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Magoosh  Quantitative Section [Hard]
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Terms in this set (170)
If everything is in ratio/percent we don't need to deal with the numbers.
If from 2004 to 2005, Vietnam's rice production increases by 25%, and all the other countries in the "Indochina" group maintain the same levels of production, then the rice production of the Indochina group would increase by what percent?
Indochina: 80 million metric tons
28% = Vietname
A) 2.8%
B) 4%
C) 5.6%
D) 7%
E) 12.5%
Well right now we are 100%
If Vietnam increases out by 25% well that's a 28%/4= 7% increase
So if nothing else changes out total output is 107%. That's an increase of 7%
Work with a table and keep variable constant per leap. and add years to a developed relationship. Then set the new relationship equal to the one you have been developing by adding or subtracting.
Another method is to treat J and T as now; T10 and J10 as ten years ago; and T+5 and J+5 as five years from now. Once this is established, use these to write out the relationships and solve for a common variable.
Ten years ago, Josh was three times as old as Tim. In five years, Josh will be 10 years more than twice as old as Tim. How old is Tim right now?
10yr: J=3T; T = Tim
So... Now: J+10 = 3T+10; Tim = T+10
Then... 5yrs from now: J+15 =
[3T+10]+5 = 2(T+15)+10
; Tim = T+15
You can factor out something and then the sum of the inside of the parenthesis is something easier to deal with.
K = sum of the integers from 1 to 500 inclusive that are divisible by 5.
Here a five can be factored out... Check it out to see a different approach
When dealing with percent increases we can directly deal with the multiplier first. The answer will be in 1._,_,_ format from which we can easily see what the percent increase of decrease is.
Suppose China's production remains more or less constant from 2004 to 2006. Suppose India is able to sustain the same percent increase in both of those years. By what percent would India's rice production have to increase from 2004 to 2005 and again from 2005 to 2006 so that it equaled China in rice production in 2006?
A) 10.0%
B) 12.8%
C) 16.3%
D) 21.5%
E) 27.4%
India: 88
China: 112
So here it would be 88(r)(r)=112
Remember your rules. You CANNOT assume that the numbers are arranged already. You have to use the information given to figure our the order first.
List A: {x, x, x, y, y, y, 3x+y, xy }
If the median of list A is 10 and 0 < x < y, what is the range of list A?
Given that 0<x<y how can we arrange our variable first?
Do not assume unnecessary things. This is not asking us to multiply things out. Simply asking what do both these things have in common?
What is the Greatest Common Factor (GCF) of 25x2 and 16y4?
We could multiply is out to get 400x^2y^4 but that is not what the questions is asking the question did not ask us to get the GCF of (25x^2)(16y^4). It asked us what the GCF is individually.
Which of the following are equal to (1/560)^4?
Indicate all correct answers.
(70^4)(1/8)^4
First write everything out and not skip steps... You are missing questions when you do this. Notice that this is the equivalent of (70^4)(8^4), which can be written out as (70 70 70 70 8 8 8 8) = 560 560 560 560 = 560^4
Also notice that (a^m)(b^m) = (ab)^m. So... (70^4 8^4)= (70*8)^4 = 560^4
You have to at least try plugging in numbers as a last resort before you make a random guess.
Triangle on a line a is at the top, b is the adjacent angle outside the left leg, c is the adjacent angle to the right of the triangle leg.
A: 180+a
B: b+c
Try to use 70, 60, 50
Try to use 90, 60, 30
Once you get a pattern notice that 47 is not divisible by 4 or less. So go to 46, and notice that the largest number that is divisible by is 2.
What is the units digit of 18^47?
46 divisible by 2 so... _4
47 then would have to be _2
This is a division problem. First eliminate the decimal because this number is misleading saying that we can have 5^10 hidden in there but that is not possible since this number multiplied out is not a whole number.
When trying to figure out if a number can be divided, write out the complete prime factorization.
Which of the following are divisors of 1.2x10^10?
Notice that when you factor this out and group everything you get:
(2^11)(3)(5^9)
Now you can clearly see what numbers this number can be divided by
Careful with these questions. You need to figure out if this is proportional or inversely proportion.
Working together, 7 identical pumps can empty a pool in 6 hours. How many hours will it take 4 pumps to empty the same pool?
If it take 7pumps 6 hrs then it must take 1 pump 42 hours
Become confident in dealing with equalities and converting things to decimals to make the easier to deal with. Also be very careful in dealing with additions or subtractions.
Something which is flatly applied to the individual parts can be applied to the whole.
The sum of the pretax costs of Item A and Item B is $300. In Alumba, each item would be charged a flat 7%. In Aplandia, Item A is subject to 5% tax and Item B is subject to 10% tax. If the tax in Aplandia on the purchase of both items is exactly $3 more than it is in Alumba, then what is the pretax price of Item A?
Write out the equations carefully
*Draw lines to see what happens first. Also do not make up numbers if you can just deal with normal given triangle relationships. 12srt(3)
The triangle in a diagram is equilateral. The smaller circle is tangent to all three sides of the triangle. The larger circle passes through all three vertices of the triangle. What is the ratio of the area of the smaller circle to the area of the larger circle?
This is very easy to solve once you make your own diagram and draw for a radius
Deal with equations rather than own numbers to make things easier to solve
The area of a circle is equal to the area of a square.
Column A
The circumference of the circle.
Column B
The perimeter of the square.
Try it out
2(pi)(r) vs 4x
we know that area of square is x^2 and area of circle is (pi(r^2))
Write out comparisons neatly.
In a certain sequence of all positive terms,
{a1, a2, a3, ...} each term equals the previous term times a constant factor. If (a1)(a5) = 900, what is the value of a3?
a1
a2=a1r
a3=a2r=a1rr
a4=a3r=a1rrr
a5=a4r=a1rrrr
(a5)(a1)=(a1rrrr)(a1)=900=(a1^2)(r^4)=(a1r^2)^2=(a3)^2=30
Best to use variables when dealing with comparisons and use arithmetic to narrow things down.
(advanced QC trick)
Gerry is three times as old as Pat.
Column A
Gerry's age 20 years ago
Column B
Pat's age in 12 years
Now: G=3P
20yrs ago Gary must have been: 3P20
12 yrs from now Pat will be: P+12
Learn to recognize patterns from two distinct patterns!
For hard you need to make connections between the information given. Do not go off on a tangent. Try to connect two pieces of information that you've solved to the max.
(algebra + geometry/triangles)
If a right triangle has area 28 and hypotenuse 12, what is its perimeter?
ab=56
a^2+b^2=144
so... if we do 2(ab)=112 we get the square of sums!
For something as small as this, write out neatly all of the possibilities.
Set K consists of all fractions of the form x/(x+2) where x is a positive even integer less than 20. What is the product of all the fractions in Set K ?
(2/4)(4/6)(6/8)...(18/20)
Another interesting coordinate plane problem.
(coordinate + fraction comparison)
Which of the following lines intersects the vertical line x = 3 between (3, 1) and (3, 2)?
Figure out the limitations and then approach
Your job should be to draw a connection between equations.
(algebra/triangles + geometry)
A right triangle has legs of 6 and x, and a hypotenuse of r. If 5r = 5x + 9, what is the value of r + x?
36=r^2x^2
5r5x=9
Suppose in the next year, 2007, College D's expenses and enrollment remain about the same, but in addition to their current revenues, they receive an additional $50,000,000 grant. This would allow them to reduce average tuition by how much?
A) $1388.89
B) $3571.43
C) $5555.56
D) $9500.00
E) $25888.89
number of students = 36,000
Think before answering. If nothing else changes, and we have more money, how much does our tuition go down by? Well it would be 50/(number of students) = how the money would be divvied up.
Prime factorization. If k is a multiple of 1440 then it must be divisible by 1440 and for that to happen, all the prime factors of 1440 must be available in 1 to n.
n is a positive integer, and k is the product of all integers from 1 to n inclusive. If k is a multiple of 1440, then the smallest possible value of n is
First, take the prime factorization of 1440 
1440 = 144
10 = 12
12
10 = (2
2 * 3) * (2 * 2
3)
2
= 144 * 10 = 12 * 12 * 10 = (2 * 2 * 3) * (2 * 2 * 3)
(2
5)
1440 = 2 * 2 * 2 * 2
2
2
= 2 * 2 * 2 * 2 * 2 * 3
3
5
1440 has five factors of 2, two factors of 3, and one factor of 5. Any number has to have at least that many of each one of those factors in order for it to be divisible by 1440.
Well, we see a 5, so we know n has to be at least as big as 5
5! = 5
4
3
2
1 = 5
(2
2)
3
2
So far, that's three factors of 2 and one factor of 3  neither of those is enough
6! = 6
5! = (2
3)
5
(2
2)
3 * 2
Now, we have the two factors of 3, which is enough, but we have only four factors of 2, not enough yet.
The next factor, 7, doesn't help us at all. Let's go up to 8 
8! = 8 * 7 * 6! = (2 * 2 * 2)
7
2
8 * 7 * 6! = (2 * 2 * 2)
7
(2
3)
5
(2
2)
3
2
Now, we have enough factors to cover all the prime factors in 1440.
Therefore the smallest value of n is 8.
Here I at least was on the right track in thinking that this has to be a multiple/divisible problem.
The Sargon Corporation, which employs both men and women, offers an optional stockoption buyin program to its employees. If 85% of the men and 77% of the women choose to participate in this plan, then which of the following could be the total number of employees?
Indicate all possible values for the number of employees.
100
200
350
460
525
640
750
880
When the GRE says 77% of the women choose X, they DON'T mean approximately 77% (for example, 10 of 13 would be 76.9%, approximately 77%). The GRE does not mean approximately but exactly 77%. We could only have exactly 77% of something if the whole were 100 or divisible by 100. In the problem, we know that the number of women must be divisible by 100.
For the men,
So the number of men must be 20 or a multiple of 20.
The total number of employees must be a number that we can write in the form
(multiple of 100) + (multiple of 20)
(A) 100 = this could be 100 women and no men, or 100 men and no women, but the first sentence tells us that Sargon "employs both men and women," so neither can be zero. This doesn't work.
(B) 200 = this could be 100 women + 100 men (100 is a multiple of 20). This works.
(C) 350 = the 50 in 350 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work.
(D) 460 = this could be 400 + 60 or 300 + 160 or etc., many combinations of (multiple of 100 + multiple of 20). This works.
(E) 525 = the 25 in 525 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work.
(F) 640 = this could be 600 + 40 or 500 + 140 or etc., many combinations of (multiple of 100 + multiple of 20). This works.
(G) 750 = the 50 in 750 can be represented neither as a multiple of 100 nor a multiple of 20. This doesn't work.
(H) 880 = this could be 800 + 40 or 700 + 140 or etc., many combinations of (multiple of 100 + multiple of 20). This works.
Remember with quadratics you cannot cancel out variable unless you know what the variable is.
If sqrt(sqrt(sqrt(3x)))) = (2x)^1/4, what is the greatest possible value of x?
3x = 4x^2 > 4x^2  3x = 0
factor and set equal to 0
So we have 2 solutions: 0 and .75
Remember that when diluting, we do not need to change the numerator; just the denominator.
A container holds 4 quarts of alcohol and 4 quarts of water. How many quarts of water must be added to the container to create a mixture that is 3 parts alcohol to 5 parts water by volume?
The concentration we want to achieve is: 3/(5+3)=.375
Know how to set up the equations. A good place to start may be by asking how many unknowns are there?
In this care there are two unknowns SD and M and they are used to set up and equal 12.5 and 8.9.
Notice that SD is just a given number. Is SD is 2 and we need to find three SDs above the mean. Then that is (2)(3)=6 in actual movement in score. Thus M+1 represents one standard deviation above the mean so that would be 2, M+2 = 4=2+2=2S, M+3=6=2+2+2=3S.
In a certain set of numbers, 12.5 is 1.5 units of standard deviation above the mean, and 8.9 is 0.5 units of standard deviation below the mean. What is the mean of the set?
M+1.5SD = 12.5
M0.5SD = 8.9
Now this should be pretty simple
Think about what you know about mean. Mean = median if the set is consecutive or evenly spread out.
Also if nothing stands out you should aim to plug in numbers.
If the average (arithmetic mean) of seven consecutive integers is k + 2, then the product of the greatest and least integer is
k2  9
k2  2k + 1
k2 + 4k  12
k2 + 6k + 9
k2 + 4k  5
k+2 is then the median which means k+1 and k and k1 are to the left of k+2
Remember that average times the number of numbers is total.
The average (arithmetic mean) of y numbers is x. If 30 is added to the set of numbers, then the average will be x  5. What is the value of y in terms of x ?
xy is total so...
xy+30/y+1 = x5
Ratios can be cross multiplied to give us equations.
If 8 tigers were added to the zoo, the new ratio of lions to tigers would be 4 to 3. How many bobcats are at the zoo?
A) 4
B) 8
C) 12
D) 24
E) 48
Lions 32%
Leopards 16%
Ocelots 20%
Tigers 8%
Bobcats 24%
L/T = 32/8; which simplified to 4:1
If we crossmultiply this we get that L=4T meaning that however many tigers there are (T) 4x that is the number of lions
Now we know that if T if increased by 8. T+8 then...
L/(T+8)=4/3, which we can crossmultiply to get 3L = 4T+32. We know that L=4T so substituting that we get 4T+32=12T and we get T=4
We know that Bobcats is 3x Tigers so 4*3=12
Distinct means different.
What is the sum of each distinct prime factor of 9999?
11(3^2)(101)
11+3+101=115
You can group multiplication to make sense of things.
(p)(q)(r)<0
(pq)^2/r<0
Column A
pq
Column B
0
Based off the second equation we know that r must be negative
From the first equation we can make two groups (pq)(r)<0. If we know that r is negative that means that p*q MUST be positive. And any positive is greater than 0. So A is the correct answer
Notice the small things like this. We can simply flip number of people to number of televisions to get what we want.
In 1955, the ratio of the number of televisions to the number of people was approximately
A) 1 to 13
B) 1 to 23
C) 1 to 26
D) 1 to 50
E) 1 to 90
Population per television is 13 (or 13:1)
televisions to the number of people then is (1:13)
Use triangles for this!
In the standard x,y plane, a circle has a radius 6 and center (7, 3). The circle intersects the xaxis at (a, 0) and (b, 0). What is the value of a + b?
But notice the shortcut, since the two triangles are congruent it will be 7+x+7x, where x is the length of the bottom leg. This just equals 14
The trick here is to realize that the numerators are going to be identical since adding a 0 doesn't do anything. But notice that 0 is a number and the second one is being divided by 51 to get an average.
Column A
Average (arithmetic mean) of integers from 50 to 1 inclusive.
Column B
Average (arithmetic mean) of integers from 50 to 0 inclusive.
The first one's sum is 50(51)/2
The second's is 51(50)/2
These are identical
Here you have to realize that 30minutes delay but arriving at the same time means that the new driving time is shorter by 1/2hour.
Set D=rt in what you need first.
Every day at noon, a bus leaves for Townville and travels at a speed of x kilometers per hour. Today, the bus left 30 minutes late. If the driver drives 7/6 times as fast as usual, she will arrive in Townville at the regular time. If the distance to Townville is 280 kilometers, what is the value of x?
regular driving time = new driving time + 1/2hour
280/x = 280/(7/6x) + 1/2
t=D/x
The 1/2 is very important it is the part that is left on the table for being late and since we arrive at the same time it means that whatever the new time is, that plus 1/2hr must equal the old time
This is another ratio problem that we can crossmultiply to make things simpler.
If 3/4 of the number of women working at Company X is equal to 2/3 of the number of men, what fraction of the employees at Company X are women?
3/4W=2/3M > 9W=8M
We want ratio of women to total. Well W/M would be 8/9
In another words there would be 8 W for 17 (8+9) employees
Learn to save time by examining the limiting factors.
System of equation of rebuilding the remainder doesn't work because the quotient is different for both the equations!
x is a positive integer less than 100. When x is divided by 5, the remainder is 4, and when x is divided by 23, the remainder is 7. What is the value of x?
Notice that we could list all the possibilities of 5 but that would be tedious and cumbersome
Notice that we need a positive integer value below 100, so... use 23 instead start with 7 and keep adding 23 to that
7, 30, 53, 76, 99
Now we only need to check 5 values of which it is easy to see that 99 must be the answer
Number and integer property.
0 is an
INTEGER
that is
EVEN
[It is neither positive nor negative.]
Remember that when we add the same value to the numerator and denominator the value gets closer to 1.
Yesterday, at a certain school, the ratio of boys to girls was 1 to 3. Today, an equal number of boys and girls joined the school. The number that joined was greater than zero and no students left.
Column A
Number of Boys to Girls Now
Column B
1/3
It's easy to see based off the rule that A will always be bigger
Remember to keep track. Remember to not double count.
Another thing to remember is that positive and negative need to cancel out.
The sum of k consecutive integers is 41. If the least integer is 40, then k =
Based off the second rule we get 40 to 0 and 141 which gives us 81, the 82nd number is 41
Check out the saved image 41
Write down what equations you know.
Big Rule: Something+x > x
Try to use what is given first.
180ab=x [top triangle]
180cd=y
Well there is another way to get x... 180bcd=x
Set these equal to each other and you get that a=c+d
Since we know that c+d=a and the column asks as to compare c+d vs a+b; it should be obvious that a+b is bigger

Another way is:
a+x=180 [x being in the bottom triangle]
c+d+x=180
Check out saved image 42
Look at the big picture!
Should be really easy once you do that and use the information given
Check out saved image 43
Remember your triangle properties and use them effectively.
Magoosh trick is to come up with two equations equal to 180
The outer angle is the sum of the opposite inner angles
We are asked to compare the opposite top angle (y) vs the outer angle
Well y+something = x. This means that by itself x ALWAYS greater than y alone

Using Magoosh we should get that
y+a+b=180
b+x=180
Set these equal to each other to get that y+a=x
Meaning that x is greater than y by itself
Remember if not told explicitly, all of these integers could be the same exact number.
If a, b, c, d, e and f are integers and (ab + cdef) < 0, then what is the maximum number of integers that can be negative?
The answer here is 5
This was tricky. If using plugandchug start with E.
Otherwise recognize that to solve this we need to add .2x top the top and x to the bottom and set that equal to (1/4=.25).
Solution Y is 40 percent sugar by volume, and solution X is 20 percent sugar by volume. How many gallons of solution X must be added to 150 gallons of solution Y to create a solution that is 25 percent sugar by volume?
1/4 = (60+.2x)/(150+x)
Write formulas and keep track.
For a certain event, 148 people attended. If all 148 had paid full admission price, the total revenue would be three times the cost of sponsoring the event. (Admission price was the only source of revenue.) As it happens, only 50 paid the full admission price, and the others paid nothing.
Column A
the total revenue
Column B
the cost of sponsoring the event
148P=3C. So C = 148/3P; that is the right hand side
Total revenue is only going to be 50P
So we are essentially comparing 148/3P to 50P and this means that 50P is always going to be greater
GRE jargon
Must be true
= the statement or equation must hold
ALL of the time
or in all cases
Remember the Mississippi rule.
A librarian has three identical copies of a single cookbook and four different novels that he wants to display. Assuming all seven books will be in a single row, how many different arrangements can he make?
Here we have (7!)/(3!)
Pick numbers and work with them. Remember is something is 20% less than someone else's score, that means that "something" is only 80% of someone else's score.
Ashley's score was 20% higher than Bert's score. Bert's score was 20% lower than Charles' score.
Column A
Ashley's score
Column B
Charles' score
A: 240 if B:200
Since B is 20% less than C that means that C=200/.8=250
Check out saved image 44
W, X, Y and Z each represent a different number. If the sum of each column is shown beneath that column, and the sum of each row is shown beside that row, then n =
Remember that the sum of all the columns and sum of all the rows must be equal
How to set this in an equation format.
Helen mixes a first solution, 4 liters of 40% concentrated sulfuric acid solution, with a second solution, 5 liters of a sulfuric acid solution with a stronger concentration, and the resultant solution is 9 liters of 50% concentrated sulfuric acid solution. What was the concentration, as a percent, of the second solution?
4(.40) + 5x = 9(.50)
We could set this equal to the 180 equations. But in this case we can solve for x and y and see what works.
Check out saved image 45
Column A
x
Column B
y
x=180w+1
y=180w1
now compare
This is a counting question of possibilities.
A knockoff website requires users to create a password using letters from the word MAGOSH. If each password must have at least 4 letters and no repeated letters are allowed, how many different passwords are possible?
At the least we must have a scenario with 4 letters: 4!
Another scenario that we must add to this is the 5 letter password: 5!
Lastly, we need to consider the 6 letter scenario: 6!
S1: (6)(5)(4)(3)=360
S2: (6)(5)(4)(3)(2)=720
S3: (6)(5)(4)(3)(2)(1)=720
The sum is 1800
The largest integer value that a remainder can be has to be one less than the divisor. Otherwise the divisor would go into the dividend.
n is a positive integer.
n is not divisible by 4.
n is not divisible by 5.
Column A
The remainder when n is divided by 4
Column B
The remainder when n is divided by 5
Test multiple scenarios values less than both the divisors and value above
Don't assume; for this write down what is available then solve.
While driving from Aville to Btown, Harriet drove at a constant speed of 115 kilometers per hour. Upon arriving in Btown, Harriet immediately turned and drove back to Aville at a constant speed of 135 kilometers per hour. If the entire trip took 5 hours, how many minutes did it take Harriet to drive from Aville to Btown?
T1+T2=5hrs
D=115T1
D=135T2
This may look like a recursive sequence but we have to think of it in terms of arithmetic sequence.
The nth term (tn) of a certain sequence is defined as
tn = tn1 + 4. If t1= 7 then t71 =
Solve this using arithmetic sequence and recognize that
Key here is to realize 1) that the xintercepts are symmetrically located. We should notice that the height is (0,k). If we plug in 0 for x (to find the xcoordinates we get sqrt(k),0 and sqrt(k),0. This means that the total base is 2sqrt(k). Plug this into the area equation to solve.
Check out saved image 57
The figure shows the graph of the equation y =kx^2, where k is a constant. If the area of triangle ABC is 1/8, what is the value of k?
See solution image 57
GRE question tip.
Be careful about questions asking about the "difference" of something. For ex: "how much shorter would it be to go from x to y directly rather than x to z to y?"
This was a triangle question and the trick answer was 5 (x to z is 4; z to y is 4)
The right answer is 75=2
Be very careful about what is being asked. We are asked for the TOTAL TIME not just the second leg of the trip. I missed this despite solving everything correctly and picking 25 instead of 30.
For the first 5 hours of a trip, a plane averaged 120 kilometers per hour. For the remainder of the trip, the plane travelled an average speed of 180 kilometers per hour. If the average speed for the entire trip was 170 kilometers per hour, how many hours long was the entire trip?
Average speed = total distance/total time
=(120*5)+(180T)/(5+T)
distance for second leg is 180T=D
n is a positive integer
Column A
(0.99)^n
Column B
0.01
Key is to note that any fraction raised to a "n" power will approach 0.
But the relationship can vary for this problem since different values of n give different relationships
For all numbers a and b, the operation & is defined by a b = a^2  ab. If xy ≠ 0, then which of the following can be equal to zero?
I. x & y
II. xy & y
III. x & (x + y)
Key is to work out the operation and then simplify by factoring and then seeing what possible numbers could make the situation zero given our limitation.
Notice for the last one that we get: x^2  x(x+y). This simplifies to x^2 x^2

xy {NOT x^2x^2+xy}
Check out saved image 62
Column A
AB + AD
Column B
DC + BC
Follow your intuition. Remember that angles do not dictate how long or short the sides of a triangle are! They can only tell us which side is shorter or bigger but the shape could be changed dramatically and the angles can still hold.
Check out saved image 63
Column A
Number of different triangles possible using the given points as vertices.
Column B
42
Notice that this is a counting problem. We have 7 points and any 3 points define a triangle but triangle dgb is the same gdb. Essentially this is a combination problem then.
7C3 will give us 35 which is less than 42
Slope gives a lot of answers.
Divisibility rules and coordinate geometry.
How many points (x, y) lie on the line segment between (22, 12 2/3) and (7, 17 2/3) such that x and y are both integers?
Solve for the slope and then see what works
I approached this right but missed it based off 5x+6
If x>0, and two sides of a certain triangle have lengths 2x+1 and 3x+4 respectively, which of the following could be the length of the third side of the triangle?
4x+5
x+2
6x+1
5x+6
2x+17
notice that 5x+6 can never work (try plugging any number for x, it will be 1 greater than our required criterion)
6x+1 and 2x+17 could world
Once you reach a point where you can't move further. Plug numbers to see what makes sense. We are comparing 8 so plug than into the matrix to see if that works or not.
In a group of 45 children, 60 percent of the children are boys, and 60 percent of the children are lefthanded.
Column A
Number of boys who are lefthanded
Column B
8
Plug and chug when you can't move forward with anything else
With quartiles our data must be organized. In this case it is from lowest score to highest. So all we have to do it count the number of that gets us to the middle of the bottom half of the data meaning between the 10th and 11th student will be where our Q1 score is.
Boxand whisker takes population and spreads it over whatever metric is being measured. So the spread is the difference between the lowest and highest metric. The population itself determines the quartiles because it would show how many fall with what quartile of the metric being measured. As long as the data is organized all we have to do it count the population into even 4 parts.
Check out saved image 67
The first quartile SAT score among the forty scores at the Newboard Free School is in which score range?
1300  1400
1400  1500
1500  1600
1600  1700
1700  1800
40 students:
120 bottom half (halfwar between this is 1011)
2140 upper half (3031)
This is a dependent probability with no replacement. With pairs or whatever the first chance is not dependent on anything. We can pick anything or in other words we have p=1 for the first pick.
A box contains 10 pairs of shoes (20 shoes in total). If two shoes are selected at random, what is the probability that they are matching shoes?
1/190
1/20
1/19
1/10
1/9
The first time we can select any shoe (20/20)
The second time our shoe has to match the first, since we already picked a show there are 19 left so our probability is (1/19)
Overall our probability is (1)(1/19)=1/19
Realize that all the digits are set except for the 3s. Basically what we want is to select 2 spots of the 7 available and not repeat.
Sid intended to type a sevendigit number, but the two 3's he meant to type did not appear. What appeared instead was the fivedigit number 52115. How many different sevendigit numbers could Sid have meant to type?
10
16
21
24
27
The combination for that is 7C2 = 7*6/(2)(1)=21
Another is to count each scenario and remember that we can go down to the lowest which is 1 so... Our first choice with the 3's next to each other gives us 6 choices our next is 5 if we skip a blank since is 4 if we skip 2...down to 1
6+5+4+3+2+1=21
Remember to write out the equations neatly.
If we do not know the exact number and let a variable equal to the actual number then we have to add or subtract from the variables.
Every person in a certain group is either a Dodgers fan or a Yankees fan, but not both. The ratio of Yankees fans to Dodgers fans is 5 to 3. If 22 Yankees fans change teams to become Dodgers fans, the ratio of Dodgers fans to Yankees fans will be 1 to 1. How many people are in the group?
y/d=5/3
y22/d+22=1/1
Special case where we can use percentage to get a RELATIONSHIP of the actual number of CDs. We don't know exactly know many CDs there are but the relationship will always hold.
Yesterday, Carl had 40 percent more CDs than Karen had. Today, Carl gave 20 percent of his CDs to Karen.
Column A
Number of CDs that Carl now has
Column B
Number of CDs that Karen now has
Try 100 and 200
Power rule.
If 8^n+1 + 8^n = 36, then n =
8^(n+1) = 8^n(8)
Percent trick. We don't have to deal with the number at all just the percent.
At the moment there are 54,210 tagged birds in a certain wildlife refuge. If exactly 20 percent of all birds in the refuge are tagged, what percent of the untagged birds must be tagged so that half of all birds in the refuge are tagged?
Let's work with an easy number of 10
2 of these are tagged and 8 aren't we need to have 5 tagged well 3/8 is 37.5% of the untagged birds
Check out saved image 74
This is a triangle rule: hypotenuse of a triangle is always bigger than the legs.
Here we are essentially comparing
leg/hypotenuse to hypotenuse/leg
clearly the second must always be bigger since quantities are fixed and the ratios remain the property no matter what the numbers are (hypotenuse/leg 1+) vs (leg/hypotenuse 1)
We can either work with our own numbers of equations.
Captown is the capital city of Maltania. If the population of Captown is 25 percent of the rest of the population of Maltania, then the population of Captown is what percent of the entire population of Maltania?
If P=100 then (100C)(1/4)=C. What is C/P?
Another approach is C/(MC)=25/100 and we need C/M
This is really tricky! We wants our answers in terms of k so do not forget to keep track of variables.
A certain barrel is 1/5 full. When k liters of liquid are added to the barrel, it becomes 2/3 full. In terms of k, what is the capacity of the barrel, in liters?
F=Capacity of barrel
(1/5)F+k=(2/3)F
k=7/15F [This is k in terms of F]
We are asked for capacity of barrel (F) to be represented in terms of k. So We need F=15/7k)
Check out saved image 77
On the GRE when we see one of the algebraic equations separate it if it is in complex fraction form and a solution will emerge.
If x + y ≠ 0, which of the following is a solution to the inequality
x=3 and y =7
x=3 and y =7
x=11 and y =9
x=9 and y =6
x=20 and y =24
x=12 and y =9
x=2 and y =16
In this case we can separate to get [(x^2y^2)/(x+y)][1/(x+y)]>1/(x+y)
Here our 1/(x+y) get canceled out and we are left with [(x^2y^2)/(x+y)]>0 and this simplifies to xy>0 which equals x>y
This should allow us to see the solutions easily
When given the algebraic we have to factor and use the information given.
x2  y2 < 8
x + y > 3
If x and y are integers in the above inequalities and 0 < y < x, what is the greatest possible value of x?
(x+y)(xy)<8
x+y>3
It's clear that the maximum value that x+y can be is 7 and xy has to be 1 we can use system of equations to figure out that the max value of x is 4
Careful about arrangements! I missed this because I arranged smallest to greatest instead of greatest to smallest.
If A, B, C and D are positive integers such that 4A = 9B, 17C = 11D, and 5C = 12A, then the arrangement of the four numbers from greatest to least is
To solve this start with A and get is alone for ex: A=9/4B meaning that A>B. Do this for all of them and see how things stack up
Answer should be DCAB
Percentiles go left to right and essentially give a percentile mark to each person.
Check out saved image 80
Suppose all students surveyed answered in integer number of hours only. Suppose, of 82 surveyed, only one respondent answered "16 hours." Within this group, the approximate percentile of this person would be:
32nd percentile
51st percentile
67th percentile
75th percentile
80th percentile
Here we want the person that is at 16 which means the first three columns added together divided by 82 will give us our percentile because that person will be above all the first three columns
65/82 ~ 64/80 ~ 8/10 ~ 80%
Make your own numbers and keep track of the variables and numbers neatly.
It takes 1 pound of flour to make y cakes. The price of flour is w dollars for x pounds. In terms of w, x and y, what is the dollar cost of the flour required to make 1 cake?
xy/w
y/wx
w/xy
wx/y
wxy
Let's say that 1 pound of flour makes 5 cakes. So y=5
Let's say that it costs 6 dollars for 2 pounds. So w=6 and x=2
that's $6 for 2 pounds which can make 10 (2*5) cakes
6/?=10/1
That's $0.6 for 1 cake
Plug in to see which option gives us this 6/(2*5)=w/(xy)
Be very careful about how you take care of the left side.
If 2^2n + 2^2n + 2^2n + 2^2n = 4^24, then n =
a^2+a^2=2(a^2); similarly... the left side of the equation is:
4(2^2n)=4^24 now you can solve. The left side simplified to 2^2+2n
Beware! The left side DOES NOT EQUAL 8^2n. We cannot add the bases.
There are two ways to tackle this.
If 2^k = 3, then 2^(3k+2) =
For the right hand side you can recognize that 2^3k= (2^k)^3.
For multiple exponents we multiply out
We can pug in 3 for 2^k
Another way to recognize is that for the left equation we can simply raise both sides to the power of 3 and get that 2^3k=27
Answer should be 108
First start by determining how many unique combos are possible and then move from there.
Integer properties: If P and Q is a multiple of r then P+/Q and P+r will be multiples of r. We can also keep adding or subtracting Q and r multiple times.
Set A: {1, 3, 4, 6, 9, 12, 15}
If three numbers are randomly selected from set A without replacement, what is the probability that the sum of the three numbers is divisible by 3?
7C3 = How many combos are possible that are unique that will be the denominator
Based off the integer rule we can see that 5 numbers are divisible by 3. So... 5C3 will be our numerator
Start by questioning what the upper bound maximum area could be. That is the reason the limits are given.
Point A (4, 2) and Point B (2, 4) lie in the xycoordinate plane. If point C lies in the first quadrant and contains the coordinates (p, q), where p < 2 and q < 4, which of the following could be the area of triangle ABC?
1.1
3.9
11.9
We have 3 conditions:
1) p<2
2) q<4
3) point is in the I quadrant
This means that the lowest point in which our point could maximize the area is (2,0). If we get the area of this triangle and then compare we should see that all three options work.
Careful about "at least 2" vs "exactly 2". If it were exactly two then our answer would be 9/64. But now we have to consider the scenario in which all the rolls could be 6s.
A weighted die, numbered one through six, has a probability of 1/4 of rolling a six. If this die is rolled three times, and each roll is independent, what is the probability of rolling at least two sixes?
P(at least 2 6s) = P(two 6s) + P(three 6s)
P(2): 3C2((1/4)^2)(3/4)
P(3): (1/4)^4
(9/64)+(1/64)=10/64= 5/32
Remember that we cannot start with 0 as the even since 0 is neither positive nor negative.
X = sum of the first 31 positive odd integers
Y = sum of the first 30 positive even integers
Column A
X  Y
Column B
30
Check out saved image
Be careful to read the statements clearly. Here is it asking for "MORE THAN 50%". That would mean that we need 2 not 1 since 1 would make it exactly 50/50.
There are 44 nations in total
What is the least number of nations currently not members of NATO that would have to join NATO so that more than 50% of the nations in Europe would be members of NATO?
1
2
3
4
5
Check out saved image 88
I tired this with formulas but quit since I thought it wouldn't work and then just relied on my made up version of D. For the formula I was on the right track.
Andy drove from Townville to Villageton at an average speed of 40 miles per hour. He then drove from Villageton to Townville at an average speed of 60 miles per hour.
Column A Column B
50 The average speed of Andy's entire trip in miles per hour.
Total time = (d/40)=(d/60)
Total distance = 2d
2d/(d/40+d/60)=48 since the d's cancel
Check out saved image 90
Column A
AB
Column B
BC
Triangle rule: An isosceles triangle will have the angle bisector as the altitude. But just because we see an angle bisector that does not mean that we have an isosceles.
check out saved image 90 solution for image of why D is the right answer
Sometimes you just have to pick numbers for variables and solve that way.
If 3x < 2y < 0, which of the following must be the greatest?
2y  3x
3x  2y
(3x  2y)
(3x + 2y)
0
I used 10=3x and 1=2y
Another example where assigning numbers for variables gets you a quicker answer.
2x + y years ago, Roberto was 3x + y years old. How many years old was Roberto x years ago?
x
4x + 2y
5x + 2y
6x + 2y
6x + y
I used x=2 and y=3 and came up with the number 14 for Robert's age x years ago
This gives us B as our answer
Be able to spot the meaning of parallel lines. Here that is key to see that parallel lines allow to equate two angles to be the same.
Check out saved image 93
In the figure below, ∠ADE = 60°, ∠EFC = 40°, and ∠DAE = 55°. If AB  CD, what is the value of x?
The angle at the top right is equal to the top angle of the bottom right triangle
Check out saved image 94
If 2 ÷ 2 ÷ 2 ÷ 2 ÷ 2 = 2^x, then x =
Here it's easier to deal with this in the format that If we divide something with something else the exponents subtract topbottom
The answer should be 3
Divide into stages of which there are 2. Next realize that this is a combination problem since order does not matter.
From a total of 5 boys and 4 girls, how many 4person committees can be selected if the committee must have exactly 2 boys and 2 girls?
(5C2)(4C2)=60
Do not forget how the line looks like. In this case it is downward sloping so the answer will be negative!
In the xycoordinate system, line k has yintercept 12 and an xintercept greater than zero. If the area of the triangular region enclosed by line k and the two axes is 30, what is the slope of line k?
Answer: 12/5
Be careful about how you organize this. Our first multiple of 5 is 85 NOT 80!
How many multiples of 5 are there between 81 and 358?
85/5= 17
355/5=71
[7117]+1=55
I started at 80 and got the answer to be 56 which is wrong since it includes 80 as a multiple but that's not part of our range
Most of the info in this question is useless and plays no role. Best way is still to deal with variable but realize that the answer is simple.
Do not try to write an equation with k as the variable that will lead to wrong answers. We don't know if k is in the terms we need. Write out how you would normally write out using your own variables and see if you can draw a connection.
If the average (arithmetic mean) of five consecutive negative integers is 2k  1, what is the difference between the greatest and least of the five integers?
4
4k
4k + 4
4  4k
4k2  4k
x+x+1+x+2+x+3+x+4
The difference between the largest and smallest is
x+4x=4
Remember that 2pir=pid. Use the variables to solve methodically.
Again use the variables given and set equal to what you want to solve for. In this case we want to solve for the diameter or D.
If ABCD is a rectangle, BC = x and AB = 2x, then the circumference of the circle, in terms of x, is
Check out saved image
x^2+(2x)^2=D^2
D=sqrt(5x^2)=sqrt(5)x
So the circumference is sqrt(5)xpi
Look out for shortcuts when available.
If (8x)/(x+1)= x, then x2 + 2x  3 =
Here we get 8 = x^2 + 2x
If we subtract 3 from both sides we get 5
Keep in mind that even a variable divided by the same variable is 1.
If w/x = 2/3 and w/y = 8/15, then (x + y)/y =
(x+y)/y = (x/y)+(y/y) = (x/y) + 1 = ?
We can easily use the relationships given to get what x/y must be
Exponent rule.
If 4^n + 4^n + 4^n + 4^n = 4^16 , then n =
Notice that the left side simplifies to 4(4^n) = 4^16
We can divided both sides. To do this correctly with exponents realize that (4^16)/(4^1) = 4^15
This is not tough but good to highlight that something at the SD 3.2 and 10.4 are inclusive so you would count those numbers if those numbers were part of the list.
If the mean of list A is 6.8 and the standard deviation is 3.6, then how many elements of list A are within 1 unit of standard deviation of the mean?
A = {2, 9, 2, 6, 9, 10, 7, 4, 5, 14}
Nothing to comment simply solve and count correctly
If the exponents are different but the exponents are the same then we can do the same things to the base as with any other number.
Check out saved image 104
If [image], then k =
8
12
16
24
36
Check out the saved image solution
The change multiplied by 3 is equal to A. Pay attention to the key word "then".
Appleton's population is 400 greater than Berryville's population. If Berryville's population were reduced by 900 people,
then
Appleton's population would be 3 times as large as Berryville's population. What is Berryville's current population?
A=B+400
A=3(B900)
Think about it theoretically. Otherwise this could be solved using numbers.
If the retail price of a shirt is R dollars, and the price including sales tax is T dollars then the sales tax, as a percent, is
Check out saved image 106 for options and answers
T=retail + tax
R= Retail
TR=tax rate
(TR/T)100 = Percent tax rate
Remember grouping things together and then multiply that by the ways that in groups can be arranged.
In how many different ways can 3 boys and 3 girls be seated in a row of 6 chairs such that the girls are not separated, and the boys are not separated?
24
36
72
144
288
2(6)(6)
I solved this first by assuming 55 are the right answer. Another way would be to keep increasing the number of candies from 1 onwards till you get to the 10 child who gets the rest.
Joan has 100 candies to distribute among 10 children. If each child receives at least 1 candy and no two children receive the same number of candies, what is the maximum number of candies that a child can receive?
10
34
39
45
55
1+2+3+4+5+6+7+8+9+____
The sum is equal to [(9)(10)/2]=45
That means that the 10th child gets 55
Realize that in questions like this, the weights of the cats could be anything. They could all weight the same!
Jack has 5 cats and 1 dog. If the dog's weight is 3 times the average (arithmetic mean) weight of the cats, then the dog's weight is what fraction of the total weight of all 6 animals?
5x/5=x
Dog = 3x
3x/8x=3/8
2k years ago Frank was 3k years old. In k years Frank's age, in years, will be
4k
5k
6k
7k
8k
Notice that Right now the age is F
F2k=3k so F=5k
In k more years the age will be 5k+k=6k
Do the formula that you come up with. Don't just sit there. Mess around with the formulas you do know.
In a group of children, the average (arithmetic mean) weight of the boys is 60 pounds, and the average weight of the girls is 48 pounds. If the average weight of all of the children in the group is 50 pounds, what is the ratio of the number of boys to the number of girls?
1/12
1/6
1/5
1/4
1/3
B/x=60
G/y=48
(B+G)/(x+y)=50 => (60x+48y)=50x+50y => 10x=2y => y=5x
B:G = x:y => x:5x => 1:5
This question has 2 scenarios. When we deal with scenarios we "add" since they are mutually exclusive.
A box contains 4 red chips and 2 blue chips. If two chips are selected at random without replacement, what is the probability that the chips are different colors?
1/2
8/15
7/12
2/3
7/10
S1: B, R
S2: R, B
S1: (2/6)(4/5)=4/15
S2:(4/6)(2/5)=4/15
S1+S2=8/15
I did this through POE, where I knew the denominator (30) had to be a divisible by the answer choices. C does not work. There is no way this is a 50/50 so A is wrong. 2/3 and 7/10 look too big so I went with B.
Probability lesson: If on the first try Ann were supposed to pick a specific ball the probability would be 1/10 but since she can pick any her probability is 10/10.
A box contains 10 balls numbered from 1 to 10 inclusive. If Ann removes a ball at random and replaces it, and then Jane removes a ball at random, what is the probability that both women removed the same ball?
1/100
1/90
1/45
1/10
41/45
This is an "and" scenario
Ann: 10/10
Jane: 1/10
(1)(1/10)=1/10
Do not assume that x y and z have to be in ascending order.
10^x + 10^y + 10^z = n, where x, y, and z are positive integers
Which of the following could be the number of zeroes, to the left of the decimal point, contained in n?
x + y
y  z
z
It's easy to see why A and B work
The only way to test c is to do a descending order where x=3,y=2,z=1
So all three work
This deals with tricky scenarios with the double O's which are repetitive.
From the letters in MAGOOSH, we are going to make threeletter "words." Any set of three letters counts as a word, and different arrangements of the same three letters (such as "MAG" and "AGM") count as different words. How many different threeletter words can be made from the seven letters in MAGOOSH?
Scenario with OOs: XOO; OXO; OOX. In each of these cases we can pick 5 of the remaining letters for X for a total of 5+5+5=15 choices
Then we have to consider just the 6 letter scenario of MAGOSH (6)(5)(4)=120
120+15=135
These one's can be confusing so just start with one statement at the time from the top.
Anne pays 150 percent more for a wholesale widget than Bart pays.
Anne's retail price per widget is 15 percent greater than the wholesale price she paid.
Bart's retail price per widget is 185 percent greater than the wholesale price he paid.
Column A
Anne's retail price.
Column B
Bart's retail price.
Pick numbers that satisfy the first statement and move on from there
Manipulation choices.
Which of the following equations is true for all positive values of x and y?
Check out saved image 117
Notice here everything is in equation form
One side of the equation must equal the other. How can you manipulate for make sure one of them is true?
Concept review: how percentiles work.
The following boxplot shows the 2012 season runs batted in (RBIs) of 280 American League (AL) batters (the top 280 batters in terms of number of plate appearances).
Check out saved image 118
FiveNumber Summary for AL RBIs in 2012:
Minimum = 0
First Quartile = 9
Median = 25
Third Quartile = 56
Maximum = 139
There is a fast way to solve this by realize that the percentile represents the number of people so total times the percentile will give you the people at or between those percentiles
9075=15%
15%(280)= 28+14=42
Areas of circle.
Check out saved image 119.
A cow is tethered to the corner of a rectangular shed. If the length of the rope is 5, and the shed has length 4 and width 3, what is the maximum area that is accessible to the cow? (The cow cannot enter the shed).
Check out solution but think in terms of circles first
Equation of a circle in the coordinate plane.
In the standard x,yplane, a circle has a center = (1, 2) and a radius r = 5. Which of the following points are on the circle?
(4, 2)
(3, 1)
(1, 6)
(1, 7)
(3, 2)
(4, 6)
(5, 1)
(6, 2)
x^2+y^2=r^2
in this case (x1)^2+(y+2)^2=25
Consecutive integers
x is a positive integer. k is the remainder when x3  x is divided by 3.
Column A
k
Column B
1
(x^3x)=x(x^21)=(x1)x(x+1)
There there are 2 consecutive integers one of their products must be divisible by 3 and if that is the case then their product must be divisible by 3 as well and so the remainder will always be 0
I did this just by plugging in numbers to see what happened and missed it only because I accidentally picked A instead of B
Concept knowledge about medians.
The median of x , y , 8 and 11 is 19.
Column A
x
Column B
23
Notice that our median is 19 in which case we know that the average of the two middle numbers must be 19
That means that two of our numbers are bigger than 19 and two of our numbers are smaller than 19
Knowing this we can place 8 and 11 to the left and know have to determine the 3rd place number which has to be 27 which is bigger than 23
Check out saved image 122
What was the approximate population of Town X in 1945?
150
750
1500
3000
6000
This isn't hard and I had the right approach but picked the wrong year to get the number of population per television
B is the right answer
Think about the whole number versus individual sums. Also "numbers" is not the same as "integers".
Check out saved image 124
Column A
p + q
Column B
5
Here the answer is D
Think about it like this let's say our
numbers
were 0.2/0.3=2/3; well in this case p+q is definitely less than 5
On the other hand let's say our numbers were 2000/3000=2/3; Well 3000 is definitely bigger than 5
How averages work.
Check out saved image 125
There are 10 employees in an office, not counting the office manager. The table shows how many employees have 0, 1, 2 or 3 pets. If the office manager also were included in the table, the average (arithmetic mean) number of pets per person would equal the median number of pets per person. How many pets does the office manager have?
3
4
5
6
7
I solved this first by looking at the answer choices
The way averages here work is total animals/total employees
The middle numbers must be an integer by itself (given the answer choices) and the fact that we will have 11 people to consider
Use one of the answer choices beginning with 7 to get the mean and see if is equals that numbers as well
Only 6 works out
How many times does each number appear in the list?
The sum of all the digits of the integers from 18 to 21 inclusive is 24 (1+8 + 1+9 + 2+0 + 2+1 = 24). What is the sum of all the digits of the integers from 0 to 99 inclusive?
450
810
900
1000
1100
0=20x
1=20x
2=2x
so... our sum is:
(0)+(20)+(40)+(60)+...(180)=900
Realize that the only choices that don't work are the one's the make the denominator 0.
If , then which of the following are possible values of x?
Check out saved image 127
Indicate all such values.
60
12
1
1
2
5
Here both sides of the equation are equal except when...
Even number list so the median is the average of the 50 and 51st digit.
What is the median number of televisions per household?
Check out saved image 128
Cannot be determined
1
2
2.5
3
In this case both the 50 and 51st are 3
Multiplier problem.
If A is the initial amount put into an account, R is the annual percentage of interest written as a decimal, and the interest compounds annually, then which of the following would be an expression, in terms of A and R, for the interest accrued in three years?
A(R)3
A(R+R3)
A(3R+3R2+R3)
3A(R)3
3A(R+R2+R3)
A
1st: A(1+R)
2nd: (A+AR)(1+R)=A+AR+AR+AR^2 (basically multiply everything by 1 and add it to everything multiplied by R)
3rd: (A+AR+AR+AR^2)(1+R)=A+AR+AR+AR^2+(AR+AR^2+AR^2+AR^3)
C is the answer
Another way is
A(1+R)^3
We don't know what sides are what and what angles are what. So, do not draw and triangle and label it and assume all triangles to be that way DEF could have different sides than DEF where DE from the first is different from DE of the second.
Two sides of triangle DEF are equal to 3. Which of the following, taken alone, would be sufficient in finding the area of triangle DEF?
The ratio of DE to EF = 1 : √2
The sum of angles DEF and EFD is 135 degrees
The sum of angles DEF and FDE is 90 degrees
Very Hard  I missed this
With the information provided in [A], there are still two possible scenarios:
DE: EF: DF = 1:√2:1
DE: EF: DF = 1:√2:√2
Notice that two of the sides are equal for each of the scenarios, since the question states two sides equal 3. Yet, the two different proportions above result in two different areas. Thus, [A] is not sufficient.
With the information provided in [B], there are still two possible scenarios:
45 degrees, 45 degrees, and 90 degrees
67.5 degrees, 67.5 degrees, and 45 degrees
[B] is not sufficient.
For [C], DEF and FDE must be the two sides that equal Since the remaining angle, EFD, must equal 90. And you can't have two angles in a triangle that both equal 90 degrees. Therefore, given the information in [C], we have a 45:45:90 triangle with sides 3:3:3√2. Since, such a triangle has only one possible area, [C] is sufficient.
Opposite angles are 90 degrees.
Check out saved image 131
Column A
AB
Column B
BC
The answer is D; you can have a triangle in which the two sides differ and a triangle in which the two sides are the same
Independent Probability: P(A and B) = [P(A)] [P(B)].
Mutually Exclusive: P(A or B) = P(A)+P(B)P(A and B)
Events A and B are independent.
The probability that events A and B both occur is 0.6
Column A
The probability that event A occurs
Column B
0.3
P(A and B) = [P(A)] [P(B)]
We use this formula when events are
independent
0.3 = P(A) P(B)
P(A) and P(B) must both be greater than 0.6; because if not than one of them will end up being greater than 1 and that is not possible since the maximum probability can only be 1 or 100%
Approximation: Meaning in this case we can use simple interest.
Simple interest you just need the annual rate and don't even have to worry about dividing it in quarters because it's the equivalent of applying a base rate to everything and adding versus adding and then applying the base rate.
Diana invested $61,293 in an account with a fixed annual percent of interest, compounding quarterly. At the end of five full years, she had $76,662.25 in principal plus interest. Approximately what was the annual percent rate of interest for this account?
1.2%
4.5%
10%
18%
25.2%
(4.5)(400)(61,293)(4)(5) = ~75,000
~60,000
~5%=3000/yr
5=15,000
60K+15K=75K
Great scenario where you can cancel off variables.
x and y are integers greater than 5.
x is y percent of x2
Column A
x
Column B
10
10*10 are the only integers greater than 5 that give 100
so C is the answer
Great case of 0 (nonnegative and nonpositive integer). And 1 being a divisor of every single integer.
If k is a nonnegative integer and 15^k is a divisor of 759,325 then 3^k  k^3 =
15=3*5 since the number is not divisible by 3 the number cannot be divisible by any power of 15
0 must = k
Remember to count 2 as a number.
How many positive integers less than 100 have a remainder of 2 when divided by 13?
6
7
8
9
10
That should give us 8
Build a table with 0 as even and 1 as odd
Always four scenarios with 2 numbers. EE;OO:EO;OE
2 variables/stages with with 2 options (E or D): (2)(2)=4. Since there are only 4 possible scenarios then we must be able to evaluate each expression to be true or false.
If x and y are integers, and w=(x^2)y+x+3^y, which of the following statements must be true?
If w is even, then x must be even.
If x is odd, then w must be odd.
If y is odd, then w must be odd.
If w is odd, then y must be odd.
I missed this
Check out saved image 137 for the solution
ABC are true
ALWAYS, ALWAYS, ALWAYS, consider the case of the remainder.
We should also consider the negative integers, in this case we don't need to.
x is a positive integer.
When x is divided by 2, 4, 6 or 8, the remainder is 1.
Column A
x
Column B
24
I missed this
Here there are on two possible values for x
Can can either be a multiple of 25 in which case A works
OR
X can be 1 in which case B is true
That being the case the answer must be D
Break into stages and see what happens.
In how many ways can 16 different gifts be divided among four children such that each child receives exactly four gifts?
16^4
(4!)^4
16!/(4!)^4
16!/4!
4^16
1st: 16C4
2: 12C4
3: 8C4
4: 4C4
This equals to C
Pay attention to what the question is asking. What will be the greatest range?
Positive integers a, b, c, d and e are such that a < b < c < d < e. If the average (arithmetic mean) of the five numbers is 6 and d  b = 3, then what is the greatest possible range of the five numbers?
12
17
18
19
20
I missed this because of a stupid reason
Well i figured out that the largest number must be 19 and the smallest 1
The greatest range must then be 18
Pay attention to the limits! Our max can only be 90 AND that no two numbers are the same. Begin by working in reverse instead of trying out the answer choices.
The average (arithmetic mean) of 4 different integers is 75. If the largest integer is 90, what is the least possible value of the smallest integer?
1
19
29
30
33
____+____+____+90=300
if not two numbers can be the same and the numbers are INTEGERS, then the next largest number must be 89, and the one after that must be 88
That leaves us with 33 being the smallest possible value
When numbers must be INTEGERS.
In a group of 200 workers, 10 percent of the males smoke, and 49 percent of the females smoke.
Column A
Total number of workers who smoke
Column B
59
We are not given the number of F or M so we have to make them up
The only solutions that works is if we have 100 males and 100 females which gives us a total of 59 smokers so the answer is C
Set the equations equal to each other.
If f(x) = 5  2x and f(3k) = f(k + 1), then f(k) =
0.5
1
3
4
6
f(3k) = f(k + 1) and we have f(x) = 5  2x
That means that 52(3k) = 52(k+1)
Solve for k=1/2
So f(k)=4
This is very, very tricky! You must test corner cases of 0 & 1 and negative numbers.
Column A
22 percent of x
Column B
2/9 of x
B would be bigger if x is positive
If x were 0 then the two columns are the same
So the answer is D
"At least" means we have to check out cases in which we can have more digits. In this case since we are testing multiple scenarios that are mutually exclusive we have to add them.
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?
9! + 10!
2 x 10!
9! x 10!
19!
20!
09=10 digits
S1: 9digits: (10)(9)(8)(7)(6)(5)(4)(3)(2) = 10! (this takes into account any combination and permutation of the digits!)
S2: 10 digits: (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 10!
So the answer is 2*10!
Total  disobeying restriction = Obeying restriction
In how many ways can Ann, Bob, Chuck, Don and Ed be seated in a row such that Ann and Bob are not seated next to each other?
24
48
56
72
96
Total = 5! = 120
Scenario is which A&B are next to each other are:
__,__,__,__,__
There are 4 possibilities with AB leaving 3 chairs left so (4)(3x2x1)=24
But there is another scenario of BA which would be (4) for BA and (3x2x1) for the rest for a total of 48
12048=72
Draw out the figure! In this case the figure is drawn with the 90 degrees being formed at the yaxis. This means that the ycoordinates are equation to each other for points B and C!
Here it doesn't even matter what quadrant the triangle is in. We know that B is 90 degrees so just pick the first quadrant and work with that since any triangle would be congruent.
The points A(0, 0), B(0, 4a  5) and C(2a + 1, 2a + 6) form a triangle. If ABC=90degrees , what is the area of triangle ABC?
102
120
132
144
156
4a5=2a+6
a=11/2=5.5
Plug this in to get the coordinates and solve for area to equal 102
GCF of 1 means that the two integers are the same or are prime numbers. So you have to check multiple cases.
If x and y are positive integers, and 1 is the greatest common divisor of x and y, what is the greatest common divisor of 2x and 3y?
Cannot be determined
1
2
5
6
Doing that should lead to A
At least case means you have to test the next options available too.
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible?
Scenario 1: (10)(9)(8)(7)(6)(5)(4)(3)(2)= 10!
Scenario 2: (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 10!
S1+S2= 2(10!)
This is pretty easy to figure out using algebra, just need to pay attention to what is being asked.
Pump A can empty a pool in A minutes, and pump B can empty the same pool in B minutes. Pump A begins emptying the pool for 1 minute before pump B joins. Beginning from the time pump A starts, how many minutes will it take to empty the pool?
Check out saved image 150 for solution, which is in variable form
A(B+1)/(A+B)
If points A and B are randomly placed on the circumference of a circle with radius 2, what is the probability that the length of chord AB is greater than 2?
1/4
1/3
1/2
2/3
3/4
Very Hard: I missed this
The first point can be placed anywhere
The probability depends solely on the second point which can be on the right or left of the point
The entire circle is 360; the space taken up by the three points (taking into account both scenarios) is equal to 120; this is the case since the radius is 2 and the chord length is 2 and the distance form the other point to the center is 2
So 240/360 is the answer = 2/3
Check out Very Hard 151 for solution
Mississippi rule works best here too.
In how many different ways can 3 identical green shirts and 3 identical red shirts be distributed among 6 children such that each child receives a shirt?
We have 6 shirts so 6! total combinations but the two items are repetitive
6!/(3!*3!)=20 gives the fastest answer
I did this another way where I figure out the combinations possible for just 3 identical shirts __,__,__,__,__,__ (not counting repeats) we have 4+3+2+1=10 ways to organize; multiply that by 2 to get 20 (to account for the green shirts)
Tricky! Must take into account negative integers.
If k is an integer and 121 < k2 < 225, then k can have at most how many values?
Total is: 6
Thinking about what is left first.
A sum of money was distributed among Lyle, Bob and Chloe. First, Lyle received 4 dollars plus onehalf of what remained. Next, Bob received 4 dollars plus onethird of what remained. Finally, Chloe received the remaining $32. How many dollars did Bob receive?
10
20
26
40
52
R=T32, where R is the remaining money. This should make sense since R is the unit we need to deal with. Once we take Chloe out of the equation we are left with the money that is "remaining"
L=4+(1/2)(T32)
B=4+(1/3)(T32)
C=32
Adding all these = T
If an integer is divisible by 2, the resulting number doesn't have to be be EVEN!
If n = 2×3×5×7×11×13×17, then which of the following statements must be true?
I. n^2 is divisible by 600
II. n + 19 is divisible by 19
III. (n+4)/2 is even
Very Hard, I missed this
It's easy to see that will not be divisible by 600 since the prime factors of 600 do not cancel out
It's easy to see that n+19 will not be divisible by 19 since to be divisible by 19, n must be divisible by 19 and that is not the case
The last one is tricky (2×3×5×7×11×13×17+4)
is divisible by 2
by 2* but III is asking for the result of (2×3×5×7×11×13×17+4)/2 which is 255257, this is odd!
I arranged this in the way it was needed by failed to realize what I was looking at.
If sqrt(17+sqrt(264)) can be written in the form sqrt(a)+sqrt(b) , where a and b are integers and b < a, then a  b =
sqrt(17+sqrt(264)) = sqrt(a)+sqrt(b)
17+sqrt(264) = a+2sqrt(ab)+b
17+sqrt(264) = a+b+2sqrt(ab)
Realize how similar the two equations are!
a+b=17
The left hand side sqrt can be simplified to 2sqrt(66)=2sqrt(ab). So ab=66
This means that a and be must be 11 and 6 since a is larger than b
The difference between the two is 5
Must check for extraneous solutions!
Notice for things that look really similar to the quadratic algebra equations.
What is the sum of all possible solutions of the equation
x + 4^2  10x + 4 = 24?
16
14
12
8
6
I missed this because I did not check for extraneous solutions.
a=x + 4
a^210a24=0
(a12)(a+2)=0
a=12 and a=2
x + 4= 12
x + 4= 2
Before we even solve these; plug in to see whether the values work.
We see that 12 does work but 2 does not. So we only need to solve using 12
x + 4=+ or  12
x=8
x=16
16+8=8
Do not forget that squares can be of negative or positive numbers.
If a and b are integers and ((ab)^1/3 (b)^1/2)^6 = 500, then a + b could equal
2
3
4
5
6
make the left look like the right and you get
a^2(b^3)=(a)^2(5^3)
here a could be 2 or 2
This is a double matrix questions, but highlights how important is it to be able to read it correctly.
At a certain university, 60% of the professors are women, and 70% of the professors are tenured. If 90% of the professors are women, tenured, or both, then what percent of the men are tenured?
Very Hard: I missed this
T ~T
M 40
F 60
70 30
We should now recognize that based off the last piece of information, 90% or 90 professors fall in the FT,F~T, and MT. This means that our remaining box M~T will have 10 members we can use this to figure out the rest and get that 30MT
30/40=75%
Number of places
How many integers between 1 and 10^21 are such that the sum of their digits is 2?
Recognize that we have 22 spaces of which 21 are available since 1 takes the 22nd spot
Here the only way to get digit sum to 2 is if we just have 2 by itself or two 1's
21 places and we can pick 2 spots = 21C2 = 210
21 places and we can pick 1 spot = 21= 21
231
Corner cases where either variables or plugging in numbers = 0 or 1
If x and y are positive odd integers, then which of the following must also be an odd integer?
I. xy+1
II. x(y+1)
III. ((y+1)^x1) + 1
I missed this: Very Hard
I is always ODD
O(E) = Even =Note ODD
The third is tricky! Usually this will be an ODD. But check for the corner case! If x=1 then this number will be EVEN
So only I must be true
Corner cases or 0s and 1s
If x is an odd negative integer and y is an even integer, which of the following statements must be true?
I. (3x  2y) is odd
II. xy^2 is an even negative integer
III. (y^2  x) is an odd negative integer
I missed this: Very Hard
I is always ODD
II is a corner case where if y=0 then this equals 0 which is not a negative numbers
III is false since we can get a positive ODD
Only I is true
How to approximate
Sarah invested $38,700 in an account that paid 6.2% annual interest, compounding monthly. She left the money in this account, collecting interest for three full years. Approximately how much interest did she earn in the last month of this period?
$239.47
$714.73
$2793.80
$7,888.83
$15,529.61
Let's round the deposit up to $40,000, and the percentage down to 6% annual. Compounding monthly means each month, Sarah will accrue 6/12 = 0.5% in interest. Well, 1% of $40,000 is $400. Divide by 2: then 0.5% of $40,000 would be $200. That would be the simple interest amount, as well as the interest in the first month. We expect the amount in the last month to be a little more than this, but certain not even as large as double this amount. The only possible answer is (A).
Total possible divided by 2
There are 10 people in a room. If each person shakes hands with exactly 3 other people, what is the total number of handshakes?
15
30
45
60
120
30 total possible but we are double counting so 30/2=15
Absolute values can be just two values positive or negative.
If x + x + y = 7 and x + y  y = 6 , then x + y =
1
1
3
5
13
Assume that x + y  y = 6, y is positive if that is the case then we get x=6 if we plug this into the other equation we get y=5 but that's a contradiction! So y must be a NEGATIVE number
Assume that x + x + y = 7, x is negative and we get y=7. But this can't be right since we determined that y must be negative. So x MUST be POSITIVE
Using this info we get (Now we know the actual identities and don't have to deal with the absolute values!)
2x+y=7
x2y=6
We solve and get that x=4 y=1
4+1=3
Area vs. size limitation. Area can range from 0+ to the maximum possible at right degree angle REGARDLESS of the size of the sides.
Two sides of a triangle have length 6 and 8. Which of the following are possible areas of the triangle?
I. 2
II. 12
III. 24
All 3 can work. Area maximized at 6 and 8 being a right triangle
Our last side must be 2<x<14
But imagine starting with a straight line at 8 if we raise our other two sides just a tiny bit (6+2.01)=8.01 we area can be as small as 0 to just a bit above 0. So two could technically work
Prime factorization is key
In the game of Dubblefud, red chips, blue chips and green chips are each worth 2, 4 and 5 points respectively. In a certain selection of chips, the product of the point values of the chips is 16,000. If the number of blue chips in this selection equals the number of green chips, how many red chips are in the selection?
1
2
3
4
5
The prime factorization of 16,000 is:
(2)(2)(2)(2)(2)(2)(2)(5)(5)(5)
We have Blue chips worth 4 so group the twos to get
(2)(4)(4)(4)(5)(5)(5)
We can now see that we have 1 red chip

Below is the mistake that I made.
We need the product of the point value of each chip not the sum of the value of the total chips
FAQ: According to the problem I have 16000 = (2
red) (4
blue) (5
green), which is the same as 16000 = (y
2) (x
5) (x
4). I can rewrite that as 400 = x^2
y. So 400 can be written as (10
10
4) or (20
20 * 1). So the number of red chips could be either 4 or 1. Right?
A: Unfortunately, no.
The question states that "the product of the point values of the chips is 16,000." So we have to find the product of the point value of each chip individually—not the sum.
Your equation is the equivalent of summing the point values within a given color and then multiplying, not multiplying the individual chip values. If there are x blue chips, that means 4^x, not 4x.
For example, say there were three blue chips. They are worth 4 points. We want the product of these chips, so 4
4
4 = 4^3 = 64. Again, not adding three 4's together or 4 * 3 = 12.
Thus, the expression:
(4^x)(5^x)(2^y) = 16000
or
(20^x)(2^y) = 16000.
But that's a very complicated equation with which to wrestle. I actually think it's a mistake to introduce variables into this problem at all. In general, in prime factorizations problems, introducing variables is not helpful.
Be mindful of what you are labeling.
If ABCD is a square with area 625, and CEFD is a rhombus with area 500, then the area of the shaded region is
125
175
200
250
275
Check out saved image 168 for the figure
I was on the right track with prime numbers. Only the squares of integers have an odd number of divisors. Here we only want to focus on the square of prime numbers which have a only 3 divisors.
How many integers from 1 to 900 inclusive have exactly 3 positive divisors?
10
14
15
29
30
I missed this: Very Hard
2,3,5,7,11,13,17,19,23,29 = 10 numbers whose squares have only 3 divisors meaning we only have 10 numbers from 199 that have 3 divisors
4,9,25,49,121,169,289,361,529,841
Notice that we can't go above 29 since the square of 30 is 900
Do not be confused by repetition: 2^2=4 is divisible by 1,2,2,4 for a total of 1,2,4 three divisors
Prime factorization if very important. I only went down to 21 and 10 and did not simplify more
How many positive integers less than 10,000 are such that the product of their digits is 210?
24
30
48
54
72
10,000 = 1,_,_,_,_ Notice that we can only deal with 4 spaces since we can't have anything bigger than 1 on the first space because that would violate our rule. That's why we don't consider the scenario of organizing 1,2,3,5,7
The prime factorization of 210 is 2,3,5,7
S1: We can arrange these numbers as 4!
S2: We can combine 2 and 3 for 6 (which is a one digit number) for a total of 3!
S3: We have to account for 1,6,5,7 for a total of 4!
so we get 24+6+24= 54
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