QUANT Flashcards
(92 cards)
1
Q
UNFOIL: (A + B)^2 Unfoiled
A
(A + B) (A+B) = A^2 + 2AB + B^2
2
Q
UNFOIL: (A - B)^2
A
(A - B) (A - B) = A^2 - 2AB +B^2
[Note neg x neg = Pos, but when factored some will not have neg sign so will be neg x pos = neg]
3
Q
UNFOIL: A^2 - B^2
A
DIFFERENCES OF SQUARES: (A + B) (A - B)
4
Q
Negative - Negative
A
-5 - -5 = -5+5 = 0 -4–5= -4+5 = 1.
5
Q
Positive - Negative
A
5 - -6 = 5 + 6 = 11
6
Q
What is the square root of 2
A
1.4
7
Q
what is the square root of 3
A
1.7
8
Q
What is the square root of 4
A
2.2
9
Q
Negative x Negative
A
Positive (Same signs = positive)
10
Q
Negative x Positive
A
Negative (Opposite signs = Negative)
11
Q
Negative / Negative
A
Positive (Same signs = Positive)
12
Q
Negative / Positive
A
Negative (Opposite signs = Negative)
13
Q
Negative + Negative
A
Negative (Keep sign)
14
Q
What can be learnt from (k^3 - k)
A
Consecutive integers when FOILED: K(K^2 - 1) ===> K^2-1 is same as (K+1) (K-1). (Another consecutive series of 2)
15
Q
What can be learnt form (K^2 - 1)
A
Consecutive integers of 2: (K - 1) (K + 1) = (K^2 -1)
16
Q
Solve x^2y^2 - xy= 6
A
Factor: xy(xy-1)=6. meaning 6 = xy x xy-1 (I.e. 1 less than xy) figure out which two numbers 1 number apart multiply to get six
2 and 3 positive or negative but same sign as 6 is positive.
17
Q
Solve X^2 - Y^2
A
Difference of squares: (x - y) (x + y). opposite signs = negative thus + and minus
18
Q
When I see a question with numbers raised to the 4th power I will…
A
Show these numbers as squares of squares in order to better manipulate the equation. i.e. 9x^4 = (3x^2)^2
19
Q
When I see divisibility problems I will…
A
Prime factorize and know that the smallest integers that are divisible will have prime factors contained by no more than needed (I.e. the exact amount of factors)
20
Q
List the first 10 prime numbers
A
2, 3, 5, 7, 11, 13, 17,19, 23, 29
21
Q
Which is greater 3/4 or 31/42 ….
A
to find this you cross multiply and keep the anchor on the numerators: 3 x 42 = 124. and 31 x 4 = 126. 126 > 124. so 3/4 > 31/42
22
Q
How can you tell how many trailing zeros there are in a number based on their prime factorisation? I.e. How many trailing zeros are there in an interger with the prime factorisation of:
2^16 x 3^15 x 4^11 x 5^19
A
Trailing zeros= ,0000 the only numbers with a 0 at the end are multiples of 10.
10 has the prime factorisation of 2 x 5
so you can get the number of 0’s by counting the number of 2 x 5 pairs.
In this case the number of trailing zeros is 19 because the max pairs are limited to 5^19 as 4^11 has some 2’s in their as well
23
Q
If a set of numbers (D) are all multiples of 3 what will this mean for their prime factorization?
A
All the numbers in that set will have a prime factorisation that includes the number 3 if they are multiples of 3. I.e. 24 = 2 x 2 x 2 x 3
24
Q
If a set of numbers (E) are all factors of 400 what does this tell us about the prime factorisation of all the numbers in set E.
A
The prime factorisation of 400 = 2^4 x 5^2. All the factors of 400 will have some combination of these same prime factors. I.e. 40 = 2^3 x 5 ….the prime factors will always be the same.
25
1.2 x 5 = ?
Technique: 5 x 1 = 5 and 5 x 2 = 10 or 5 x 0.2 = 0.10 move the 1 over the decimal to get 6.0. 5 x 1.2 = 6
26
2 buses and a van were used to take a class of kids to school:
3/5 of the kids went on the first bus
2/3 of the remaining kids went on the second.
The rest of the kids took the Van.
When the 2nd bus broke down half the students on the second bus boarded the first.
What fraction of the class was onboard the first bus?
When I see a fractions qn with multiple splits......I will use smart numbers with the least common denominator:
2/3. 3/5. Smart Number (LCD) = 3 x 5 = 30
30 STUDENTS = 3/5 1st bus (18)
12 KIDS REMAINING 12 X 2/3 2ND BUS (8)
2nd bus breaks down and half the students go to first bus (8 x 0.5 = 4)
FIRST BUS 18 + 4 = 22
FIRST BUS FRACTION = 22/30 = 11/15!
27
What happens if I divide this inequality: -3x > 6 by -3 on both sides?
When multiplying or dividing both sides of an inequality by a negative number you must change the direction of the inequality symbol.
An inequality however cannot be divided by an unknown variable as this could be zero.
28
How to tell if a number is divisible by 8? i.e. 23,556 or 32
A number is divisible by 8 if it divides into two 3 times or if the last 3 digits is divisible y 8. I.e. 556 can be divided 3 times by 2 so yes. and 32/2 = 16, 16/2 = 8 so yes.
29
1. How to tell if a number is divisible by 6?
2. How to tell if a number is divisible by 5?
3. How to tell if a number is divisible by 4?
1. A number is divisible by 6 if it is divisible by 2 and 3. (A number is divisible by 3 if the sum of its digits is divisible by 3- 393= yes because 3+9+3= 15 which is a multiple of 3)
2. A number is divisible by 5 if the last units digit ends in 0 or 5.
3. A number is divisible by 4 if it divides into 2 twice. I.e. 8/2 = 4, 4/2= 2 so yes.
30
Will the product of the following be odd or even?
69 x 70 x 71
As they are consecutive integers one of them will be even and any odd number multiplied by an even number is even.
31
X = 69 x 70 x 71. Is x divisible by 3 or 2?
They are consecutive integers meaning there will be factors of 1,2 and 3 regardless so x will be divisible by both 3 & 2.
32
56 x 25
you can double and half twice:
56 x 25 = 28 x 50
again
28 x 50 = 14 x100 = 14000
33
How to square 40^2
For multiples of 10 squaring = take out the first number =
4^2 = 16
and then put it in front of 2 zeros (As 10^2 = 100)
40^2 = 1600
34
For numbers that end in 5 how do we square them?
35^2 = ?
75^2 =?
25^2= ?
65^2 =?
115^2 = ?
1. Note: Any number ending in 5 will have the units digit of "25" i.e. 5^2 at the end. - so we are only looking for the number in front of that.
2. To find 35^2 we know that 30 <35 <40 so......30^2 < 35^2 < 40^2
Which is the same as 900 < 35^2 < 1600
so that means the first digits cant be 3x3 as this is 9 which is too small OR 4 x4 Which is 16 which is too big. But it can be 3 x 4 =12
so 35^ = 1225
and 75^2 = 5625
25^2 = 625
65^2 = 4225
115^2 = 13225
35
How can we find 8^2 using 7^2
Try 69^2 also
N^2 = (N-1^2) + (N-1) + N.
8^2 = 7^2 + 7 + 8
Note you can go backwards as well as upwards 69^2 can be found using 70^2. (JUST REVERSE ALL THE SIGNS)
N^2 = (N+1^2) - (N+1) - N
69^2 = 70^2 - 70 - 69
= 4900 - 70 - 69
= 4761
36
3/7 x N = 5/4 x T
I solate N.
When moving a fraction across an equals sign you have to invert it or take the reciprocal.
N = 7/3 x 5/4 x T.
N = 35/12 x T
37
Mental Maths:
1. How to add 2digit numbers in your head : 56 + 32
2. How to subtract 2 digit numbers in your head:
i ) 46 - 32
ii) 54 - 47
1. Addition: You can separate the tens and the units digit to simplify the calculations 56 + 32 = (50 + 6) + (30 + 2) = (50 + 30) + (6 + 2) 88
2. Subtraction: You can do the same as addition ONLY if the number being subtracted has both tens and units digit smaller:
46 - 32 = (40 + 6) - (30 + 2) = (40 - 30) + (6 - 2) = 10 + 4 = 14
3. Subtraction.2: To subtract numbers where the number being subtracted is bigger, you can add a constant to both ends to simplify the calculation: 54 - 47 ---- 54+3 - 47 + 3 = 57 - 50 = (50 +7) - (50 + 0) = (50 - 50) + (7-0 )= 7
38
What is the algebraic interpretation of
I. |X|
II. |X - 5|
III. |X + 3|
|X| = The distance of X from the origin (usually from zero)
|X - 5| = The distance of X from (POSITIVE) +5
|X + 3| = The distance of X from (NEGATIVE) -3
39
What does the algebraic expression |x - 1| > 4 mean on a number line
This means that starting from the number POSITIVE 1 on a number line the distance between +1 and X is greater than 4 so this also means that x < -3 OR X > 5
40
What is (2.5 x 10^17) ( 6 x 10^-8)
= 2.5 x 6 x 10^9
= 5 x 3 x 10^9 (Doubling and halfing trick)
=15 x 10^9
41
What is 3.6 x 10^-8 divided by 10^3
This can be rewritten as (3.6 x 10^-8) x (10^-3)
= 3.6 x 10^-11
For some reason we only divide the last part
42
What is 1/9 as a decimal?
0.111111 so this means 5/9 is 0.555555
43
What is 1/7 as a decimal?
0.143
44
What is 1/6 as a decimal ?
0.16667 so this means 5/6= 0.833. (5 x 1.6 = 8~)
45
What is 2/3 as a decimal?
0.666
46
What is 3/4 as a decimal?
0.75 because 1/4 is 0.25 so x3 = 0.75
47
What is 1/8 as a decimal?
0.125
48
What is the reciprocal of -7/2?
-2/7
49
What is 1/(1/10)
This is the reciprocal of 1/10 = 10
50
What is 4/17 x 17/4 ?
The product of any fraction with its reciprocal = 1
51
Show x - 1/4 as an improper fraction
= X x 4/4 = 4x/4 (Express x as a fraction)
=4x/4 - 1/4 = 4x -1/4 (Now subtract)
Answer: 4x-1/4
52
What happens to a fraction when the same number is added to both the numerator and the denominator?
This is the law of CENTRALITY - the fraction will move close the 1 i.e. same number / same number (I.e. 3/3)
53
What happens to a fraction when different numbers are added to the denominator?
The law of CENTRALITY- the fraction will move closer the fraction it was added to I.e. For the fraction 8/10 if 2 is added to the numerator and 3 is added to the denominator the fraction 8/10 will move closer to the fraction 2/3.....this means it will get smaller as 8/10 was already bigger than 2/3 so moving closer to a smaller fraction means you get smaller.
54
MATHs w PROPORTIONS:
Solve for x :
12/5x = 8/15
What are the 3 rules?
Answer: x = 9/2
Note. Proportions are those where the equals sign is between 2 fractions- In these cases the rules are as follows:
1) You can cancel factors vertically in the same fraction (I.e. on one side of the equals sign)
2) You can cancel factors horizontally across fractions (I.e. across the sign but on the same level)
3) You CANNOT cancel factors DIAGONALLY! (I.e. Across the equals sign from bottom to top- you have to cross multiply to solve at this stage)
55
If the price of a stock rose 20% and then dropped 50% and then increased 40% what was the percentage change over those 3 months?
GEOMETRIC MEAN X Multipliers:
Convert to multipliers and do geometric mean:
1.2 x 0.5 x 1.4
= 1.2 x 0.7
= 0.84
56
16 * 8 = ?
= (10 + 6) (8)
= (80 + 48)
= 128
57
How can you tell if a number is divisible by 9?
Exactly the same way as the divisibility rule for 3- sum up all the digits and if the sum is divisible by 9 then its divisible by 9.
58
How many prime numbers are there between 80 and 90
Rule: To find out if any number less than 100 is a prime number - all you have to do is test if it is divisible by any of the prime numbers less than 10 (I.e. 2,3,5,7)
The only numbers that are not divisible by (2,3,5,7) are 89 and 83
59
How many factors does 8400 have?
Trick: To find the number of factors:
Step.1 = Prime factorize
8400= 2^4 x 3 x 5^2 x 7
Step.2 = Pull out only the exponents of the prime factorization (Including the exponents of 1)
= (4,1,2,1)
Step.3 = Add 1 to each exponent
= (4,1,2,1) -- (5,2,3,2)
Step.4= Multiply them together to get the number of factors
= 5 x 2 x 3 x 2 = 60
8400 has 60 factors
60
How do you find the odd prime factors or even prime factors for a given number?
For odd prime factors- you basically calculate the number of factors but leave out the even prime factors from the count.
For even prime factors - You calculate the total number of factors and then subtract the odd prime factors
61
How can you tell by a prime factorisation of a number if that number is a perfect square?
If the exponents of the prime factorisation are all even numbers then the number is a perfect square. (I.e. 360 = 2^6 x 3^4 x 5^2 - since all the exponents are even = perfect square)
Note: This is because when you find the total number of factors you add 1 to each exponent. Adding 1 (Odd number) to an even number will make it odd. Then multiplying all those odd numbers will = an odd number.
And only perfect squares have an odd number of factors.
62
What is the GCF of 720 and 1200
GCF - USE PRIME FACTORISATION GCF:
Prime factorise
720 = 2^4 x 3^2 x 5
1200= 2^4 x 3 x 5^2
Highest power of 2 in common = 4
Highest power of 3 in common = 1
Highest power of 5 in common = 1
GCF = (2^4) (3^1) (5^1) = (2) (3) (2) (5) = (24) (10)
=240
63
What is the LCM of 12 and 75?
LCM - USE PRIME FACTORISATION LCM:
Prime factorise
12= 2^2 x 3
75= 5^2 x 3
GCF = 3
LCM
12 = 3 x 4 (Find a number that multiplies with the GCF to equal)
75 = 3 x 25
Therefore
LCM = 3 x 4 x 25 = 3 x 100 = 300
64
What are the rules that determine whether adding or subtracting even/odd numbers will be even/odd?
When adding/subtracting "Likes" = Even (I.e. Even + Even or Odd - Odd)
When adding/subtracting "Unlikes" = Odd (I.e. Even + Odd = Odd)
65
When do we now if the product of a group of numbers will be even or odd?
The only time a product of numbers is odd is if they are all odd numbers.
Even 1 single even number will make the entire product even
66
Simplify:
(8y^2 + 10y - 2) - (3y^2 + 2y - 6)
RULE: When simplifying expressions in brackets:
i) If the bracket has a "-" sign infront convert all the signs within the bracket to their opposite i.e. -(3 + 4 - 6) = 3 - 4 + 6)
ii) If the brackets have a + sign - just remove the brackets and simplify as normal
5y^2 + 8y + 4
67
Simplify:
1. (3x^3 y^2) (7x^5 y^6)?
2. (15x^6 y^12)/3 ?
3. (2xy) (3xz) (4yz)
Rule for Multiplying or dividing monomials:
Only multiply or divide the respective coefficients !
1. 21x^8 y^8
2. 5x^6 y^12
3. 24X^2 Y^2 Z^2
Note: THIS DOES NOT WORK FOR ADDITION OR SUBTRACTION
I.e. 2x^2 + 2x DOES NOT equal 4x^2
68
What is the prime factorisation of 9975?
When I see prime factorisations of large ugly numbers I will....
Use (-1 rule) and factor out using difference of squares method
9975 = 10000 - 25
= (100^2) - (5^2) Difference of squares (a^2 - b^2)
= (100 + 5) (100 - 5) Difference of squares
= (105) (95)
= (5 x 21) (5 x 19)
=(5 x 3 x 7) (5 x 19) Prime factors
69
Simplify:
0.999951/0.993
Use difference of squares:
0.999951/0.993
= 1- 0.000049 / 1- 0.007 (Notice that the numerator is a square)
= 1 - 0.0007^2/1-0.007 (Notice the numerator is difference of square)
= (1 -0.007) (1 + 0.007) /1-0.007 (Remember 1 squared is just 1)
= (1 + 0.007) (As the 1-0.007 cancels out)
Answer = 1.007
70
Simplify the complex fraction:
x/2 + 5/4 over x/3 + 3/2
When simplifying complex fractions multiply by the LCM across all the denominators
LCM = 12
= 12 (X/2 + 5/4) = 6X +15
=12(X/3 + 3/2) = 4X + 18
= 6X +15 / 4X + 18
71
What are the rules for subtracting inequalities ?
You can combine inequalities (Addition) if they are pointing in the same direction but you CANNOT do the same for subtraction. Unless they are pointing in opposite directions I.e. The range - Big-Small > Big - Big
72
Express the following as an absolute inequality:
|X - 7| < 3
This essentially means that the positive integer of 7 is less than 3 away on a number line than a given number X.
So 7 - 3 = 4
AND 7 + 3 = 10
Answer: 4 < X < 10
Note if its a
Positive numbers: |X - 5 |
Negative numbers: |X - (-3)| = |X + 3|
73
Express the following as an absolute value inequality:
5 < X < 17
When I see dual inequalities I will....
Find the midpoint! So we can know what distance from we are calculating.
(5 + 17)/ 2 = 22/2 = 11
Distance away:
17 - 11 = 6
5 - 11 = 6
|X - 11| < 6
74
Advanced Substitution Algebra:
Solve the following:
(2x - 1)^2 + 5(2x - 1) - 24 = 0
SOPHISTICATED Questions like this use the kind of algebra that creates its own variables (Letters) and then solves them.
1. This is clearly a quadratic so make it so
2. Substitute (2x-1) [As it appears twice] with your own variable "u"
= u^2 + 5u - 24 = 0
=u -8 AND u = +3
Solve for u now
-8 = 2x - 1 ---- -7/2
3 = 2x - 1 -----2
75
ADVANCED SUBSTITUTION ALGEBRA.2
Solve for k:
3/1- (8/(7+k)) = 15
CREATE YOUR OWN VARIABLES AND SOLVE FOR THEM
1- (8/(7+k)) = "A"
3/A = 15
A = 1/5
IF 1 - (8/(7+k)) = 1/5
Then this means 8/(7+k) = 4/5 (Delta of 1/5)
if 4/5 = 8/7+k
then 8/10 = 8/7+k
10 = 7+k
k = 3
76
Interpret algebraically: "Twice A is 100 less than 3 times B"
In Algebra
"Is" or "Are" is represented by the "=" sign
So the phrase translates to:
2A = 3B -100
77
How should you tackle distance word problems with multiple travellers?
Each traveller needs their own D=RT equation
78
List the rules for tackling distance problems where 2 vehicles are travelling ?
1. Opposite directions (1 East, 1 West) = Addition
Same Direction (Both East/West) = Subtract
2. Solve D=RT for the gap itself and apply the gap values to the question depending on whether the gap is expanding or decreasing
3. Gap is increasing for examples if the cars are moving away from each other - but you still add rates because opposite = Addition
79
How many days are there from April 8th to the end of april?
30 Days in April mean that 30 - 8 = 22 (This is wrong)
You have to add 1 to include April 8th as the above only shows the difference in the endpoint.
30-8 + 1 = 23 days
INCLUSIVE COUNTING
80
Simplify (3^5)^5 ?
Answer= 3^15
Think about it (3 x 3 x 3 x 3 x3) ^ 5 means 5 times = 3^15
81
Use the distributive law to apply to the following:
1. (ab)^N
2. (a/b)^N
The distributive law applies to Multiplication and Division of powers NOT to addition and subtraction.
1. (ab)^N = (a^N) (b^N)
2. (a/b)^N = a^N/ b^N
82
What is the units digit of 57^123
Step 1: Find the pattern of the units digits
step 2: Find the frequency of repetition
step 3: Scale the pattern to the desired level using the multiples
1. 7 ^ 1= 7
7^2 = ...9 (7 x 7 units digit only)
7^3= ....3 (7 x 9)
7^4 = ......1 (7 x 3)
7^5 = ......7 ( 7 x 1)
etc ....7,9,3,1....
2. Repeats every 4th time [ This means all exponent multiples of 4 will have the same units digit]
3. 57^123 units digits = 7^123 = units digit 3
(This is because since the frequency is 4 that means we need to find the number near the desired number that is a multiple of 4 which is 120 so 120 = units digit 1 (4th) then 121= 7, 122= 9, 123 = 3)
83
1. What is the square root and 4th root of -8 ?
This is impossible. We cannot take an even root of a negative number, we can only take an odd root of a negative number.
(Even roots = 4th root, square root etc.)
84
1. Will the cube root of -9 be positive or negative?
Any odd root of a positive number is positive and any odd root of a negative number is negative.
85
Is the square root of K + the square root of Q the same as the square root of K + Q?
NO! similar with the distributive law of squares. Roots distribute over multiplication and division but they do not distribute over addition and subtraction
86
Combinatronics
A librarian has 5 identical copies of book A, two identical copies of book B, and a single copy of book C. In how many distinct orders can he arrange these 8 books on a shelf?
When I see a counting problem with "identical" items included...
I will know that these have to be stripped out via division (Mississipi rule- because identical items are technically NOT distinct)
Total counting list = 8! (5+2+1)
8!/5! x 2! x 1! (The identical items must be stripped out this way)
= 8 x 7 x 3
= 8 x 21
= 168
87
Combinations - Eliminating Repetitions
From a set of 10 different items, Lisa is going to select three as a gift for someone. How many different sets of three items can she pick?
When I see a counting problem with repetitions...
I will remove the repetitions by dividing them out to subtract them
Notice how if we use 10 x 9 x 8 we would be counting some sets 6 time repeated. So we have to divide by 3!
= 10 x 9 x 8/ 3!
= 10 x 3 x 8/ 2
= 10 x 3 x 4
= 120
88
Combinations
An amusement park has 12 different rides. A coupon gives its holder access to any 3 of these rides for free. How many sets of three rides are possible?
When I see a problem asking me to select a set from a bigger pool...
I will use nCr - N choose r
In this case nCr = 12 choose 3
N= Size of pool (Counting) 12 x 11 x 10
R = Set = 3! = 3 x 2 x 1
nCr= 12 x 11 x 10 / 3 x 2 x 1
= 2 x 11 x 10
= 220
89
Suppose in a game, the probability of outcome A is 0.6, the probability of outcome B is 0.7, and the probability of A or B is 0.9. What is the probability of A and B happening at the same time?
The Golden formula in probabilities is:
(Note this is only for non-mutually exclusive probabilities)
P(A or B) = P(A) + P(B) - P(A and B)
= 0.9 = 0.6 + 0.7 - P(A and B)
=0.9 = 1.3 - P(A and B)
= 0.4
90
If a machine can empty 5/6 of a pool in an hour- how many hours will it take to empty the entire pool?
Take the reciprocal - if 5/6 of a pool can be emptied in an hour then it takes 6/5 of an hour to empty the entire pool.
Rule: If a fraction of a task can be completed within one unit of time then the entire task can be completed in the reciprocal unit of time of the fraction.
91
When I see a very complex fraction I have to simplify I will....
I will multiply both the numerator and the denominator by the common denominator
92
