purse than in mine
Mine *1.2 = her purse
mine is the base since it’s after than
sum of even integers btwn 100 and 200
avg = 300/2 # int is 200-100=100 /2 (bc even) so 50 +1 =150*51
sum of integers btwn 100 and 200
avg = 300/2 # int 200-100=100+1 (no /2 bc it's not even or odd) =150*101
200% more than him - to get to the value of “him”, you need to
take the DIFFERENCE, not just 2x
(him - you / him) = 200%
when you see a figure where things are rotated, beware
counter clockwise or clockwise!!
divisor is the
smaller number that you can divide by
Ex: 2 is a divisor of 10
If you’re picking numbers and question asks to find a % of %, use
1000, not 100
Ex: 16% for 1 yr, 9% for another yr. You need a denominator but you’re given that the 2nd denominator increases by 20% (% of % )
when you have multiple calculations ex: 3x2x2, avoid careless
do each one separately
3x2 = 6
6x2 = 12
when you’re estimating / checking answers, make sure to look at
- range the answer should be in (elim too high / low)
- draw comparisons with existing (OG machine vs new)
- round up and down to estimate
- odd vs even hint
- redraw figures to visualize
- if ask for range, take middle value
- if the answer choice is exactly the number in the problem (likely eliminate)
compound interest (semiannual) vs simple interest
compound interest will be a little bit higher than regular (apply interest more times)
weighted avg estimate - if you have a bunch of numbers with a lower score, the only way to increase your average with 1 score is
if that 1 score is really high
a sphere inside a cube means the radiuses are
radius of sphere = 1/2 side of cube
when you plug in smart numbers, remember to think about
what makes sense - if 2 machines work together, their hours should be LESS than the 1 machine
careless in division - better to write it out
78 / 3 (actually cross out so you don’t put 29 instead of 26)
200% of 1 is
1+2(1)=3
to check: (3-1)/1=200%
when you put the remainder over the divisor ex: 9.12 where remainder is 9
that means 12/100 = 9 / divisor
don’t forget that (10^2)^4 is
10^8 not 10^6
when you have a crazy root (ie 1+rt2) in an equation like x^2+x+1 , you can always
PLUG IN that crazy root
Word problems - alternate methods:
BSEB
- Backsolve, smart #s, estimate, brute force
- Don’t forget to draw arrows for relationships
Geometry - coordinates - if you redraw, make sure
you draw ACCURATELY as possible, if you mess up, redo. It’s better than carelessly making a mistake
if you see the word “ANY” in a question about list of numbers
Ex: sum of ANY 3 numbers is 12
be wary - ANY means every single possible variation HAS to be equal to 10…that’s very restrictive
hint: likely all the numbers have to be the same
Trick when you’re trying to find the total number of 2 variables and you’re given a ratio (B=1.5C) AND the total is btwn 21-28
Take the TOTAL so 1.5C + C = 2.5C or 5/2C and that must be equal a number in the range.
In this case, that total number must be divis by 5
“at a particular time” signifies that at this 1 point in time, everything must happen
can they process 500 calls if there are x machines?
this isn’t is 500 divisible by x. It’s is x >=500 because it has to process it ALL AT ONCE
if you’re given an inequality like A+C >10 and you need to find C, you can assume that ….
more than 1 pair will work so it’s not enough info. Even if there’s a complicated 324A+4598C = 230492834
Also way too time consuming to find…and you don’t have time
Trick to DS per Connie
figure out what the Q is REALLY asking
- trick: fill in as much as you can
when you have a border, don’t forget to LABEL all 4 sides
so if the inside is 8x10, then it becomes 8+2w x 10+2x, it’s not just 8+w
Is 1 prime?
Is 1 a factor?
No, prime starts at 2 (2 is only even prime)
1 is a factor
Number Properties 0 is an integer? Even or odd? Pos or neg?
0 is an integer
Even
Neither pos or neg
of Factors- don’t forget what?
Trick to find # of factors for really big number?
Don’t forget 1
ALLL pairs so for 25 it’s 1. 25, 5, 5
Trick: 1440 = 2^53^2 * 5
Get all exp - 5, 2, 1
Add 1 to each
Multiply = 63*2= 36
when you see 3 groups in overlapping set
x1 = only 1, x2= in 2 of the circles, x3 = in all 3
1) x1 + 2x2 + 3x3 = sum of the 3 groups that include overlap
2) x1 + x2 + x3 + N (for neither) = total population excl overlap
Can solve for any of these x’s
when you see APPROX in the question, be prepared for ugly math
so estimate!
or guess if you don’t have time
Probability of A or B occuring is
P(A) + P(B) - P(A and B occurring)
How many options 2 ppl out of 10?
- no repeating
- order matters (AB and BA are different)
Ex: no. 140
Probability to get women if 50% are women
\_\_ \_\_ (slot machine) 10*9
Prob of women:
Numerator is 54
Denom is now 109
You have total number, need to choose 2 of them. Order doesn’t matter (AB and BA are the same)
Ex: no. 187
10!/(8!*2!)
Includes people not chosen because not being chosen is also a combination
Don’t forget that PS, you should always
eliminate when you can
BSEB
Don’t forget that DS, you should always
test cases
how many times a triangle intersects the circle, think of
times when the triangle TOUCHES the circle (intersect just means overlap so if they share a point, then it intersects)
For geometry, if you have to solve for an angle,
write out as many angles as you know like 180-R before solving
what’s the prime factor of 2^3 * (123) + 1
When you see the +1, it indicates that the prime factor must be greater than 3 since without +1, it would be 3
246 = factor out 2
2^3 * (123)
If the question asks “Did one of the members do something” interpret as
Did AT LEAST 1 do something, not EXACTLY 1 does something
To speed up testing cases,
start with proving the bottom row, flip or do the opposite to prove the top row
when you’re given 2 variables and asked what their greatest prime factor is, think
Test the variables with crazy prime factors like with 7 or 11 or 13 and a regular like 2
10,000 initial amount, 4% annual interest, compound QUARTERLY. What’s amount after 6 mths
Q1 = 10,000(1+.01) = 10.000+100=10,100 Q2 = 10,100(1+.01) = 10,100+101=10,201
Formula = 10,000(1+4%/4) ^ (4*1/2) = 10,000(1+.01)^2
Trick: 10% of 10,000
1% of 10,000
Remove a 0 so 1,000
Remove 2 0’s so 100
