Quant Flashcards
Area of a trapezoid
Area = ½ (a+b)*h (90 degree heigh)
Area of a circle
area = pi * r^2
Circumference
Circumference = 2πr or π*diameter
Length of the third side of a triangle
Third side rule: the third side of a triangle is always greater than the difference but less than the sum of the two sides
Proportions of a 30-60-90 triangle
1x; x√3; 2x (hypotenuse is 2x)
Proportions of a 45-45-90 triangle
1x, 1x, x√2 (hypotenuse is x√2)
Area of a equilateral triangle
s2*√3 / 4 or (s/2)^2 * √3
Area of a parallelogram
b*h but height has to be 90 degrees from the base
Inscribed angles
All inscribed angles that cut out the same arc or arcs of equal length are equal in measure so the angles are all equal
Inscribed central angles
Any inscribed angle that cuts out the same arc as a central angle is exactly one-half the measure of that central angle
X^-y / X^-z
X^-y-(-z)
8^-12 / 8^-9
8^-12-(-9) = 8^-3
(8^-11)(8^-5)
8^-16
(x^y)(x^z)
x^y+z
X^-y / X^z
X^-y-z
(X^-y)^z
X^(-y*x)
X^y / X^z
X^y-z
X^y / X^-z
X^y-(-z)
((x^y)(x^z))^a
(x^ya)(x^za)
(X/Y)^-z
Y^z/X^z
X^-y
1 / X^y
(x)(x^a)(x^a)
x^2a+1
(X^a)^b
X^ab
2^3 * 2^-5
2^-2
X^0
1
4^2 * 3^3 / 12^2
Find common bases so the denominator looks like the numerator: 4^2 * 3^3 / 4^2 * 3^2 = 3^3 / 3^2 = 3^1 = 3
2^(3)^2
2^9 (do what’s in the brackets first)
Exponentials: find common bases
When there is a equation with prime number on one side and non-prime on the other, break the non-prime down to it’s prime base that matches the prime on the other side of the equation - so 75^y = 5^4 becomes (3*5^2)^y=5^2 = 3^y * 5^2y = 5^4 - now realize that for 5^4 to equal 5^2y then y must be 2 and substitute y=2 into the rest of the equation to solve
When exponents are added or subtracted (such as 3^8 + 3^7 - 3^6 - 3^5)
Exponents need to be multiplication so turn addition/subtractions into multiplication by factoring. Take the smaller number in the equation and factor it and use the difference between the factored exponent and the added/subtracted exponent. For example, 3^8 + 3^7 - 3^6 - 3^5, 3^5 is the smallest number so use that to factor: 3^5(3^3 + 3^2 - 3^1 - 1) you’ve taken 3^5 out of all the other equations and remember to have the 1 at the end to represent the number outside the brackets. Then do the math inside the brackets so 3^5(32) or 3^5(2^5)
Factoring exponents when not all whole numbers are the same for example: 2^5 + 4^4
Find the common base: break the non prime exponent so it matches the prime exponents: so 4^4 is (2*2)^4 = 2^4 * 2^4 = 2^8
Adding or subtracting negative exponents for example: 2^-5 +2^-6 + 2^-7 + 2^-8
Factor as normal but within the brackets use positive exponents (because when you multiply by the negative exponent outside the bracket, it’ll become a negative number) and remember to use the smallest number which is the largest negative exponent. For example: 2^-8(2^3 +2^2 + 2^1 +1) = 2^-8(15)
When exponent bases are the same, for example: 3^x-1 = 3^13
When the bases are the same, they can be removed because you only need to exponents to find what x is. So x-1 = 13
x = 14
What is (2x+5y)^2
4x^2+20xy+25y^2
What is (3y-2x)^2
9y^2-12xy+4x^2
What is 25x^2+30xy+9y^2
(5x+3y)^2
What is (16x^2)-1
(4x+1)(4x-1)
What is (5y+2x)(5y-2x)
25y^2-4x^2
What does (x+y)(x-y) =
x^2-y^2
What does x^2-y^2 =
(x+y)(x-y)
What is 49y^2-9x^2
(7y+3x)(7y-3x)
Is -a^4 positive or negative?
Positive, any negative number raised to an even exponent is positive
Is -a^5 positive or negative?
Negative, any negative number raised to an odd exponent is negative
How to set up ratios
One over the other = the ratio
30-60-90 triangle: what is the length of the side opposite 60
x*sqrt3
What are the two common side ratio triangles
3:4:5 and 5:12:13
Three element ratio problems - how to set up?
set up the ratio as a:b:c = x and then apply numbers to the ratios so you can work out what x is (for example, If Jim ate four times as many hot dogs as Leon but 1/2 as many hot dogs as Billy) it would be L:B:J and L+L4+L8=78 or L13=78 and L=6
What is (x-5)^2
x^2-10xy-25
What does x= in x^2=9x
9 or 0
Sequences; when you are given a formula/constant such as q # r = (q+3)(r-2) and another formula such as 1 # z=-8
Then just substitute the second formula into the first. So 1=q and z=r so (1+3)(z-2)=-8 = 4(z-2)=-8 = z-2=-2 = z=0
When dealing with exponents remember to break down non-prime number. For example 5^2/10^3 is
5^2/2^3*5^3
When there are exponents being divided such as 5^55/2^30 * 5^30 how can you simplify?
Minus the exponents of the denominator from the numerator so you can remove the number in the denominator: 5^25/2^30
Geometry: don’t assume a right triangle is a 30-60-90 triangle.
It may be a 45-45-90 triangle or 3:4:5 or 5:12:13 - all you know is that one angle is 90
Geometry: don’t assume where the letters on, for example on a XYZ triangle.
N/A
When dealing with equations that have a mixture of exponents and non-exponents, convert the one without variable to match the form of the variable. For example, 2^a * 3^b * 5^c = 270,000,000 is:
2^a * 3^b * 5^c = 27 * 10^7 then always break non-prime exponents into prime: 2^a * 3^b * 5^c = 27 * 2^7 * 5^7 so you you know that a=7 b=3 c=7
With speed/distance questions remember to write out all the information available to you, such as “He rode 2 hours at a average speed of 40m/h”
Then you know he rode 80 miles
If an algebra problem or sequence is confusing, try giving the variable a random number.
N/A
Difference of squares, if there is a number with an exponent larger than ^2 then half the exponent when it’s distributed, for example (2^8)-1 is
(2^4+1)(2^4-1) = (16+1)(16-1) = 17*15 and 17 is the prime factor
How can 6^8−3^8 be simplified
3^8 * 2^8 - 3^8, then factor the common terms so 3^8((2^8)-1)
Remember to add permutations when there are multiple options. If person X can’t sit at the edges of the 5 seats then it’s x _ _ _ _ or _ _ _ _ x
so it’s 4! + 4! = 48
Collision problems:
Combine the two speeds and divide by the distance to find how long it will take to meet. If the trains/cars are leaving at different times then take the distance travelled by the earlier entity before the later one leaves and take that off the total distance, but remember to add that time to the final answer.
Trick to finding the number of unique factors
Express the number as a product of its prime factors (24 is 2^3 * 3^1) then drop the bases and add 1 to each exponent (4 and 2) then multiply the exponents together (4*2=8 which is the number of unique factors in 24)
Finding the prime factors of a integer
Create a number tree and stop the tree at a prime number then add all the prime together (for example if there are two 2s, then it’s 2^2)
Don’t forget about negative numbers and 0 in DS questions
N/A
Calculating the number of unique pairs
Take the total number of options and multiply it by the total minus 1 and divide by two
10^8 / 10^7 is
2^8 * 5^8 / 2^7 * 5^7 = 10
Sequences: if the question ask for the first 5-10 terms then just do the math. For example, what is the 7th term of this sequence if the first term is 4: An = 2A(n-1)-3
A(2)=(2*4)-3 =5 A(3)=(2*5)-3 =7 A(4)=(2*7)-3 =11 A(5)=(2*11)-3 =19 A(6)=(2*19)-3 =35
Sequences: if the question asks for a term that’s far away, do the first few terms and find the pattern. For example, what is the sum of the first 700 terms of An = A(n-1) - A(n-2) if the first term is -2 and the second is 2
A(1) = -2 A(2) = 2 A(3) = 2 - -2 = 4 A(4) = 4 - 2 = 2 A(5) = 2 - 4 = -2 A(6) = -2 - 2 = -4 A(7) = -4 - -2 = -2 A(8) = -2 - - 4 = 2
Now just find the how many times the sequence goes into 700 but notice the sequence adds to 0 so you need to be find that 6 goes into 700 116 times with a remainder of 4. The first four terms of the sequence add to 6 so that’s the answer
If numbers are within brackets without variables, then perform the equation within the brackets before expanding it. For example, what is -1(-1-2)^3
-1(-3)^3 = -1(-27) = 27
If converting a smaller measure into a larger one (such as seconds into years) then
Put digits into the same format (so if seconds into years then make sure both numbers are in seconds) and then divide the larger number by the smaller
If dividing a smaller number by a larger one (2/4) the try to multiple them to a hundred and add a decimal point (2/4 multiply by 25 so it’s 50/100 then add a decimal points to the numerator =.5)
N
Look for the quadratic formula
X^2 + bx + c = 0
X-/+x)(x-/+x
DS - try to rework equation to mirror the question stem. For example z/y=7 could also be
Z=7y
