Quant Flashcards
Special triangle 1
3, 4, 5 or 6, 8, 10 or 1.5, 2, 2.5 or any multiple of that
Special triangle 2
5, 12, 13 or 10, 24, 26
Isosceles triangle
x, x, xrt2
30, 60 90 degree
x, xrt3, 2
Other odd spec triangles`
8, 15, 17 or 7, 24, 25 or 9, 40, 41
Similar triangles, given 2 parallel sides
If share angle, then they are similar
*corresponding angles and prop sides; area is sq of proportion of sides
Similar triangles, given 2 parallel lines with vertical angle
Vertical angles are the same, corresponding angles are the same
*corresponding angles and prop sides; area is sq of proportion of sides
Similar triangles, both have right angles
If share angle, then the last angle must have same measurement
*corresponding angles and prop sides; area is sq of proportion of sides
Consec int: avg # odd consec int
always integer
Consec int: avg even # consec int
not an integer
If # of int is odd, sum is
divisible by n
If # of int is even, sum is
not divisible by n
Product of K consec int is divisible by
Product of K evenly spaced int is divisible by
K!
(because if you divide the evenly spaced by K!, you’ll get the set of consec int - figure out # of terms in set and if # is a multiple of K!)
3 consec int
if middle is odd, then 2 evens on the side (one divis by 2, other divis by 4), always divis by 3
2 consec int
must be even int, so product ALWAYS even
E +/- E
E
O +/- O
E
E */ E
E
O */ O
O
Sum of 2 primes = E or O?
E, unless one’s a 2 and thus it’ll be O
OEOEO (5)
O bc 1 pair of O makes E and single O make O
EOEOEO (6)
O bc 1 pair of O makes E and single O makes O
If unit digit is 0, 1, 4, 5, 6, 9
it’s a perfect square
if unit digit is 1, 5, 6**
Any power of that number has same unit digit
Sum of DISTINCT prime is odd
then it’s a perfect square (49–> 7 which is odd)
Perfect square (# total factors?)
Odd # of total factors (1, number, number)
10!+7 is a multiple of
7 because 7 is in both 10! and 7
Evenly spaced set, mean
= median (doesn’t matter if it’s odd number or even number of set)
= (1st + last) / 2
a^x*b^x
(ab)^x
x^2=-9
no real solutions!
x^2-x
0
1/x > 1/y
when +x< +y
1/x > 1/y
when x<y and both are negative
Area of equilateral triangle
rt3/4 *s^2
Area of rhombus
D1D2 / 2 (D is diagonal) or bh
*Must know D, can’t assume rhombus has 90 deg
Diagonal of a square
x rt2
Diagonal of a cube
x rt 3
Square maximizes ?? (V or perimeter)
Volume
Rectangle maximizes ?? (V or perimeter)
Perimeter
Total average speed
2 S1 S2 / (S1 + S2)
*distance must be the same or same time with diff speeds
1/A + 1/B = 1/T
To complete 1 task, 1/T to finish task in 1 hour
Plug in: if square roots
Plug in 1 or -1
Represent 10s digit
10x+y
If there’s no shared primes, then GCF is ? LCM is ?
GCF = 1, LCM= product
1 even integer in a set, product is divis by?
2 even integers in set, product is divis by?
2
4
Never forget __ as a factor
1
If asks for ratio and doesn’t specific integer,
it could be anything
5! (what’s a multiple)
10
2^4
16 (think 4^2)
2^5
32
2^6
64 (think 4^3)
2^7
128
2^8
256 (think 4^4)
2^9
512
2^10
1024 (good to know as proxy to 1000)
3^3
27
3^4
81
3^5
243
5^3
125
5^4
625 (think 25*25)
1/6 ~
16.7%
1/7 ~
2/7
3/7
14%
28%
42%
1/9 ~
2/9
3/9
11%
22%
33%
3!
6
4!
24
5!
120
6!
720
7!
5040
What’s LCM for 6 and 4
Write out primes: 2, 3 and 2, 2 so together it’s 2, 3, 2 that will work for both
What’s GCF for 6 and 4
2
Only ___ inequalities
ADD! no -, *, /
Sum of consecutive integers cubed
Is always a perfect square (hint if ask for sum, make chart with numbers and sum so 1+8+27 so sum is 1, 9, 36
Square root of cumulative sum (of consecutive integers cubed)
Is always the sum of consecutive integers
1,9,27 so 1,1+2+3 etc
D, E more likely in
C more likely in
middle
end
interior angles sum
(n-2)*180
Diagonal of a rectangular solid
SQ RT OF (x^2+y^2+z^2)
sphere volume
(4/3) pi r^2
Sphere area
4 pi r^2
Surface area of cylinder
2 pi r^2 (wrap around) + 2 pi rh (2 circles on top and bottom)
x=-b+/- sq rt (b^2-4ac) / 2a
if b^2-4ac is >0
if b^2-4ac is =0
if b^2-4ac is <0
2 x-int
1 x-int
0 x-int
Parabola f(x)= ax^2+bx+c
If a is large
If a is small
graph is narrow
graph is wide
1/11
.09999
2/11
.18
basically something /11 is
multiple of .09
18^2
324
19^2
361
17^2
289
16^2
256
3^9)^(3^9)
3^(93^9) bc multiply exponents but can multiple 93*9
3^(3^23^9)=3^ (3^11), don’t just say 3^(93)^9 bc you’re exponentially counting
of 2’s or 3’s in set
Start w bigger factor and write out numbers - then can simplify so write 3^3, 325 etc
2 is evens but also need to check division by 4, 8 etc bc there’s so many 2s
If stuck
Plug in numbers!!
Manipulate numbers so if ask for 47*49 try
(48-1)(48+1)
If multiply by consec integ, product always
Product always a factor of ea consec integ
Manipulation of exponents often clue -
Need to factor out if its 1 difference so 9^a = 10*9 - 9^2 bc you factor out 9 and get (10-1)
Gcf
Lcm
Product of Every factor in middle - even if There are repeats
Product of all factors (don’t double count ones already in middle); can also calc by taking the product of the numbers / GCF
Sum of consec int
Median *n and you can test if median is int when n is odd, median isn’t int when n is even
first term: A, last term: A+(n-1)d; so sum is [A+A+(n-1)/d]n/2 —-> so n/2[2A+(n-1)d]
N(n+1)/2 only when starts with 1
If you’re given 2+4+6, factor into 2(1+2+3)
Patterns # of pairs*avg where avg=(1st +last) / 2 and n is ((last-first)+1)/2
Divis by 4
Last 2 digits divis by 4
Divis by 6
If even and divis by 3
(X-1)(x-4) which #s are positive
Negative - btwn 1 and 4
Pos - all else
Perpendicular slopes
Negate reciprocal
3, -1/3
Sum of consec integers - how to get a variation without trying crazy numbers
If you’re set is 1 2 345 then try -1,0,1,2,3,4,5,6
Helps prove insuff if number in set is important
If you want to find min,
Ex: x _ 55 _ 2x+20
max the rest to get min x
x 55, 55, 2x+20, 2x+20
Probability - desire outcome for AB
AB and BA possibilities so rmbr to multiply by 2
- calc probability of AB
- calc total sequences (AB, BA so 2)
- multiple 2* probability of AB
Probability - select 2G and 2B out of 3B and 3G
Can use perm/comb:
Numerator use 3C2 * 3C2 because want G AND B, and think like a team
Denom use total ways so 6 total C 4, also think like team bc don’t matter AB or BA
Probability if want to choose 2 girls out of 5, 10 ppl total
5/104/9 or think of it like perm bc not “team” so 5!/3! so 54 in numerator
Probability tips
- identify max / min to check and elim answer
- committees / teams - think combination bc don’t want to dbl count
- arrange books / ppl in line - think permutation bc order matters need to count all
- # of ways / total <- perm/comb may help find
Group grid
Use Venn if there’s no outside area (not A or B or AB)
Use group grid if there’s 4 possib
Identify max / min to eliminate / check answer
Triangle
length of largest side is < sum of 2 sides
3 circles in venn diagram
- sum all 3 =total and create a bunch of equations with each circle then add /subtract equations
- a+b+c-(all groups of 2) - 2*(group of 3)+ group of none
* ** don’t forget *2
- make sure the groups of 2 are also “clean numbers” and you don’t double count with group of 3
Rhombus vs square
Rhombus: 4 equal sides, opposite sides and angles parallel, diagonals bisect; DOESN’T HAVE TO HAVE 90 DEG
- is a parallelogram, not a square
Square: 4 equal sides, opp sides parallel, all 90 deg angles, diagonals bisect
- is a parallelogram and rhombus
Trick: Estimate
Replace decimal / roots w fractions and vice versa or estimate
- if frxn, can elim by looking at >1 or <1
- if geom, set limits to how big / small something can be
- if overestimate, answer is less
Trick: pick values
pick 0 or 1
if see “except ?s” find how other answers are the opposite
Trick: plugging
don’t plug if it doesn’t fit in obv place, elim choices first
2 eq, 2 var
if you see 3 var need 3 eq (even if 2 of the eq only has 2 variables bc you can solve for the 2 var first and get last)
Ineq th
A>b and b>c then a>c
x^a + x^b
BEWARE it’s not x^(a+b)
x^a* y^b
BEWARE it’s not x*y^(a+b)
3^a*2^a
(3*2)^a
Wtd avg
Find distance between end points and average, use that proportion and flip it to get the proportion of the weights for each end bucket
prime v factor
factor = 1,2,3 --> don't forget 1 as a factor prime = only 2, 3
1/8
.125
3/8
.375
5/8
.625
7/8
.875
D=RT
Work together, Opposite direction - add rates
Same direction - subtract rates
Work separately - add distance and time, not rates
Tip: find the gap
Random pick
Ineq - guess C or E
Middle choose ?
End choose C
Median and mean if they equal, formula is
(A+b+c+d)/4 =(b+c)/2
Once you solve you get a+d=b+c
So much easier to prove than w numbers
When you think about remainders,
the divisor (bottom number) must be at least 1+remainder in order to get that remainder - remainder can never be > divisor
See set of numbers
Think even odd, 2x+1
Rt 3 estimate
1.7
3^k+3^k
2*3^k not 3^2k
A^6-b^6 can also be thought of as
|a|=|b| so once you know one is pos/ neg, can tell you a lot
Divis by 6
Needs to be even and a factor of 3 ( bc divis by 2 and 3)
If a number has a factor
Always divis by factor! No need to waste time thinking of others
When triangles are similar, all sides have the same
Ratio! So easy to solve if you know one ratio
Tangent on circle
2 tangents lines form an angle, minor arc = central angle, outside angle is 180-x bc below
That angle is supplementary to the angle formed inside the circle (kite like shape)
2 chords inside circle form angle x and minor arc=2x
When comparing inequalities for fractions
Must make sure the base is 10 not 1/10 bc that changed the direction of inequality
Increase by 1/3
Means multiply existing by 1/3
10x the previous number
Think 10^n because its likely to be 1010…
When you pull out 5^(k-1)-5^k
Can do 5^k(5^-1 -1) so its 1/5-1 inside
Rt3
1.7
Rt 5
2.25
Smaller? -2.3 or -3
-3, think to the left in number line
Has less than twice as many x as y
X<2y
4^y+4^y
2*4^y
WRONG
4^2y (exponents make it too big, there’s just 2 that you’re adding, only if it’s a multiplication)
x^2 - x<0
0<1
absolute value equations, to check, you MUST
plug soln back into ORIGINAL eq
reciprocal of inequalities:
if you know the sign of variables, can you flip the inequality x<y
the reciprocal is 1/x ___ 1/y
if both positive or negative flip
if opposite signs, don’t flip
squaring inequalities:
do you flip the sign of inequality
if both are positive - don’t flip
if both are negative, flip when you square
if opp signs, or unclear - CAN’T FLIP, don’t know
direct vs inverse
easier to just combine equations so
Direct: y1/x1 = y2/x2
Indirect: y1x1=y2x2
Calc # of terms
Last number - make sure it’s one that is included
First number - is not included in set
last - first + 1 = # of terms
sum between 2 numbers
avg (of the 1st and last) * # of terms (be careful how you calc # of terms - pay attn to even/odd, incl last but not first)
max / min of parabola
o Y=a(x-h)^2+k then max is (h,k)
o Or, solve for –b/2a to get x, then plug in x to get y
4/x < -1/3
if x is neg
12>-x (switch sign, but don’t add a negative)
|z| <1
-1re faster
when you see a number ^2 is < itself
x^2<x
then x must be pos fraction
0<1
max / min length for triangle / polygon
triangle: a-b < ? < a+b
polygon: max is ? < sum of all known sides
min is largest given side - sum of rest (but no min if largest given side is not bigger than the sum of rest)
12*18
216
18*30
540
if A>B, then if A increases by x% and y decreases by x% is net increase/ decr?
increase bc A>B initially
Repeating decimals
if denominator is 9, 99, 999 look at numerator for repeating digits
Ex: 23/99=.2323
Ex: 1/9=.11111
Tens digit
use 10x+y - helps you figure out factoring or other tricks
Ratios if you have 10% and final amount is 70%
don’t find 100% then 70%, go straight to 70%
x/y=.1/.7
Number of terms in a set (evenly spaced)
(last-first)/increment + 1
** add 1 after you divide out increment
4b=5a, b must be a mult of ___ and a must be a mult of ____
b must be a multf 5 and a must be a mult of 4
rt6
2.5
rt 2.5
1.6
of distinct factors
count every number even repeats
Ex: 49= 1, 7, 7 so 3 total
Angle in circle - central angle is ___ when it’s corresponding arc is x
what’s inscribed angle
x (where angle touches center)
if inscribed angle (angle doesn’t touch center but rather is on the circle), then x/2
rt 6=
rt 5=
rt 2.5=
rt 1.6 =
- 5
- 25
- 6
- 2 (just gets closer to 1)
Benchmark: rt1 =1 rt2= 1.4
alternate strategies
- plug in answers - back solve
- smart numbers
- estimate / benchmark
2 similar triangles, ratio is a:b what’s Area ratio
a^2: b^2 for triangles
If it’s a circle, and radius is a:b then Area is a^2:b^2 too
Area - 1:4 then radius is 1:2
13n / m
which cannot be a divisor of 35x and 20y
m must be 13 or 13 * a part of n or just n
– think how can I cancel out denominator using factors in numerator
same concept, you should try dividing 35x / choice 20x and 20y / choice 20x. 35x/20y=7/4
Add / subtract multiples of 3
Add / subtract non multiples of 3
Add / subtract multiple and non multiple
will still be divis by 3
may be divis by 3 (19+13=22 NO; 19+14=33 YES)
will NOT be divis by 3
If x/N and y/N are integers
x+/-y is also divis by N
If m contains 2, 3 as factors
n contains 2,5 as factors
m*n contains —-
223*5 as factors
of factors for complex numbers like 9450
2 * 3 * 5^2 * 7
- add 1 to ea exponent: 1+1, 1+1, 2+1, 1+1
- Multiple together: 48 factors
Perf squares have ___ # of total / distinct factors
Perf squares can be expressed as ____ number of prime factors
The p could have ___ number of factors
Sum of perf square factors is ___
odd
even
odd or even
odd
Divisors means
factors
Prime factors vs factors
Prime factors/ factorization: break out 12= 2^2*3
Factors: 1, 2, 3, 4, 6, 12 (think pairs)
DRT- if you travel twice as fast and the difference in time is 15 min, then what’s original time? D is same
Think ratios: R: 1:2 T: 2:1 so T is 30:15 –> diff is 15
- 2T (where this is the orig since it takes longer)+T=15 so 2T=30
Or solve D/2x=T-15 and D/x=T get T=30
When you see question that asks what the percent change or percent incr/decr
Make sure you find the actually change so new -old / old not just the difference!!
Percent question, be careful
Make sure you write exactly what question asks, if its “of all bulbs turned on what’s the percent…” Make sure you circle xx/ bulbs turned on. Don’t solve for xx/all bulbs
Percent terms
- when to use 100-x%
reduce by x%
% fewer than x
