Quant Flashcards
When I do smart numbers
I’ll avoid weird numbers like 0, 1 and fractions and the numbers in the problem
If I want to check divisibility by 9
I check to see if sum of digits is divisible by 9
What are the first 5 prime numbers?
2,3,5,7,11
What are the 6th to 10th prime numbers
13, 17, 19, 23 ,29
If I want to check divisibility by 8
I can divide by 2 three times
or
I can check if the last 3 digits are divisible by 8
If I want to check divisibility by 6
I need to make sure it’s divisible by 2 & 3
If I want to check divisibility by 4
I can divide by 2 twice
or
I can check if the last 2 digits are divisible by 4
If I want to know if a # is divisible by 3
I sum the digits of the number (Integers only) and check if it’s divisible
If N is a divisor of X & Y
Then N is a divisor of X + Y
When checking divisibility of a large number by another large number
I can check if it’s divisible by a factor pair of the divisor
ex 6750 div by 18
6750/2 works
and
6750/9 works
2*9=18 so it works
√ 2
approx 1.4
√ 3
Approx 1.7
11^2
121
13^2
169
14^2
196
15^2
225
16^2
256
25^2
625
4^3
64
5^3
125
10^3
1000
It is a zero per power
When do you Test Cases in Problem Solving?
You use it when the problem asks a must be or could be question
When do you Test Cases in Data sufficiency
Use it when several unknowns and it’s a Y/N question
x^2 - y^2=
(x+y) (x-y)
x^2 - 2xy + y^2 =
(x-y) (x-y) or (x-y)^2
x^2 + 2xy + y^2
(x+y) (x+y) or (x+y)^2
When should you use smart numbers?
Problem solving only for variable expressions, relative values in answers or the problem never gives a real #
When should you work backwards?
Problem solving only when answers are nice real numbers or if there is only 1 variable
If I see
x^2 - x < 0
or
x^2 < x
They both say that 0 < x < 1
If I see XY < 0
I know X and Y have opposing signs
If I see XY > 0
I know X and Y have the same sign
(a + b)/c =
a/c + b/c
If two denominators share a factor
It is faster to find the least common multiple instead of cross multiplying.
ex: 6&8 least common multiple of 24
-(3)^2=
-9
(-3)^2+
9
3^4=
81
x^2=16 x=
4^2 or -4^2
x^3 * x^4 =
x^7
always add the exponents for multiplication
x^3 + x^4=
Can only be changed through factoring
x^3(1+x)
(x^y)/ x^2 =
x^(y-2)
x^-2 =
1/x^2
(x^2)^4 =
x^8 you multiply with exponents
(ab)^3=
(a^3)(b^3)
(x/y)^2 =
(x^2)/(y^2)
Prime factorization of 18^3 is
(2^3)(3^6)
18= 29 =2(3^2)
(2*(3^2))^3= (2^3)(3^6)
(2^3)*(3^3)=
(6^3)
Only works if they have the same exponents
How do you estimate a root of a non perfect square ex:70?
Look for the perfect squares nearby and estimate
ex: 64^0.5 = 8 and 81^0.5= 9 so 70^0.5 is approx 8.5
What happens when you square root a number between 0 & 1?
You get a larger number because division by a number less than 1 and larger than 0
What is the main difference in cube root vs root main in regards to negativity
You can cube root a negative number but not normal root one.
8^2/3 =
(8^1/3)^2 = (2)^2 = 4
√x * √y =
√xy
√x/√y =
√x/y
When doing the radical of a large number with no obvious perfect square factor, what tool can be used?
Prime factorization
ex: √360= √(2 * 2* 2* 3* 3 * 5) = √((2^2) * (3^2) * 2 * 5) =2 * 3 * √10 = 6√10
√((3^10)+(3^11)) =
√((3^10)(1+3)) = √(3^10) * √4 = (3^5)2
m^(5y) * m^(y+5) =
m^(5y+y+5) = m^(6y +5)
1/8 in decimal is =
0.125 or 12.5%
3/8 in decimal is
0.375 or 37.5%
5/8 in decimal is
0.625 or 62.5%
7/8 in decimal is
0.875 or 87.5%
When multiplying decimals what technique should be used
Multiply as if no decimals then add it in after by counting how many digits were behind the decimal originally
ex: 0.75 x 0.2 = 75 x 2 x .001=0.150
Times 0.001 cuz three behind decimals like 0.150
How to structure a x is y% of what number?
Z = unknown y= in decimal
x = yz so x/y = z
ex: 16 is 2% of what number?
16 = 0.02z so 16/0.02 = z
1600/2 = z = 800
When I see “x in terms of y”
I know to isolate “x” and get it in the value of “y”
ex: x= 4 + 3y
If I see a variable is an exponent
I know to factor bases into primes
ex: 3^x= 27^4 -> 3^x = 3^3^4
3^x = 3^12 so x=12
What are the 3 exceptions for bases of exponents
1, 0 and -1 because they always lead to the same number (except -1 which can lead to 1 or -1 depending if even or odd)
When using the substituting method
I know to isolate the variable not mentioned in the question.
When a question asks for “x+y =”
I don’t need to find each individual solution just the combined x+y value
When I come across a difficult multiplication like 102*301
I know I can distribute the multiplication using foil
ex: (100 + 2) * (300 +1)
= 30,000 + 100 + 600 + 2
= 30,702
When I see a quadratic with a number before the x^2
ex: 2X^2 4x + 4
I know to first factor out the number and then tackle the quadratic. Same thing for a minus sign as well
Ex: 2( x^2 + 2x + 2)
If I see a problem like x^2 = 4
I know to square root both sides and use the negative and positive results
Ex: x = 2 & x= -2
When I see a hard square like 306^2
I know I can do:
300^2 + 2 * 300 * 6 +6^2
Because of (a+b)^2 = a^2 + 2ab + b^2
When I want to do percent larger
I know to use the formula ((a-b)/b) * 100
When I see a straight line
I know its the shortest distance between two points
when I see two or more intersecting lines
I know their middle angles add up to 360
When I see two lines intersect I know the angles opposite
are equal to each other. aka Vertical Angles
When a transversal line cuts through two parallel lines
I know the angles are the same for both lines
I know that when you add any acute angle with any obtuse angle
That both angles add to 180 degrees
When I see the symbol ||
I know the lines mentioned are parallel
ex: MP ||AB
When I see a parallelogram
I know that the opposite sides and opposite angles are equal
When I see a shape with 4 sides
I know the angles inside add to 360 degrees
When I see a shape with 5 sides
I know the angles add to 540 degrees
When I see a shape I know I can calculate the amount of angles with
(n - 2) * 180 where n = amount of sides
When I cut a 4 sided shape with a line connecting opposite sides
I know it makes two triangles with 180 degrees each
To get the area of a trapezoid
I know to do (B1+B2)/ 2 * height
To get the area of a parallelogram
I know to do Base * Height
When I see a question asking for the surface area of all faces
I know it’s asking for the sum of all faces
When someone is asking for the surface area of a cube or rectangular solid
I know it has 6 faces
Volume for a 4 sided object is:
Length * Width* Height
When I’m asked about fitting a 3D object into another 3D object
I know that knowing the volume of each is not enough. It depends on dimension
When trying to find the third side of a right sided triangle
I know to use the Pythagorean Theorem
a^2 + b^2 = c^2
When I see two sides of a triangle are equal
I know that their angles are also equal
Triangles sides are bounded by what two restraint
- A side of a triangle can’t be more than the sum of the two other sides.
(t1+t2) > t3 regardless of which side!! - A side of a triangle can’t be less than the sum of the difference of the other two sides
(t1- t2) < t3 regardless of which side!!
Every right triangle is composed of what?
Two legs (a & b) and a hypotenuse (c)
The right angle is formed by the legs a & b
When I see a triangle with two of the sides 3 - 4 - 5
I know it’s a right triangle.
9 + 16 = 25 for the Pythagorean theorem
Multiples include:
6 - 8 - 10
9 - 12 -15
12 - 16 -20
When I see a triangle with two of the sides 5 - 12 - 13
I know it’s a right triangle with the values:
25 + 144 = 169 for the Pythagorean theorem
Multiples include:
10 -24 -26
When I see a triangle with two of the sides 8 - 15 -17
I know it’s a right triangle with the values:
64 + 225 = 289 for the Pythagorean theorem
When I see a triangle with two of the three sides equal that has a right angle
I know its a 45 - 45 - 90 triangle
When I see a 45 - 45 - 90 triangle
I know I can find the sides through this formula:
Leg : Leg : Hypotenuse
x : x : x√2
When I see a square with a given diagnoal
I know I can use the 45 - 45 - 90 ratio to find the sides of the square which are equal to the leg of the
45-45-90
When I see a equilateral triangle (all 3 same sides) cut in the middle to form two other triangles
I know it turns into two 30 - 60 - 90 triangles
When I see a 30 - 60 - 90 triangle
I know I can find the sides using this ratio
Leg : Leg : Hypotenuse
30° : 60° : 90°
x : x√3 : 2x
When I see a exterior angle of a triangle
I know I can find the interior angel through
180° - exterior angle = interior angle
I also know that the two other interior angles = to the exterior angle
When I think about the base of a triangle
I know it’s important to remember that each side of a triangle can be the base
When I see a triangle that the sides are multiples of each other
I know that they have the same angles
When I see triangles with the same angles
I know that they are multiples of each other
When I see a line pass through the center of a circle
I know it’s the diameter of the circle
When I have the radius of the circle
I know I can find the diameter, circumference and area of a circle
What are the formulas of a circle if radius is equal to r
Radius = r
Diameter = 2r
Circumference = 2 * r * π
Area = π * r^2
When I need to estimate a value with π
I make π = 3
When I see a center angle and an inscribed angle
I know the inscribed angle is equal to 1/2 of the center angle.
If I see an inscribed triangle where one side is the diameter
I know that the triangle must be a right triangle
When I see a question involving quadrants on a coordinate plane
I know the quadrants are ordered as follows:
Q2|Q1
Q3|Q4
I know that if I have two points on a line and I want to find the slope
I need use the formula
Rise/Run or (y2-y1)/(x2-x1)
It is not important which point is x1 or x2 as long as it’s consistent
When I see a question asking for an x intercept
I know it’s the point where y = 0
When I see a question asking for a y intercept
I know its the point where x = 0
When I see it’s a linear equation
I know it cannot have terms such as x^2, √x or xy
Linear equations are usually represented in the following form
y = mx + b
Always reorganize to isolate y
When I see an angle BAC
I know that A is the mid point
When X & Y are positive , if 3x < 2y
Then x < y and same goes for other situations where the larger quantity has the smaller multiplier
When I want to check divisibility by 11 in a 3 digit number
I know to see if the hundred digit and the single digit summed equal the tens digit.
ex: 165/11 = 15
1 + 5 = 6
When I have a 3 set problem
I know to use the formula:
Total = Group A + Group B + Group C - (AB) - (BC) - 2(ABC)
What is 1/11 in decimals
0.0909090
What is 2/11
0.1818
What is 3/11
O.272727
4/11
0.3636
How do I find x/11 if x = 1 to 10
= to 0.9x repeating
Ex: 8/11 = 0.72727
10/11 = 0.9090
2^5
32
2^6
64
4^3
64
3^4
81
1/6 in percent is
16.66% or 16&2/3%
1/9 in decimals is
0.11111
What happens if you add 1 to both the numerator and denominator of a fraction
The fraction gets closer to 1
