Quant 3.0 Revisit List Flashcards
You flip a Heads/Tails coin 7 times.
How many different ways can you flip 5 heads?
How many ways can you arrange exactly 5 heads and 2 tails?
7!/((7-2)!2!)
Think about equation after new people move into the town
Don’t forget the muplier!
(13x + 15)/(2x + 120)
The setup gives us the following two equations:
3w + 2p = 35.20
2w + 3p = 30.80
(where w=sand p=soup)
goal: solve for w
What is the first step?
Multiply equations and then Subtract equations to solve for one variable! OFten, may need to multiply by a negative number.
Remember, the goal is to REMOVE a variable, this will help you find what numbers to multiply by! in this case 3, and -2 make w dissappear so you can solve for p, which then lets you solve for w.
How many different equations do . you need to solve this absolute value?
2.
The absolute value will produce the same value in ++ and –; and -+, +- scinarios:
see example here:
My Q:
I think this equation is wrong because if the number of tickets is less than 100, arnaldo gets punished.. how do we write an equation that accounts for this condition? I realize this equation works properly for t > 100, but the question doesn’t guarantee that. I have encountered this dilemma a few times. How should I think about it?
Ben’s Note: The equation only applies for t>100 because there wouldn’t be a bonus otherwise. For t<100 the money earned is just 11t. But the questions on GMAT dont trick you this way – your concern is not a tested trap.
WE CAN SPECIFY t > 100 in the equation’s application.
Solve for r:
remember, when you take the square root of something under the radical, you get the absolute value of something as a result!
When the sqrt(x) operation is being applied to a variable, the sqrt() returns | x | !!!
sqrt of a number is always positive. Ex: sqrt(25) is 5, not |5| !!
sqrt of a squared variable gives absolute value –> because the variable could be negative, and we get 2 answers
Does a tangent line “intersect” a graph that it shares a point with (or is tangent to) ??
YES. at that one point.
What is the first and last numbers in the sequence?
Remember to look for this trap!
1002…1866
even + even =
odd + odd = ?
even + odd + even + odd = ?
EVEN!
unlike multiplication:
odd*odd*odd*odd = odd (all in a sequence must be odd)
even*any_sequence*2*8*3*9*13*17 = even (single even in multiplication sequence makes all even!)
What is the units digit of the answer, even or odd?
Is the answer negative or even?
Answer: -1997
Hope you were thinking “difference of squares” right away.
Factor this bad boy
Rewrite with a base of (3/5)
Anser options:
1,3,5,6,14
E - Neither!
Note: whenever I choose a DS answer, because of “2 equations, 2 variables, therefore must be C”, Check to make sure both equations are UNIQUE equations. Do a verification: see if I can manipulate S2 to look like S1.
In this case, the trap was that both equations were not unique, they were the same equation, therefore, when S1 and S2 are combined, you end up with 1 unique equation and 2 unknowns.
Do we neglect numerators?
No! They make a denominator that appears to be non-terminating, terminating!
Probably need paper, but should be able to derive main takeaway
Given two points, we know the relationship is LINEAR.
This means we can have two equations of y = mx+b, solve for m and b.
remember 3^150 = 3*3^149 = 3*3*3^148! therefore
9*3^148 - 3^148 = 8*3^148
ALSO, just remember simple exponent math. 3^2*3^148 = 3^150… by ADDING exponents
Think: How to set up equations to solve for x,y,x and a,b,c ?
Given: x + y < z, and x, y, z are integers greater than zero
Is x < z AND y < z ?
Y/N?
YES, both are less than z because their SUM is less than z and none of them are negative!
If 73! has 16 zeroes at the end, how many zeroes will 80! have at the end?
Remember, Any multiple of a perfect square adds TWO of the prime factor!
75=3*25=3*5*5
Therefore, 80 and 75 add 3 total 5’s to the heap! Answer is 19 zeroes at the end!
Let X be an integer,
if
a = max value of X,
b = min value of X
and:
-22 < X < 18
what is a-b?
since -22 < X < 18
Max value of X = 17, Min value of X = -21
a-b = 17 - (-21) = 17+21 = 38
When inequalities exist, watch out for this trick!
1024 is what percent greater than 256?
1024 is what percent of 256?
let y=1024, x=256
1024=4*256
y=4x
if y = 4x, what percent greater than x is why? Answer: y is 300% greater than x.
What percent is 1024 of 256? 400%.
Think about the setup!
Overall group = 8 people including George and Nina
Restrictions: Nina is not included and George is included.
This means we have on position (George) filled already and we need to select 2 people from the remaining 7 people.
But Nina is not to be included, hence we need to select 2 people from the remaining 6 people.
Number of ways: 6C2 = 15
Total number of ways of selecting 3 people out of 8 = 8C3 = 56
Probability of selecting the desired group = 15/56
Correct Option: C
Probalby can’t do in your head, but my question is: What is Y ?
Hint, given two variables and a total….
CANNOT DIVIDE IF WE DONT KNOW THE SIGN OF Z BECAUSE WE DONT KNOW IF WE SHOULD FLIP THE INEQUALITY!
1/x will have Digits(x) -1 leading zeros.
ex: 1/16, 16 has 2 digits, therefore 1 leading zero.
1/8, 8 has 1 digit, therefore 0 leading zeroes.
etc.
How to simplify?
even +/- even = ?
odd +/- odd = ?
else = ?
even +/- even = even
odd +/- odd = even
else = odd
Once I know the distance,
I can solve for endpoint.
27*(10^-3) equals x*(10^-4)
270!
27*(10^-3) equals 270*(10^-4)
take away a 10, add a 10!
(1-b) = (-1)*(b-1)
Ex:
First step?
Should know 13^2 = 169 and 17^2=289 and plug in immediately. These two are probably not going to reduce.
(11 choose 2) * (9 choose 2)
55*36 = 1980
Imagine I reduce my answer down to:
sqrt(45)… which answer do I choose?
need to take sqrt(45) and turn into sqrt(9*5) = 3*sqrt(5)
Finish simplifying answer!
What are whole numbers?
non-negative integers.
ex: integers from 0 to infinity.
Do translation step
Since the remainder is 8 when n is divided by 9, it follows that n = 9q + 8 for some integer value of q.
ALSO: Note that q must be even because n = 9q+ 8 is even. Test each answer choice.
p = 7.42/7
what is p?
Tempting, but don’t do
7.42/7 = 1.42. INCORRECT.
742/7*10^-2 = 106*10^-2 = 1.06
boom.
How to factor?
(probably can’t do in head.. but can follow solution)
How to factor numerator? (first step)
odd*odd*odd*odd = ?
even*odd*odd*even*odd = ?
even*(bunch of other numbers) = ?
Remember, when you have one 5 and one 2 in a multiplication ==> that is 10*blah, and 10*blah will ALWAYS have a units digit of 0!
Regardless of the value of x, the units digit for xy will always be zero if y=10*blah, which it does.
See the hint?
Each answer option had sqrt(3), therefore this should make your mind think: 1 : sqrt(3) : 2 triangle!
Find an attack point, what is it?
It’s helpful within roots to look for known squares. sqrt(275) = sqrt(5*5*11) = 5*sqrt(11), so this could be a starting point!
Let’s say I solve this equation out for both cases and I get the following:
x=-3, x=4, x=3, x=-4, then what is the next step?
Check solutions to absolute value!
remember that I’ve only encountered POTENTIAL solutions. Need to plug back into absolute value to verify which solutions ACTUALLY make sense!
What is a factor of every number?
1.
don’t forget 1 !
29, 5, 2, 1, 29*5, 29*2, 5*2, 5*29*2
therefore, 8 is the answer
The question says: find a value for a - b.
I have 2a-b =5
Can I ever calculate a value for a-b with the information given?
NO!
1/sqrt(n) = ?
sqrt(n)/n
Together gives us: x=m^3=n^4=a^12
All odd powers of 9 end in 9. All even powers of 9 end in 1.
9^0 = 1, 9^1=9, 9^2=81, 9^3=729…
Can you solve for x and y?
YES! because we know the relationships.
- 2x = -80, x = 40;
- 40y = -40, y = 1;
Think of initial equations and setup
Remember to be careful on word problems.
Do not be cocky.
How to simplify?
d1, d2=dollars per hour person 1,2 worked
h1, h2 = hours person 1,2 worked
(2/4)d1*h1>d2*h2
2 > 4
by dividing variables that CANNOT BE ZERO out! Damn!
Remember you can divide because not zero.
What do you know from this rephrase?
hint: since x and y are integers, the solution space is limited bigtime!
x must be 0,
y must be either -1 or 1.
What is the greatest common factor of n and n+1 ?
1.
Combo value question, given to solve for (a - b) combo. What should I
goal: try and re-work the question to look like the combo i’ve been given to solve for: (a-b).
In this case, that required multiplying by -2. I need to see this!
how do you break this absolute value into two solvable equations?
x - 1 | > 2
how do you break this absolute value into two solveable equations?
1) MULTIPLY NON-ABS VALUE Part of EQUATION BY -1, (FLIP INEQUALITY)
x - 1 < -2
-x < -3
(REMEMBER to flip the inequality, REMEMBER do NOT multiply ENTIRE equation by -1. Whatever is inside the absolute value stays intact.
2) remove bars:
x - 1 > 2
x > 3
x - 1 | > 2
Remember to always factor out common factors! I will be rewarded.
When x is divided by y, the quotient is z with remainder r.
Set up equation!
What equation comes to mind?
n(n-1)/2 come to mind?
Great. Well, remember that n is the number of people (18 in our case),
(n-1) in the equation above is just a placeholder for #of_people_to_shake_hands_with.
2 = 2
therefore:
18*15/2 is the answer, 135
How to factor? and how many solutions
4 solutions! what a dope factor!
How to solve?
x = -x
x = -x
2x = 0
x equals 0.
HACK: Pattern repeats every 8 numbers, up to 96!
The easiest way is to create table:
Could also think about it this way:
There are:
12 multiples of 8 in 1…96
24 multiples of 4 in 1…96
48 multiples of 2 in 1…96
unique multiples of 8 in 1…96: 12
unique multiples of 4 (not shared with 8) in 1…96: 12
unique multiples of 2 (not shared with 8 and 4) in 1…96: 24
Think about how all it takes is 3 2’s (because 2^3 = 8) for (n)(n+1)(n+2) to be divisible by 8
3^3 = ?
27!
I frequently mix up 3*3 and 3^3.
Be cautious!
What is the equation for nickels and dimes?
N = 2D
Remember to be PRECISE in reading.
In DS, what do I do if S1 or S2 can be manipulated to look just like information given in question stem?
An answer choice is not sufficient if it does not give any new information!
If question stem asks “what is k in the equation x^2 + kx + 16 = 0” and Statement 2 can be manipulated to be x^2 + kx + 16 = 0, then statement 2 is insufficient!
Be on the lookout for this trap!
1/3.5 = 1/(7*5), we know (1/7 never terminates), so C is out.
since sqrt(3) does not repeate or terminate, therefore 1/sqrt(3) does not repeate or terminate.
Answer is D, 1/sqrt(3)
Can we simplify to ad > cb ?
NO!
We can’t because if we multiply or divide an inequality by a negative number, we must flip the inequality sign.
We don’t know what variables are positive or negative, therefore, we CANNOT simplify!
only way we can do the simplification is if we know that all variables are postive!
(x - y)^2
x^2 - 2xy + y^2
REMEMBER:
minus sign only goes in front of the 2xy term! not in front of the y^2 term
Is there any number with only 1 factor?
YES! 1.
Also: one is a factor of all numbers, and all numbers are multiples of 1. (note: 1 is not a multiple of all numbers)
May not be able to solve, but review
Riddle me this: in S1, what values can k take on?
hint: k is a positive integer.
Since K is a positive integer (k > 0)
and since x is between 0 and 1000,
x = k^6 means that k can either be 1^6, 2^6, or 3^6, aka 1, 64, or 729.
watch out for this constraint! a common trick.
answer is C because S2 tells us that k must be even. the only even value is 64, boom!
The answer is D because the 30 cents per minute charge only applies to each additional minute beyond the first, not to the first minute itself. So there are only n-1 additional minutes.
Identify: a) type of problem, and b) trick, then flip
type=triple overlapping sets
trick: A and B and C must be accounted for 3x when you do the subtraction because it is overlapped 3x. Think about the geometry!
What is a “near miss” for this?
Cannot divide out x without getting rid of solutions to the equation!
Even powers being subtracted ==> SHOULD THINK DIFFERENCE OF SQUARES!
B.
Choices A and C are essentially identical, and should be regarded with suspicion. Upon further review, if b were to equal 0, then a need not necessarily equal b. Try numbers: a = 1, b = 0, c = 2, then (1)(0) = (0)(2), but 1 does not equal 2. Because b could equal zero, answer choice D is also incorrect, as the two sides of the inequality would both equal zero, and therefore be equal. Answer choice E is also not necessarily true, as demonstrated by the trial numbers above. Answer choice B is correct, as it is true both if a = c, or if b is equal to zero.
Simplify the inequality:
y^2 > 16
Simplify the inequality:
y^2 > 16
sqrt(y^2) > sqrt(16)
break into two:
y > 4
y < -4
Note: the equality changes direction for a negative solution to the square root
What should jump to mind?
“How many two digit integers have exactly five positive divisors?”
Note: divisor is the same as factor! Don’t be confused by this!
The only numbers with an odd number of divisors are perfect squares (for example, 9 has three divisors: 1, 3, and 9). Since this problem asks about two-digit numbers, trial-and-error can help as there are only 16, 25, 36, 49, 64, and 81 to try to factor.
note: I tried to do:
x(x^2 + 2x +1) = -2(x^2 + 2x + 1)
cancel out common terms..
x = -2
Problem with this is that I am dividing out solutions to the equation! cannot do!
Do initial setup equation!
Think about setup.
We have total and a percent.
fraction * value = total. What are columns?
(1-x^16) = ?
(1-x^8)(1+x^8)
What is obvious about statement 2 ?
Statement 2 is identical to the stem question.
Since statement 2 does not give us any new information, statement 2 is therefore insufficient
Even / Even = ?
EVEN OR ODD!
Ex: 36/4 = 9, 36/2 = 18
solve: n^2 > 144
n^2 > 144
sqrt(n^2) > sqrt(144)
plus case: n > 12
minus case: n < -12
n | > 12
How to attack / simplify?
Attack the denominators!
Why is statement 1 insufficient?
Because it is asking for a VALUE. not a variable!
what is the “x > 1” trick?
How to solve?
What is the definition of perpendicular?
Slope is negative recripical!
What if I reduce this to 56/120, which option should I choose as the answer?
56/120 =
Probably can’t do in head, but look at technique used…
technnique used: setting equations =0 and simplifying that way.
is it:
(w-2)*2w=160^2
or
(w-2)*2w=160
?
(w-2)*2w=160
Units are sq ft. 160 is not squared!
(w-2)*2w -160 = 0
2w^2 - 4w -160 = 0
2(w^2 - 2w - 80) = 0
2(w - 10)(w + 8)=0
w = 10 or -8, w must be postive, so w=10.
does 0 equal -0?
yes!
zero is the only number that is equal to its opposite!
also, 0 is a multiple of all numbers.
B. One of the more often-missed portions of Venn Diagram logic is this: If you have a problem without a neither component (or if the neither component is already known), and you’re given a total for “Group X ONLY”, then the remainder of the total must be “Group Y”, where Group Y consists of “Y only” and “both”. In this question, you know that the total must be 100% and there is no neither component. So when statement 2 gives you that 40% of people only take spin, the remaining 60% must then take either yoga only or yoga and spin, and since both components add to the total number of yoga participants, you now know that 60% of people take yoga.
D.
B!
Can’t solve it all out, but understand the general process:
Answer is D.
hint: given a+b, you can think of combo-esque DS solution
answer me this: is two alone sufficient?
YES!
Arc AB : Arc BC : Arc CA
3: 1 : 2
We know this because it is the same as the inner angles of the triangle! how incredible!
How do we tell if a point is on a line?
When you plug in the point into the equation, the equation maintains equality.
What is the proper rephrase?
Is statement 2 sufficient?
Is line AB parallel to CD?
AKA: do they have the same SLOPE!
Great rephrase. Great.
Key to this problem is to recognize the number of unique **PRIME** factors. I have made this mistake before, but I need to remember that unique_prime_factors != unique_factors.
unique_factors has a different equation.
Also, key to recognize that 3^2 has only one prime factor: 3. it has 3^2 –> (2+1) = 3 factors, namely: 1,3,9. but only one prime factor: 3. This is true for all prime numbers that are squared! or even prime numbers that are cubed! etc.
Simplify:
(-2)^(-2)
(-2)^(-2) ==> (-1)^-2 * 2^-2 = 1/2^2 = 1/4
Note: if the negative sign was outside of parenthesis of even exponent then it would be negative. Ex:
-(2)^-2 = -1/2^2 = -1/4
What tactic should jump to mind?
DRAW SHAPE AS SKITTYWAMPUS as possible!
will help me stretch shape out.
Answer:
B
real numbers = not imaginary number (sqrt(-1)). That simple.
B.
then what is
ceil(0)
ceil(0.1)
ceil(0.9)
ceil(1)
ceil(1.1) ?
ceil(0) = 0
ceil(0.1) = 1
ceil(0.9) = 1
ceil(1) = 1
ceil(1.1) =2
C. 60 degrees.
What are the desired units? What are the source units?
mm^3 to m^3.
Remember it is CUBED!
I missed this because I copied down = instead of > on S1. Copy correct equation!!!
how to solve? what is the trick?
note: need tor recognize:
xy = xz construct.
x(y-z)=0
x=0 or (y-z)=0 => y=z are solutions
Value questions on DS must return a ____.
NUMBER!
Just because in S1 you know it is x, doesn’t mean S1 is sufficient.
C!
when you divide x, you divide out a solution in S1! x=0 or x=2
S2: x=2 or x=-2
Together:
x=2. BOOM.
What is the rephrase?
Hint: can we divide by a variable?
Answer is each alone.
What is the trick these bastards trying to get me on?
hint: look at units of two statements!
I can rephrase the given info to solve!
e!
|x|/x
