Math Flashcards
What is the approach for working backwards from answer choices?
- Start with ans B) (set up a small table to help organize math).
- If it’s wrong, check ans D). Identify the pattern if it’s wrong (which direction do I need to go).
- Check the remaining ans choices in order of that direciton
Is it easy to get from X/Y to XY or X - Y to X + Y?
No –> insufficient info on its own.
How many different equations do you need to solve for:
a) 1 unknown
b) 2 unknowns
c) n unknowns
Caveats though?
- what if you have ratios of the same equation?
- what if you have ratios of the same equation but unequal totals?
a) 1 unknown –> 1 equation
b) 2 unknowns –> 2 DIFFERENT equations
c) n unknowns –> n DIFFERENT equations
> # 1) If two equations have coefficients that are just RATIOS => they are the SAME EQUATION (because you can factor out the factor and divide) –> same line, infinite # of solutions#2) If a1X + b1Y = C1 and a2X + b2Y = C2
and a1/a2 = b1/b2 =/ C1/C2 ——-> NO SOLUTION (parallel lines)
#3) If the variable CANCELS OUT –> NO SOLUTION
SOLVE via elimination –> add or subtract equations or multiples of equations.
HOWEVER you can still try to solve for the COMBO rather than individual values.
> analyze each equation to see how they are SIMILAR (i.e., difference between each term equals 2)
ADDITIONALLY, you might be able to solve for two variables using ONE equation if there are special constraints
e.g., 3x = 5y (values must be less than 30 and cannot equal 0) –> Multiple of 3 and 5 less than 30 –> 15 = 15
e.g., 5t + 7v = 53 –> all primes
Fractions raised to even exponents, how do they behave?
Draw a number line from -2 -1 0 1 2
A) Fraction less than -1 (-3/2)
–> Value is larger (positive)
B) Fraction between 0 and -1 (-1/2)
–> Value is larger (positive)
C) Fraction between 0 and 1 (1/2)
–> Value is smaller (positive)
D) Fraction is greater than 1 (3/2)
–> Fraction is larger (positive)
Fractions raised to odd exponents, how do they behave?
A) Fraction less than -1 (-3/2)
–> Value is smaller (more negative)
B) Fraction between 0 and -1 (-1/2)
–> Value is larger (less negative)
C) Fraction between 0 and 1 (1/2)
–> Value is smaller (positive)
D) Fraction is greater than 1 (3/2)
–> Fraction is larger (positive)
“Greatest” prime factor of a number?
Break up the number into a PRODUCT of prime numbers (via prime tree)
> combine anything that is addition or subtraction
Greatest prime factor is the factor that is the largest
What are the values of x?
x^3 < x^2
Any nonzero number (integer, real number) less than 1
x^3 - x^2 < 0
x^2(x - 1) < 0
Roots: x = 0 (no switch in sign) and x = 1
x < 1 but x =/0
In ratio problems, does the multiplier have to be an integer? When is there an integer constraint?
Integer constraints exist when the actual figures must be WHOLE numbers
e.g., whole number of shirts, cats, dogs.
What is x = sqrt(16)?
What is the sqrt(x + 3)?
What is sqrt((x + 3)^2)?
x = sqrt(16) = 4
NOT +/- 4 (GMAT would have given you x^2 = 16)
sqrt(x + 3) means that x + 3 is POSITIVE
sqrt((x + 3)^2) could mean that x + 3 > 0 or x + 3 < 0
»> need cases
1.4^2 = ?
~2
1.7^2 = ?
~3
14^2 = ?
196
15^2 = ?
225
16^2 = ?
256
25^2 = ?
625
sqrt(2) = ?
~1.4
Helpful tip: 2/14 is Valentine’s Day
sqrt(3) = ?
~1.7
Helpful tip: 3/17 is St. Patrick’s Day
3^3 = ?
27
4^3 = ?
64
17 ^ 27 has a units digit of?
CONCEPT: Last Digit Shortcut
> (For product or sum of integers): Units digit is influenced ONLY by the units digit of the BASE (drop any other digits)
–> drop the 1, look at only 7^x
Next, find the PATTERN
7^1 = 7
7^2 = units 9
7^3 = units 3
7^4 = units 1
7^5 = units 7
7^6 = units 9…
Pattern - every 4 powers, the unit digit is 7.
Find which one now:
27/4
= 6 R 3 —> choose 7^3 or 3rd placement
(If R = 0, choose 7^4 or 4th placement)
17^27 has a units digit of 3!
How to quickly solve this:
If a ticket increased in price by 20%, and then increased again by 5%, by what percent did the ticket price increase in total?
Choose smart numbers (for successive percent changes)
- It is difficult to do: x(1.2)(1.05) / x
- OR convert them to fractions (1.5 = 3/2)
e.g., x = 100
20% increase of 100 => 120
5% of 120 = 6
so final number = 120 + 6 = 126
% increase = 26%
What is 0.000000008^1/3 ?
1) rewrite as an integer * power of 10
= (8 * 10^-9)^1/3
2) Figure out the # of decimal places = # of decimal places in original * exponent
–> 9 * (1/3) = 3 decimal places to the right of the decimal
= 2 * 10^-3
= 0.002
Repeating decimals:
What is 3/11?
What is 10/11?
What is 1/3?
What is 1/3 + 1/9 + 1/27 + 1/37?
RULE: for any denom equal to a power of 10 minus 1 (9, 99, 999, 9999), the numerator dictates the repeating digits.
> num must be less than denom
3/11 –> 27/99 = 0.27272727
10/11 –> 90/99 = 0.90909090
1/3 –> 3/9 = 0.3333
508/999 —> 0.508508508
How do you determine whether something has terminating decimals?
e.g., 0.4, 0.375?
How do you then find the nonzero digits?
1) Rewrite the decimal as a fraction (ratio of integers)
2) Simplify the fraction
3) Then break up the denom into prime factors!
–> denoms contain only 2s or 5s (NOTHING else)
why?
- The fraction is divisible by 10
e.g., 0.375 = 3/8 = 3/(222)
**to find the nonzero digits in terminating decimals:
1) Find the number of 10s in the denom –> don’t affect the nonzero digits
- e.g., 1/(2^3 * 5^7) = 1/(10^3 * 5^4)
2) Use nice fractions to convert into decimals
- e.g, 1/10^3 * (1/5)^4
= 10^-3 * (0.2)^4
= 10^-3 * (2 * 10^-1)^4
= 10^-3 * (16 * 10^-1)
therefore 1 6 are the nonzero digits
|a| < b as an inequality
then:
-b < a < b
Units digit of powers
1
Always 1 (special)
Units digit of powers
2
4 cycles:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6
2^5 = 2
Exponent divisible by 4 –> 2^4 unit (6)
Anything else –> remainder dictates which one to use
e.g., R = 1 –> 2^1
Units digit of powers
3
4 cycles:
3^1 = 3
3^2 = 9
3^3 = 7
3^4 = 1
3^5 = 3
Exponent divisible by 4 –> 3^4 unit (1)
Anything else –> remainder dictates which one to use
e.g., R = 1 –> 3^1
Units digit of powers
4
2 cycles: (special)
4^1 = 4
4^2 = 6
4^3 = 4
4^4 = 6
Odd exponents = 4
Even exponents = 6
Units digit of powers
5
Always 5 (special)
5^1 = 5
5^2 = 25
5^3 = 125
Units digit of powers
6
Always 6 (special like 5)
6^1 = 6
6^2 = 6
6^3 = 6
Units digit of powers
7
4 cycles:
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
Exponent divisible by 4 –> 7^4 unit (1)
Anything else –> remainder dictates which one to use
e.g., R = 1 –> 7^1 = 7
Units digit of powers
8
4 cycles
8^1 = 8
8^2 = 4
8^3 = 2
8^4 = 6
8^5 = 8
Exponent divisible by 4 –> 8^4 unit (6)
Anything else –> remainder dictates which one to use
e.g., R = 1 –> 8^1 = 8
Units digit of powers
9
2 cycles (special like 4):
9^1 = 9
9^2 = 1
9^3 = 9
9^4 = 1
Odd exponent = 9
Even exponent = 1
What is the units digit of:
(6^6/6^5)^6
FIRST SIMPLIFY before dropping all other digits!!
(Same base, exponent rules!)
= 6^36/6^30
= 6^6
Units = always 6
Find the length of non-terminating decimal
e.g., 3/7
Long division is probably the fastest.
List all perfect squares
0** (0^2)
**1 IS A PERFECT SQUARE (but not a prime number)
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
625
Why can’t you divide both sides by w here?
3w^2 = w
Because w could be equal to 0! (Cannot divide by 0)
Instead, rearrange and solve as a quadratic equation:
3w^2 - w = 0
w(3w - 1) = 0
How do you solve this?
If (z + 3)^2 = 25, what is z?
No need to expand! Recognize the perfect square
Square root both sides:
z + 3 = +/- 5
z = 5 - 3 = 2 OR
z = -5 - 3 = -8
Factor this:
x^2 - y^2
or x^2 - 4
Difference of squares
(x + y)(x - y) —> SPECIAL PRODUCT (different signs)
e.g., 9x^2 - 16
= (3x + 4)(3x - 4)
e.g., a^2 - 1
= (a + 1)(a - 1)
e.g., x^2 - 4
= (x + 2)(x - 2)
Recognize special ones:
(a + b + c + d)(a + b - c - d) = 16
ANY EVEN POWER can be converted to a difference of squares
a^8 - b^8
Factor this:
x^2 + 2xy + y^2
M: 1
A: 2
(1, 1)
(x + y)^2
= (x + y)(x + y)
Tips:
> Look for perfect squares
> recognize x and sqrt(x), and reciprocals
–> x - y = (sqrt(x) + sqrt(y))(sqrt(x) - sqrt(y))
Factor this:
x^2 - 2xy + y^2
M: 1
A: -2
(-1, -1)
(x - y)^2
Tips:
> Look for perfect squares
> recognize x and sqrt(x)
Expand this:
(a + b)^2
a^2 + 2ab + b^2
e.g., x^2 + 6x + 9
= (x + 3)^2
What does “root” mean in this situation?
If -4 is a ROOT for x in the equation x^2 + kx + 8 = 0, what is k?
Root in conjunction with a quadratic equation means SOLUTION
A: k
M: 8
x = -4 is a solution
x + 4 = 0
(4, 2)
K = 6
Or algebraically by subbing in x = -4:
(-4)^2 + (k)(-4) + 8 = 0
k = 6
What are sequences?
A collection of numbers in a set ORDER according to a specific RULE
> ORDER MATTERS IN A SEQUENCE (don’t treat it like a set where you can change the order from least to greatest)
e.g., {1, 4, 9, 16, 25} –> each number is called a TERM
Rule is An = n^2 (based on the term’s POSITION)
Special type of sequence (Recursive)
e.g., An = An-1 + 2 (based on the PREVIOUS items
Tips:
> write out the first five terms of sequence to try to find out the PATTERN x 2(e.g., find the TERMS AND find the SUM, if asked about sums, or DIFFERENCE of terms if asked for difference)
> use a number line for sum/subtraction questions (e.g., ranges of the sum of the first 10 terms)
The population of a certain type of bacterium triples every 10 minutes. If the population of a colony 20 minutes ago was 100, in approximately how many minutes from now will the bacteria population reach 24,000?
Exponential Growth:
Size = a1 * (3) ^ (time/10min)
Helpful to create a table: (since “ago” makes time different to track)
20 mins ago —-> pop = 100
10 mins ago —> pop = 1003 = 300
NOW –> 900
10 mins –> 2700
20 mins –> 8100
30 mins –> 24,300 **about 30 minutes
In some cases, you might need to pick a SMART NUMBER as the starting point.
OR formula:
Population = A(rate of increase or decay)^(time/increment in time)
The amount of electrical current that flows through a wire is inversely proportional to the resistance in that wire. If a wire currently carries 4 amperes of electrical current, but the resistance is then cut to one-third of its original value, how many amperes of electrical current will flow through the wire?
Inverse proportionality Q:
yx = k (constant) –> products
or y = k/x
AR = AR
4R = A(1/3R)
4*3 = A
A = 12
The max height reached by an object thrown directly upward is directly proportional to the square of the velocity with which the object is thrown. If an object thrown upward at 16 feet per second reaches a maximum height of 4 ft, with what speed must the object be thrown upward to reach a max height of 9 feet?
Direct proportionality Q: Think ratios
y/x = k
or y = kx
y/x = y/x
h/v^2 = h/v^2
4/16^2 = 9/v^2
CROSS MULTIPLY
v = 24
Jake was 4 1/2 ft tall on his 12th birthday, when he began to have a growth spurt. Between his 12th and 15th birthdays, he grew at a constant rate. If Jake was 20% taller on his 15th birthday than on his 13th birthday, how many inches per year did Jake grow during his growth spurt
(12 inches = 1 ft)
Linear Growth problem = Arithmetic Sequence!
Arithmetic Sequence Approach:
An = a1 + (n - 1)*d —-> goal is to find d
a1 = 4.5
a2 = 4.5 + d —> height at 13
a4 = 4.5 + 3d = 1.2a2 –> height at 15
sub in a2 into a4 and solve
d = 0.5 feet = 6 inches
How to solve “Symmetry” problems
e.g., which of the following functions does f(x) = f(1/x)?
Method 1) Sub in 1/x into the function to see if it simplifies to f(x)
> f(x) changes for EACH answer! (don’t compare to the incorrect equation!)
Tip: use =? to compare LHS to RHS
Method 2) Test cases –> see which function yields the same output for both x and 1/x
(easier)?
What is the tens digit of n? n > 0
(1) The hundreds digit of 10n is 6
CONCEPT - any multiple of 10 has a units digit equal to 0
e.g., 10, 20, 30, etc
So if 6 is in the hundreds digit of 10n, then 6 is in the tens slot of n. Sufficient
Also:
10n = 6 _ _
n = 6 _
When you see digits/place values in a question, what should you think of?
Digit constraint!
- Digits must be between 0 (or 1) and 9!! (Might not be mentioned explicitly in the question).
0 <= digit <= 9
e.g. 3P5 + 4QR = 8S4
and Q = 2P
- We know that Q = 2P <10 or P < 5!!
- For unknown addition or subtraction relating to VALUE, it is helpful to write out expressions with place values:
e.g., 54 = 105 + 14
ab = 10*a + b - Sometimes equations can be helpful to show FACTORS of digits
e.g., 11(B - 9A) = C —> C is divisible by 11 (C = 0)
What does this mean?
x^2 - x < 0
x^2 < x
x is a fraction between 0 and 1
0 < x < 1
> number raised to an even exponent is SMALLER than original number only works if the fraction is between 0 and 1
Is this allowed?
xy < 9
x < 9/y
NO - we don’t know the sign of y so we don’t know if we have to flip the sign
x < 9/y or x > 9/y
Applies to multiplication and division involving variables
What are “compound inequalities”
Multiple inequalities combined into one inequality (two or more inequality signs)
e.g., -4 < x < 4
TIP: Rearrange the inequalities so that the symbols point in the same directions
> not exactly the same thing as adding two inequalities (still get one inequality with one inequality sign)
How do you manipulate compound inequalities?
e.g., 2 < x < y
Perform operations on EVERY TERM (+ - * /)
e.g., multiply every term by y (assume y > 0)
2y < xy < y^2
Can you combine two inequalities?
e.g., a < c and d > b
Yes, but only ADDITION (you CANNOT SUBTRACT or divide inequalities)
> applies to compound inequalities too (e.g., 8 < x < 10)
1) Rearrange the inequalities so that the symbols face the same direction
e.g., a < c
b < d
2) Then add each side
a + b < c + d
Solving max/min problems and inequalities
e.g., if -20 <= 2 y <= 8 and -3 <= x <= 5, what is the maximum possible value of xy?
Or min | a - b |
0) UNDERSTAND what it means to max or min the value of something
> how high can xy go? (infinity)
> how low can | a - b| go? (0)
1) Simplify inequalities, wherever you can
2) Look at the EXTREME ends of each range
- pay attention to whether values can equal 0 or be negative.
3) Consider different scenarios that can lead to max or min values
In this example, max xy = 30
Solve:
x^2 >= 9
Two cases:
x >= 3
AND
x <= -3
A | > B as an inequality
A > B
OR
A < - B
Solve:
A retailer sells only radios and clocks. If there are currently exactly 42 total items in inventory, how many of them are radios?
(1) Retailer has more than 26 radios in inventory.
(2) Retailer has less than twice as many radios as clocks in inventory.
Linear inequality word problem
42 = r + c
r= ?
(1) r > 26
NS - r can be anything from 26 to 42
(2) r < 2c
NS
OR algebra:
sub c = 42 - r (equality) into r < 2c (inequality)
r < 2(42 - r)
r < 84 - 2r
3r < 84
r < 28 (not sufficient)
TOGETHER:
26 < r < 28
r = 27 (sufficient - C)
Can you take reciprocals of inequalities?
e.g., If x < y, can I make it into 1/x and 1/y?
If you DON’T know the sign of x and y, you CANNOT take reciprocals
Otherwise, the sign of x and y determine whether to flip the inequality or not.
> FLIP the sign UNLESS x and y have DIFFERENT signs (+- or -+)
> a / b > c / d AND b / a > d / c ONLY WHEN the two sides have different signs.
e.g., -6 < 2 (different signs, don’t flip the inequality)
1/-6 < 1/2
SPECIAL CASE: dealing with absolute values and positive powers –> always positive
Can you square both sides of an inequality?
Depends on if you KNOW the signs of both sides of the inequality (just like with reciprocals).
Why?
- because if there’s a variable, squaring it means multiplying that side with a variable
Rule of Thumb: If the signs are UNCLEAR or One side is Positive and One side is Negative, then you CANNOT SQUARE
Otherwise:
A) If both sides are known to be negative (e.g., x < -3), then squaring both sides needs to come with a FLIP of the inequality sign
x < - 3
x^2 > 9
B) If both sides are known to be positive (e.g., x > 3), then squaring both sides does not need to flip the inequality sign
x > 3
x^2 > 9
SPECIAL CASE: dealing with absolute values and positive powers –> always positive
If 4/x < 1/3, what are the range of values for x?
One variable inequality: Sewing approach
> cannot multiply both sides by x because we don’t know what the sign of x is!
Move 1/3 to the left side and do sewing approach:
(x - 12)(3x) > 0
x < 0 , x > 12
x^2 >= 9
What is x?
x >= 3
OR
x <= - 3 —-> x is negative, so flip the sign and add neg sign
There are enough available spaces on a school team to select at most 1/3 of the 50 students trying out for the team. What is the greatest number of students that could be rejected while still filling all available spaces for the team?
This question has a HIDDEN INTEGER CONSTRAINT (people cannot be split into fractional parts)
Max # accepted students = 50/3 = 16.667 –> 16 (not 17)
Max # of rejected students = 50 - 16 = 34 (NOT 33!!)
Was the number of books sold at Bookstore X last week greater than the number of books sold at Bookstore Y last week?
(1) Last week, more than 1000 books were sold at Bookstore X on Saturday and fewer than 1000 books were sold at Bookstore Y on Saturday.
(2) Last week, less than 20% of the books sold at Bookstore X were sold on Saturday and more than 20% of the books sold at Bookstore Y were sold on Saturday.
Linear Inequality Word problem:
Q: Is Nx > Ny?
(1) We are given only info about Saturday and nothing else
nx > 1000
ny < 1000
(n represents saturday)
NS
(2) nx < 0.2Nx
ny > 0.2Ny
We cannot figure out the actual number of books sold, Nx and Ny.
NS
(3) TOGETHER
- COMBINE the inequalities
1000 < nx < 0.2Nx
—> 1000 < 0.2Nx
Nx > 5000
0.2Ny < ny < 1000
—> 0.2Ny < 1000
Ny < 5000
So Nx > 5000 (C)
»> Ny < 5000 < Nx
Is a/b < c/d?
(All are positive numers)
(2) (ad/bc)^2 < ad/bc
Divide both sides by ad/bc (positive number)
ad/bc < 1
a/b < c/d
Try logic approach:
Annika hikes at a constant rate of 12 mins/km. She has hiked 2.75 km east from the start of a hiking trail when she realizes that she has to be back at the start of the trail in 45 mins. If Annika continues east, then turns around and retraces her path to reach the start of the trail in exactly 45 mins, for how many km total did she hike east?
Concept: Rate (unit over time) * time = distance
–> or set up ratios: Rate = Distance/time
Distance Travelled East = Total distance travelled divided by 2.
Total distance travelled = 2.75 + distance in 45 mins
= 2.75 + (45/12)
= 2.75 + 3.75
= 6.50
Distance Travelled East = 6.5/2 = 3.25km
Relative Rate problems (e.g., bodies move toward each other, bodies move away from each other, bodies move in the same direction)
How best to solve?
A: Two people move toward each other - one at 5mph and the other at 6mph
B: Two people move away from each other - one at 30 mph and the other at 45 mph
C. Two people move in the same direction - one at 8 mph and the other at 5 mph
Find the COMBINED rate at which the distance between the bodies changes
A: Distance decreases at a rate of 5 + 6 = 11 mph.
Extension - time to cover distance between –> use combined rate.
- if the answer isn’t as pretty –> use draw out technique to narrow choices down (see when two people have passed)
B. Distance increases at a rate of 30 + 45 = 75 mph.
C. Distance decreases (faster one is chasing slower one) at a rate of 3 mph
Gap = (combined rate)*time
Working Together problems - how do you solve?
e.g., Machine A can complete 1/3 of the task in an hour, Machine B can complete 1/2 of the task in an hour.
ADD (or subtract) the RATES to get combined rates
e.g., Rate A + Rate B = Total Rate
1/3 + 1/2 = 5/6 of the task gets completed every hour.
**to find total time to complete the task –> (5/6)*time = 1
time = 1/(5/6).
What if you have two statements for DS that give you the SAME info?
Ans is either D (each work) or E (neither work).
- ans cannot be A, B, or C
*Sometimes you have to simplify further to see if the equations are identical or different!
Al and Barb shared the driving on a certain trip. What fraction of the total distance did Al drive?
(1) Al drove for 3/4 as much time as Barb did.
(2) Al’s average driving speed for the entire trip was 4/5 of Barb’s average driving speed for the trip.
Rate Question:
Both statements individually are insufficient to calculate distance!
> while Al drove for a LONGER time, he could have had a faster speed and covered more distance than Barb
> while Al drove at a slower speed than Barb, he could have driven longer and covered more distance
Together sufficient –> 3/8
(because it is a RATIO)
Mary, working at a steady rate, can perform a task in m hours. Nadir, working at a steady rate, can perform the same task in n hours. Is M < N?
(1) The time it would take Mary and Nadir to perform the task together, each working at their respective constant rates, is greater than m/2
(2) The time it would take Mary and Nadir to perform the task together, each working at their respective constant rates, is less than n/2
Work Problem:
M < N is asking whether Rate of M is faster than Rate of N (so Mary is spending LESS time than Nadir).
We also know:
combined time (t) < M
combined time (t) < N
Is m/2 < t < n/2 ?
(1) m/2 < t
Algebraically:
Combined Rate = 1/m + 1/n
Combined time = 1/(1/m + 1/n)
Statement: 1/(1/m + 1/n) > m/2
SOLVE: (Hidden positive number constraint).
n > m (sufficient)
(2) t < n/2
Statement 1/(1/m + 1/n) < n/2
SOLVE:
m < n (sufficient)
SENSE CHECK:
- if the combined time is greater than m/2, it means that Nadir is SLOWING Mary down (or spending more time). M < N
- If the combined time is less than n/2, it means that Mary is FASTER (or spending less time). M < N
If 0 < a < b < c, which of the following statements must be true?
I) 2a > b + c
II) c - a > b - a
III) c/a < b/a
MUST BE TRUE PS with no measurements - you can choose any test case that’s valid
Or Solve Algebraically:
* a, b, c must be positive numbers.
Manipulate the inequality in question to get to the statements OR use test cases
I) test case fails
ii) b < c –> SUBTRACT a from both sides
b - a < c - a (statement is true)
III) b < c –> divide both sides by a
b/a < c/a (statement is false)
Write -5 <= x <=3 as an inequality with absolute values.
Distance between -5 and 3 is 8. So middle value is -1.
If you add 1 to every term:
-5 + 1 <= x + 1 <= 3 + 1
-4 <= x + 1 <= 4
| x + 1 | <= 4
Strategy for overlapping sets and Double Matrix approach
If given #s –> use the given #s
If given percents (and asked to find a relative figure) –> Use 100 as the total
If given fraction s–> Use a common denominator as the total (e.g., 1/3, 1/4 –> choose 12)
Also PAY ATTENTION TO WORDING –>
“10% of all cars are red and have heated seats” (Intersection of red/heated seats - 10% of TOTAL)
vs. “10% of cars with heated seats are red” (intersection of red/heated seats - 10% of HEATED SEATS)
Average problems involving two average formulas:
Sam earned a $2000 commission on a big sale, raising his average commission by $100. If Sam’s new average commission is $900, how many sales has he made?
of sales made = n + 1
Eq’n 1: Old Avg = 800 = Sum/n
Eq’n 2: New Avg = 900 = (Sum + 2000)/(n + 1)
Solve for n:
800n = Sum –> sub into eq’n 2
900(n + 1) = 800n + 2000
900n - 800n = 2000 - 900
n = 11
How to calculate the median?
*Always arrange the data in increasing order
ODD Number of Values:
> Median is positioned at the middle
> Position: # of values/2 = fractional value ROUND UP
> Median is a fraction with DENOM = 2
e.g., 13/2 = 6.5 ROUND UP = position 7
**Median is a number in the set.
EVEN Number of Values:
> Median is the average of the two middle values
> Starting Position: # of values/2
> Ending Position = Starting Position + 1
e.g., 10/2 = median is average of # at 5 and # at 6.
**Median is does not have to be a number in the set (unless two middle values are equal).
Can you tell what the numbers are in a set and how many numbers are in the set from the average and standard deviation?
e.g., average is 10 and SD is 2
Usually No
- Multiple possible combinations of numbers can result in the same average and same SD
Exception: SD = 0 –> means all the numbers in the set are EQUAL
- Can figure out what numbers are in the set (e.g., avg is 10, meaning all the numbers is equal to 10)
- However, you still cannot figure out how many instances of 10 are in the set.
Types of Standard Deviation questions?
When adding a new term to a list, how do you INCREASE standard deviation? DECREASE?
Likely won’t be asked to calculate specific SD = sqrt( sum of squared deviations / N)
1) Changes to SD resulting from transformations to the set –> how does the data move relative to the mean? (Farther, closer, or stay the same?)
> constant difference to all terms = no change in SD
> factor applied to all terms = change in SD by the factor
2) Comparisons of SDs in multiple sets –> which set has numbers that are furthest away from the mean?
> dot method
Helpful to draw a picture (number line!)
When adding a new term to a list, how do you INCREASE standard deviation? DECREASE?
> Increase by adding a new term that is GREATER THAN +/1 standard deviation from the mean
> Decrease by adding a new term that is LESS THAN +/- 1 standard deviation from the mean
e.g., Original mean is 500, st dev = 50.
Add new set with average 500, st dev 25
> will decrease standard deviation –> values are closer to the mean 500.
What is closest to the value of 0.19/0.021?
7
8
9
10
- Ans choices are close to one another –> need to do a more precise division
= 190/21
217 = 147
218 = 168
219 = 189 (closer to 190) –> 9
2110 = 210
If you have more than 1 unknown in a set, can you still figure out the value of the median or whether the median is larger or smaller than a certain value?
Possibly, check three cases:
1) Unknown is less than middle value of the set
2) Unknown is the middle value of the set
3) Unknown is greater than the middle value of the set.
Weighted average problems - do you need to calculate precisely what the WA is to figure out if the WA > or < a number?
No –> compare relative to the simple average
> which end pulls the average more?
WA = (Data1# of 1s + Data2# of 2s)/(# of 1s + # of 2s)
> if there are more 1s than 2s, then the WA will be pulled toward Data1
> To estimate percentages (PS) –> Ask yourself whether the average is closer to the extreme end or simple average (50/50)
e.g., Brand A has 40% millet and Brand B has 65% millet. Customer purchases a mix of the two types of birdseed that has 50% millet, what percent of the mix is Brand A?
> Simple average of millet is 52.5% (greater than 50% in mixture)
> Mixture has more of Brand A, and difference from simple average is not that high –> Ans: 60% (not 85%).
Alternatively do the visual approach: Wa > Wb
Wa = 15/25 = 3/5 = 60%
Tickets to play cost $10 for children and $25 for adults. If 100 tickets were sold, were more adult tickets sold than children’s tickets?
(1) Avg revenue per ticket was 18.25
(2) Revenue from ticket sales exceeded 1800
Weighted Average Problem
Simple Average of ticket prices = (10 + 25)/2 = 17.5
(1) 18.25 > 17.5 –> more A than C
Sufficient
(2) Simple Average of Revenue (assuming all 100 tickets were either A or C) = (10010 + 10025)/2 = 1750
1800 > 1750 –> More A than C
Sufficient
A truck is filled to 1/4 of its maximum weight capacity. An additional y pounds are added such that the truck is now filled to 7/8 of its capacity. In terms of y, what is the maximum weight capacity of the truck, in pounds?
> What is Y?
Y represents the ADDITIONAL weight added to the truck that causes it to go from 1/4 to 7/8
In other words, 2/8M + Y = 7/8M
Y = 5/8*M (Change in Weight!)
M = 8Y/5
Commission is equal to 5% of sales over 1000 –> translate into equation
5% * (Sales - 1000)
Translate percent into decimal: 33 and 1/3%
33.333%
(33 + 1/3)/100
(100/3)/100
= 1/3
In Western Europe, x bicycles were sold in each of the years 1990 and 1993. The bicycle producers of Western Europe had a 42 percent share of this market in 1990 and a 33 percent share in 1993. Which of the following represents the decrease in the annual number of bicycles produced and sold in Western Europe from 1990 to 1993?
Meaning of the question: A total of x bikes are sold each year from domestic and foreign producers!
% of domestic production changed from 42% to 33% = 9% drop.
ans: 9% of X
WE DON’T CARE about the other years in between 1990 and 1993. So the DECREASE in the annual number of bikes produced and sold in Western Europe is just equal to 9%x
If something is 15% more in the past than now, does that something also equal 85% of old value?
No
Better to write it out:
15% more in the past -> Dec = 1.15Jan
85% of the past –> Jan = 0.85Dec —> 1/0.85 = 1.18 (not 1.15)
How many integers are there from 14 to 765, inclusive?
COUNTING Consecutive Integers –> ints that go up by 1
(Last - First)/increment + 1 —> because the lower extreme is subtracted out and not included in the count.
= 765 - 14 + 1
= 752
How many even integers are between 12 and 24, inclusive?
COUNTING Consecutive Multiples
(Last - First)/increment + 1 ——> adjust the Last and First bounds to be inclusive
= (24-12)/2 + 1
= 7
For short ranges, it may be easier to list the terms of the pattern and count the number of integers.
More generally: Consecutive ints (increments of 1)
SUM = n + (n + 1) + (n + 2) + …
How many multiples of 7 are between 10 and 80 (not inclusive)?
COUNTING Consecutive Multiples
(Last - First)/Increment + 1 —> **using least and greatest multiples of 7 as Last and First Numbers
=(77 - 14)/7 + 1
= 10
What is the arithmetic mean of {3, 6, 9, 12}?
CONCEPT: EVENLY SPACED SET (multiples of 3)
Mean = Median = (First + last)/2
= 7.5
What is the arithmetic mean of {1, 6, 11, 16, 21}
CONCEPT: EVENLY SPACED SET (increments of 5)
Mean = Median = (First + Last)/2
= (22)/2
= 11
What is the median of a set containing the integers from 20 to 50, inclusive?
CONCEPT: EVENLY SPACED SET (increments of 1)
Mean = median = (first + last)/2
= (70)/2
= 35
> if not inclusive, adjust first and last so that they include it
How do you figure out the sum the following set:
200 to 300, inclusive?
of terms = (Last - First)/1 + 1
CONCEPT: Evenly Spaced Set (increment of 1)
–> SUM = Average * # of terms
Average = Median = (First + Last)/2
= (300 + 200)/2
= 250
= (300 - 200) + 1
= 101
SUM = 250 * 101
= 25250
> if not inclusive end points, adjust first and last so that they include it (for both avg and # of term calculations)
What is the value of SUM(100 to 150) - SUM(125 to 150)?
Recognize the overlap.
Real problem is: SUM of 100 to 124 (not 125!)
= 2800 (using Evenly spaced set tips)
Workers are grouped by their area of expertise and are placed on at least one team. There are 20 workers on the Marketing team, 30 on the sales team, and 40 on the Vision team. Five workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams. 4 workers are on all three teams. How many workers are there in total?
3 sets –> Use Venn Diagram and work from the inside out!
Don’t forget to subtract overlap for each of the overlapping parts!
Ans: 10 + 23 + 29 + 1 + 5 + 2 + 4
= 74
What is the PRODUCT of the 5 consecutive integers divisible by?
5! = 5 * 4 * 3 * 2 * 1 = 120
CONCEPT - For any number of consecutive integers, the product of k consecutive integers is always divisible by k factorial.
> Any series of 5 consecutive integers will contain at least one integer that is a multiple of 5, 4, 3, and 2.
Is the sum of k consecutive integers divisible by k?
If k number of consecutive integers is ODD –> yes
- e.g., the sum of 5 consecutive integers is a MULTIPLE of 5, so the sum is DIVISIBLE by 5 (k)
If k number of consecutive integers is EVEN –> No
- e.g., the sum of 4 consecutive integers is NOT a multiple of 4, so the sum is NOT divisible by 4 (k)
Scheduling problems - how many days after the purchase of Product X does its standard warranty expire? (1997 is not a leap year)?
(1) Bought it Jan 1997, expired March 1997
(2) Bought it May 1997, expired May 1997
Scheduling problem - consider extreme possibilities
Gist of the problem –> don’t know the exact date the product was purchased!
(1) Shortest Warranty period –> Bought Jan 31, warranty, Feb 1 to March 1 (29 days later = 28 days in Feb + 1 day in March)
Longest Warranty period –> bought is Jan 1, warranty from Jan 2 to March 31 (30 days in Jan + 28 days in Feb + 31 days in March = 89 days)
NS
(2) Shortest Warranty period –> Bought it May 1, warranty due May 2 (1 day)
Longest Warranty period –> Bought it May 1, warranty from May 2 to May 31 (30 days)
NS
Together - NS
If r, s, and t are consecutive positive multiples of 3, is rst divisible by 27, 54, or both?
*Reason it out using previous concepts with multiples, divisibility, and consecutive multiples
rst = 27(m)(m + 1)*(m + 2)
- rst contains 4 3’s
- at least one of the numbers is EVEN
27 can be broken into its factors with the three 3’s
54 can be broken into its factors with three 3’s and one 2
When it is 2:01 pm Sunday afternoon in Nullepart, it is Monday in Eim. When it is 1:00 pm in Eim, it is also Wednesday in N. When it is noon Friday in N, what is the possible range of times in Eim?
Scheduling Question - consider smallest and largest times
> however, the second statement restricts the largest time difference, so there is no need to consider the largest time difference imposed by statement 1 (saves time!)
Statement 1 –> Eim is at least 10 hours ahead of N (min diff)
Statement 2 –> Eim is at most 13 hours ahead. (max diff)
So if it is 12pm Friday in N, Eim’s time is between 10pm Friday and 1am Saturday.
Is 11/301 > 3/100?
DON’T DO LONG DIVISION
Just simplify by cross multiplying and evaluating the inequality:
11*100 > 3 * 301 —> so true
Is the average of n consecutive integers = 1?
(1) n is even
(2) If S is the sum of n consecutive integers, then 0 < S < n
Consecutive integers and statistics:
(1) if n is even, then the mean = median is NEVER an integer value (won’t be a term in the set).
> Therefore, avg cannot equal 1
> Sufficient
(2) Test cases show that always false –> number of terms will be EVEN to make this statement true
> Therefore, avg cannot equal 1 (will always equal 0.5)
> Sufficient
Is k > 3n?
(1) k > 2n
NOT SUFFICIENT
if n = 1 – statement 1 says k > 2. If k = 3, then k > 3 is false.
Otherwise, if k = 4, both inequalities are true.
CONCEPT –> inequality DS questions need more simplification!
Divisibility Rule by 4
e.g., 64, 68, 72, 23456
If the integer is divisible by 2 twice
OR
For larger numbers, the LAST TWO digits are divisible by 4
Divisibility Rule by 8
e.g. 56
If the integer is divisible by 2 three times
OR
For larger numbers, if the last three digits are divisible by 8
Divisibility Rule by 9
e.g., 108
If the sum of the digits is divisible by 9 (kind of like divisibility rule for 3).
**powers of 9 are all divisible by 3
What integer is a factor of every integer?
1
What are the first 10 prime numbers?
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Tip for identifying higher level primes:
> see if it is divisible by 3 and 7 (and obviously must be odd)
What is the Factor Foundation Rule
If a is a factor of b, and b is a factor of c, then a is a factor of c.
> An integer is divisible by its factors and the factors of ITS factors.
For a^k to be a factor of b^m, a must be a factor of b AND k <= m
If 80 is a factor of r, is 15 a factor of r?
FACTOR BOX:
SOMETIMES YES, SOMETIMES NO
Partial prime box for r:
2 2 2 2 5 …?
> We DON’T KNOW if there are other prime factors in the box
No example: r = 80
Yes example: r = 80 * 3 = 240
How do you get a negative number? How many negatives and positives must be multiplied together?
To get a negative number, you need an ODD NUMBER OF NEGATIVE SIGNS (watch out for 0!)
> regardless of the # of positive numbers
To get a positive number, you need an even number of negative signs
Is the product of all the elements in set S negative?
(1) All of the elements in set S are negative.
(2) There are 5 negative numbers in set S
Properties of a product of numbers:
is there an ODD # of negative numbers?
(1) NS –> don’t know how many negatives
(2) NS –> while there are an odd number of negative numbers in set S, there could be 0 in the set. This would make the product = 0.
TOGETHER –> If all the elements are negative and there are only 5 negative numbers –> Yes, the product is negative.
Sufficient (c)
Recall: 0 is NEITHER POSITIVE NOR NEGATIVE
If x and y are both integers and x/y = odd integer, is x odd?
Not necessarily!
Concept: There is NO GUARANTEES for division (unlike Multiplication).
e.g., 60/20 = 3
21/7 = 3
Is the integer X odd?
(1) 2(y + x) is an odd integer
(2) 2y is an odd integer
Odd/Even properties
x is an integer
(1) 2y + 2x = odd
for this to be true, y must be a fraction with 2 in the dnom, and 2y must be odd.
NS
(2) Nothing is given about x
Also same info as statement 1
NS
TOGETHER:
2y + 2x = Odd
Odd + 2x = Odd —> x can be odd or even to make 2x even.
NS
When I See an integer p with a factor n, such that 1 < n < p …
p is NOT a prime number (composite number) –> p has more than two factors (itself and 1)
When I See x and y are integers and x + y > 0…
At least one is POSITIVE!
When I See x and y are integers and x + y < 0…
At least one is NEGATIVE
When I See x and y are integers and x/y > 0
x and y have the same sign
Is 6x always even?
Yes
= 23x —> 2 is a factor!!
Same with any even integer * x
If p, q, and r are integers, is pq + r even?
(1) p + r is even
(2) q + r is odd
Even/Odd properties
(1) even –> p and r are either both even or both odd
NS –> need to know q
oe + o = o
oo + o = e
e.g., 23 + 4 = Even
32 + 1 = Odd
(2) odd –> q and r, one is odd one is even
NS –> need to know p
oe + o = o
oo + e = o
eo + e = e
e.g., 23 + 4 = even
3*3 + 4 = odd
Together:
> p and r have the SAME TYPE (either both even or both odd)
> q and r have the OPPOSITE TYPE
pq + r
oe + o = o
eo + e = e
32 + 5 = odd
23 + 4 = even
E - not sufficient even together
The length of a certain rectangle is a multiple of 18 and the width of the rectangle is a multiple of 12. Which of the following cannot be the perimeter of the rectangle?
60
72
84
96
108
72 –> this is NOT a factor question. This is a MULTIPLE question.
2(L + W) = P
Lay out the multiples of 18 and 12.
18, 36, 54, 72, 90, 108
12, 24, 36, 48, 60, 72, 84, 96, 108
Divide each of the potential perimeters by 2 –> then see if the combo can be reached.
How do you solve questions asking to find the number of possible arrangements of a group?
Combination Q or Permutation w Repetition: Anagram Grid
Top row –> number of people
Bottom row –> number of categories
= n! / duplicate! * duplicate! —> see if order matters!
e.g., 7 people that can get 1 platinum medal, 1 gold medal, 2 silver medals, or 3 bronze medals: Permutation with repetition.
7! / (1!1!2!*3!)
= 420
A local card club will send 3 representatives to the national conference. If the local club has 8 members, how many different groups of representatives could the club send?
Arranging group –> Combination Q –> Anagram Grid
Top: 1 2 3 4 5 6 7 8
Bot: Y Y Y N N N N M
Cm,n = C8,3 (from 8 choose 3)
= 8! / (3! * 5!) = 56
OR Slot method / # of duplicates
= (8 * 7 * 6) / (6) —-> 6 duplicates because the same 3 people comprise of 321 = 6 arrangements = 3!
The yearbook committee has to pick a colour scheme for this year’s yearbook. There are 7 colours to choose from. How many different colour schemes are possible if the committee can select at most 2 colours?
Combinatorics –> Multiple Decision Q (Combination Q) –> Anagram Grid with a twist (“at most” 2 colours)
1 colour OR 2 colours:
1 Colour Options: 7! / (1! * 6!) = 7
2 Colour Options: 7! / (2! * 5!) = 21 —–> C7,2 (from 7 choose 2)
7 + 21 = 28 options.
A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim?
Permutation
Order MATTERS –> There are NO duplicates (same type question)
Two groups (Male AND Females)
Males = 3!
Females = 3!
Together: 3! * 3! = 36 options
What is the probability of rolling two cubes and getting 1 one both?
1/6 * 1/6 = 1/36
–> out of 36 possible combinations, (1,1) occurs ONLY ONCE!!
Whereas a combination of 2 and 5 can happen twice, such as (2,5) and (5,2)
A magician has 5 animals in a magic hat: 3 doves and 2 rabbits. if the magician pulls two animals out of the hat at random, what is the chance that the two will be the same type of animal?
P(two doves) or P(two rabbits)
P(two doves) = P(dove on first) and P(dove on second | dove on first)
= (3/5) * (2/4)
= 3/10
P(two rabbits) = P(rabbit on first) and P(rabbit on second | rabbit on first)
= (2/5) * (1/4)
= 1/10
P(two doves) or P(two rabbits) = 4/10
Is the result of adding or subtracting a MULTIPLE of N and a NON-multiple of N a multiple of N?
NO - the result is a NON-Multiple of N
e.g., 18 - 10 = 8 (non-multiple of 3)
18 and 10 are both multiples of 2, so the result, 8, is also a multiple of 2
Recall: adding or subtracting a MULTIPLE of N and a MULTIPLE of N = Multiple of N
Is the result of adding or subtracting two NON-multiples of N a multiple of N?
Not sure - could either be a multiple of N or a non-multiple of N
e.g., 12 + 13 = 25 –> adding two non-multiples of 5 result in a multiple of 5
Is the integer of z divisible by 6?
(1) the GCF of z and 12 is 3
(2) the GCF of z and 15 is 15
Factors:
In other words: Is 6 a factor of z?
(1) GCF = 3
12 factors: 1, 12, 2, 6, 3, 4
z factors = 1, 3… –> z does not have factors with 2
Therefore, z is NOT divisible by 6. Sufficient
OR Prime Factor columns: GCF (lowest powers)
If GCF = 3^1…
12 = 223 = 2^2 * 3^1
Therefore, z = 2^0 * 3^1
(2) GCF = 15
Prime Factor columns: GCF (lowest powers)
If GCF = 3^1 * 5^1…
15 = 3^1 * 5^1
Therefore, z = at least 3^1 * 5^1 (could have a 2)
A
What is the GCF and LCM of 12 and 40?
Use Prime Column Tables
12 prime factors as powers: 223 = 2^2 * 3^1 * 5^0
40 prime factors as powers: 222*5 = 2^3 * 3^0 * 5^1
GCF = lowest powers = 2^2 * 3^0 * 5^0 = 4
LCM = largest powers = 2^3 * 3^1 *5^1 = 120
Consider the number 2000.
1) How many unique prime factors are there?
2) What is the length? (Total number of prime factors that when multiplied add to 2000)
3) How many total factors are there?
(1) Unique prime factors –> prime factorization, count unique primes
2000 = 2^4*5^3 —> 2 unique primes
(2) Length: Add exponents = 4 + 3 = 7
(3) total factors –> PRIME factorization, product of exponents plus 1 (includes 1 as a factor)
(4+1)(3+1) = 54 = 20
> kind of like finding the area of a table
Properties of perfect squares in their prime factorization form
> the PRIME factorization of perfect squares contains ONLY EVEN POWERS
the total number of factors is ODD (due to adding 1 to each power before multiplying)
Properties of “perfect” powers - how to tell if a number is a perfect square, cube or something else?
Look at the number in prime factorization form - look at the EXPONENTS
> all even (multiples of 2) –> perfect square
all multiples of 3 –> perfect cube
all multiples of 4 –> fourths
If k^3 is divisible by 240, what is the least possible value of integer k?
Factors
> k^3 = three identical factor sets of k
Prime Factorization of 240: 2^4 * 3^1 *5^1
> we have 3 2’s –> k must be divisible by 2
one left over 2 –> in order for k^3 to be divisible by 240, k must also two factors of 2 (plus, the k’s are all identical)
we have incomplete factors of 3 and 5, but along the same logic as above (kkk each with identical factors), 3 and 5 must be present too
So, least value of k is a product of its prime factors = 223*5 = 60
What is 10! + 11! a multiple of / divisible by?
CONCEPT: Multiples of N added or subtracted together produces a result that is also a Multiple of N
10! –>a multiple of integers from 1 to 10
11! –> a multiple of integers from 1 to 11
So common factors include – 1 to 10
Therefore, N! is a multiple of all the integers from 1 to N
ALSO all numbers are a multiple of 1
What is the remainder of 20 divided by 3 in DIFFERENT FORMS
What about 3/20?
Integer Form: 20/3 = 6 R 3
Fractional From: 20/3 = INTEGER + FRACTION = 6 + 2/3
Decimal form = 2/3 = 0.6667 = Remainder/Divisor
3/20 = 0 integer + 3/20 = 0 R 3
When positive integer x is divided by 5, the remainder is 2. When positive integer y is divided by 4, the remainder is 1. Which of the following values CANNOT be the sum of x and y?
12
13
14
16
21
TEST CASES for REMAINDER questions
Remember: Dividend = Divisor*quotient + remainder
Dividend/divisor = quotient + remainder
x 2 7 12, 17 (goes up by 5 = pattern)
y 1 5 9 13 (goes up by 4 = pattern)
find different combos
12 = 7 + 5
13 = 12 + 1
14 = ?
16 = 7 + 9
21 = 12 + 9
If b/4 is an integer and a = b + 4, what is the GCF?
b/4 is an integer => 4 is a factor of b; b is a multiple of 4.
a = b + 4 => a is also a multiple of 4
CONCEPT: For any two POSITIVE CONSECUTIVE multiples of an integer n, n is the greatest common factor of those multiples, so the greatest common factor of a and b is 4.
A) How many different words can you make from SSSEEE?
B) What about SSSETG?
C) What if you have to pick items to put in a box where order doesn’t matter?
e.g., SSSE but you have 5 S’s to choose from, 3 E’s to choose from.
Combinatorics Question
> Can you think of it as a Y/N arrangement? (Combination)
> are you asked to ARRANGE different items WITHOUT DUPLICATES (Permutation) —>
> Are you asked to ARRANGE different items with some duplicates? (Permutations w Repetition) —> “identical” or otherwise says there are duplicates
> Are you asked to create a “box” of different items where order doesn’t matter? (Not an arrangement)
A) How many ways can you ARRANGE SSSEEE?
> Arrange LETTERS (permutation with repetition)
> n! / duplicates! * duplicates !
6! / (3!)(3!)
= 20
B) For SSSETG?
= 6! / 3! (order six positions, divided by 3! ways to order S’s)
C) Combination This question is different than ARRANGING different items. You are asked to choose items to put into something where order doesn’t matter. The “arranging” part comes in when you are picking which items to put in the box - you don’t arrange the resulting items in the box.
Approach –> do item by item
= 3 S AND 1 E
= C5,3 AND C3,1
= [5!/(3!2!)] * [3!/(1!2!)]
Six people sit beside each other in six adjacent seats at the movie theatre. If Jean and Mark refuse to sit next one another, how many seating arrangements are there?
Combinatorics with Constraint –> Glue Method
Permutation
> Arrange LETTERS
= Total # of Ways to Arrange - # of Ways to Arrange Sit Together
= (6!) - (2*5!)
= 480
5! because we treat Jean and Mark as one group (stuck together).
2*5! because each 5! ways could be in the order JM or MJ
A miniature gumball machine contains 7 blue, 5 green, and 4 red gumballs, which are identical except for their colors. If the machine dispenses three gumballs at random, what is the probability that it dispenses one gumball of each color?
of ways to arrange BGR = 3!
Combinatorics and Probability (Domino-effect Q)
Probability (B and G and R) –> no replacement each time!
= (7/16) * (5/15) * (4/14) –> FOR ONE CASE, BGR
= 1/24
Total probability
= 1/24 * (3! cases)
= 1/4
Five A-list actresses are vying for the three leading roles in a new film. The actresses are J, M, S, N and H. Assuming that no actress has any advantage in getting any role, what is the probability that J and H will star in the film together?
Combinatorics and Probability (Domino-effect Q)
P(JH and other)?
= [P(JHM) + P(JHS) + P(JHN) ] * 3! ways to arrange each
= (1/51/41/3) + (1/51/41/3) + (1/51/41/3) * 3!
= 1/20 * 3!
= 3/10
or P(JH and other)
= (1/51/43/3)*6
= 3/10
Triangle inscribed in a circle and one of the sides is also the diameter of the circle
Triangle is a RIGHT TRANGLE (90 degree angle is opposite to the diameter)
Area of Parallelogram
Base * height
> Height –> shortest distance between base and opposite side, segment is perpendicular to base.
Recall that the two parallel sides are EQUAL in length
**max area of a parallelogram is a rectangle (perpendicular sides)
Area of Trapezoid
1/2 * (base 1 + base 2) * height
In other words, take the AVERAGE of the two bases and multiply it by the height
Common Right Triangles
- Without special angles * 3
- With special angles * 2
RATIOS (remember they can be scaled)
W/o special angles:
3-4-5 —–> key multiples (6-8-10, 9-12-15, 12-16-20)
5-12-13 —-> key multiples (10-24-26)
8-15-17
W special angles:
1-1-sqrt(2)»_space;>isosceles right triangle has two 45 degree angles (45-45-90 degrees)
1-sqrt(3)-2»_space;> half of an equilateral triangle (30-60-90 degrees)
What does this mean?
MN || OP
Line segment MN is PARALLEL to line segment OP
Surface area of rectangular solids and cubes
Sum of area of each side (6 sides!)
Cube –> area of one side * 6
Rectangle –> area12 + area22 + area3*2
How many books, each with a volume of 100 in^3, can be packed into a crate with a volume of 5000 in^3?
NOT SURE –> when dealing with fitting 3D objects into other 3D objects, knowing the respective volumes is NOT ENOUGH
> need to know shapes
e.g., 100 in^3
= 20 * 5 * 1
= 25 * 4 * 1
= 5 * 5 * 4
If two angles in a triangle are equal, what does that say about the sides of the triangle?
CONCEPT: Angles correspond to their opposite sides
If two angles are equal, then the sides opposite to them are also equal in length
Other related concepts:
> Shortest side is opposite from the smallest angle
> Longest side is opposite from the largest angle
Triangle has the following angles: 45-45-90
What are the ratios of the sides?
Where to find this triangle in other polygons?
Isosceles Right Triangle
1x - 1x - x*sqrt(2)
Half of a square is an isosceles right triangle
Triangle has the following angles: 30-60-90
What are the ratios of the sides?
Where to find this triangle in other polygons?
1x - x*sqrt(3) - 2x
Half of the equilateral triangle is a 30-60-90 degree triangle
(so the height of an equilateral triangle has length x*sqrt(3))
Which quadrants does the line 2x + y = 5 pass through?
Sketch the line. Make sure you find the x and y intercepts!
y = -2x + 5
Points:
(0, 5)
(-1, 7) –> Quadrant II
(1, 3) –> Quadrant I
(10, -15) –> Quadrant IV
What is the circumference of a circle?
(1) The area of a circle is 16*pi.
(2) diameter is 8
As long as you have at LEAST ONE of the following, you can solve for the rest!
> Area of Circle (pir^2)
> Circumference of a Circle (2pir = pid)
> radius, r
> diameter, d = 2r
How do you calculate the area of a “sector” in a circle? Length of an arc?
Area of a Sector (pie-shaped wedge)
= % * Area of the Circle
= (Central Angle/360) * Area of the Circle
Length of an arc
= % * Circumference
= (Central Angle/360) * Circumference
How do you calculate the inscribed angle?
Recall: Central Angle is formed when one of the vertexes is at the CENTER of the circle (and each segment is the radius)
Inscribed Angle = 1/2 * central angle
Both share a common arc (same two vertices)
***explains why an inscribed triangle with one side equal to the diameter must be a RIGHT TRIANGLE
> two arcs, two different central and inscribed angles
Volume of a cylinder and Surface Area of a cylinder
V = pi*r^2 * h
= area of circle * height
SA = 2(pir^2) + 2pi*r * (h)
= area of two circles + area of rectangle
Three gnomes and three elves sit down in a row of six chairs. If no gnome will sit next to another gnome and no elf will sit next to another elf, in how many different ways can the elves and gnomes sit?
Permutation with same types (no duplicates because we treat each gnome and elf as unique)
e.g., GEGEGE
EGEGEG
= 3! ways to arrange the Gnomes AND 3! ways to arrange the Elves * 2 starting positions
= 2 * (3! * 3!)
= 72
Can you find the height or side of an equilateral triangle given its area?
E.g., Area = 36*sqrt(3)
YES
Half of the equilateral triangle is a 30-60-90 degree right triangle.
Let’s call one side y and the height h.
18sqrt(3) = 1/2(1/2y)(h)
36sqrt(3) = (1/2y)(h) —–> Recall ratio of sides! (x-xsqrt(3)-2)
REWRITE HEIGHT AS A FUNCTION OF Y
36sqrt(3) = (1/2y) * (1/2y)sqrt(3)
36 = 1/4*y^2
144 = y^2 —–>y must be positive
y = 12 (one of the equilateral triangle’s sides)
Height = (1/2y)sqrt(3)
= 1/212 * sqrt(3)
= 6sqrt(3)
or directly: Area of an equilateral triangle = s^2*sqrt(3) / 4
What are the characteristics of a rhombus?
How do you calculate the area of a rhombus?
Basically a square that’s been squished (or a diamond)
> equal sides
Diagonals BISECT one another at 90 degrees (perpendicular bisectors) but are not equal in length
Area = (diagonal1 * diagonal2)/2
For a given perimeter, what type of QUADRILATERAL will MAXIMIZE area?
For a given area, what type of QUADRILATERAL will MINIMIZE perimeter?
SQUARE
e.g., P = 36
MAX area –> square with each side equal to 36/4 = 9
Other types of dimensions: 2x14, 6x12
e.g., A = 100
MIN Perimeter –> square with each side equal to sqrt(100) = 10 (P = 40)
Other types of dimensions: 50x2 (P = 104), 25x4 (P = 58)
For a given perimeter (or two sides of a triangle or parallelogram or rhombus), how do you MAXIMIZE area?
Two sides must be PERPENDICULAR to each other
*be wary of questions where perimeter is incomplete (e.g., one side of a fence is a wall) –> cannot use this approach.
Formula for calculating the area of an equilateral triangle?
[s^2 *sqrt(3)]/4
Just need to know the length of each side OR height to calculate area! (don’t need to know height AND side!)
This is because an equilateral triangle is comprised of two special 30-60-90 degree right triangles.
What is the ratio of the areas of two similar polygons (triangles, quadrilaterals, pentagons etc.)?
SHORT CUT: If the ratio of sides/heights/perimeters is a/b, then the ratio of areas will be (a/b)^2 or a^2 : b^2
CONCEPT: Similar polygons have PROPORTIONAL corresponding side lengths and EQUAL angles
e.g.,
Triangle A is a right triangle with two sides equal to 9 and 12.
Triangle B is a similar right triangle with two corresponding sides equal to 3 and 4.
Ratio of sides a to b = 9:3 = 3:1
Ratio of areas a to b = 9:1
Check:
Area of Triangle A = 9120.5 = 54
Area of Triangle B = 340.5 = 6
54:6 = 9:1
What is the length of the diagonal of a cube equal to?
side*sqrt(3)
Derived from Pythagorean Theorem where the hypotenuse equals the diagonal, one leg equals the edge of the cube (s) and the other leg equals the diagonal of the square (s*sqrt(2))
Half of the diagonal is side/2 * sqrt(3)
> from the CENTER of the cube
How do you find the equation of the perpendicular bisector of a line segment?
e.g., Line segment has two points, (2, 2) and (0, -2).
Slope of the perpendicular bisector = Negative Reciprocal Slope
(Rule applies to ALL PERPENDICULAR LINES)
Slope of Line Segment = -4/-2 = 2
Therefore, slope of the perpendicular bisector = -1/2
To find the y intercept, first find the point at which the lines intersect (or the MIDPOINT of the line segment)
=> midpoint coordinates are the AVERAGE of the x and y coordinates = (x1 + x2)/2 and (y1 + y2)/2 = (1, 0)
y = -0.5x + 0.5
x = 9^10 - 3^17 and x/n is an integer. If n is a positive integer that has exactly two factors, how many different values for n are possible?
UNDERSTAND:
> testing factors
> two factors = prime
> Rephrase Q: How many prime factors does x have?
PLAN –> need to break down x into its prime factors = prime factorization.
(don’t just rely on Odd - Odd = even concepts to get to 2 and multiples of 3 method to get to 3. There could be more primes that are missing!)
x = (3^2)^10 - 3^17
= 3^20 - 3^17
= 3^17(3^3 - 1)
= 3^17(26)
= 3^17(2)(13)
Three primes
a b c
+ d e f
———–
x y z
If a, b, c, d, e, f, x, y, and z each represent different positive single digits, what is the value of z?
(1) f - c = 3
(2) 3a = f = 6y
Digits Question:
(1) z can be many values
9 - 6 = 3 —> 9 + 6 = 15 –> z = 5
8 - 5 = 3 –> 8 + 5 = 13 –> z = 3
NS
(2) y must be 1 (otherwise, f > 10)
y = 1
a = 2
f = 6
2 b c
d e 6
——–
x 1 z
WRITE OUT in algebraic form: PAY attention to UNITS digit
200 + 10b + c + 100d + 10e + 6 = 100x + 10 + z
196 = 100x - 100d - 10b - 10e + z - c
196 = 100(x - d) - 10(b + e) + 1(z - c)
Focus on the units column: (because others are powers of 10 and have units digit equal to 0)
6 = z - c
(z, c) = (9, 3), (8, 2), (7, 1)
However, the last two pairs contain integers that have already been used!
z = 9 (sufficient).
In year x, it rained on 40% of all Mondays and 20% of all Tuesdays. On what percentage of all the weekdays in year x did it NOT rain?
(1) During year x, it rained on 10% of all Wednesdays.
(2) During year x, it did not rain on 70% of Thursdays and it did not rain on 95% of all Fridays.
Q: # of weekdays no rain / total number of weekdays = ?
(1) We know % of Mondays, Tuesdays, and Wednesdays that have rain. We are missing Thursdays and Fridays => Not Sufficient
(2) Like 1, we are missing, Wednesdays.
(3) Together: Even if we know the % of days that do not have rain, we do not know how many weeks had no rain (i.e., days in which there are no rain could fall on different weeks!)
=> Not sufficient
E
Is y > 7/11?
(1) 1/5 < y < 11/12
(2) 2/9 < y < 8/13
In other words, is y outside of the ranges?
7/11 > 0.5 –> we can only focus on the upper bounds.
(1) 11/12 versus 7/11
We know that 7/11 < 8/12 and 8/12 < 11/12
So, 7/11 < 11/12
=> NS (y is in the range)
(2) 8/13 versus 7/11 ==> ugly comparison, CROSS MULTIPLY
Is 8/13 < 7/11?
Is 88 < 91? => Yes
So 8/13 < 7/11
=> S (y is not greater than 7/11)
Each digit in the two-digit number G is halved to form a new two-digit number H. Which of the following could be the sum of G and H?
153
150
137
129
89
Digits Question: Values –> algebraic form
G: _x__ _y__ —-> G contains even digits
H: x/2 y/2
G + H =?
MAX value -> Since G and H are each two digit numbers, G + H < 88 + 44 = 132
(Eliminates 153, 150, and 137).
MIN value –> G + H > 22 + 11 = 33
Write out algebraic form:
10x + y + 10(x/2) + y/2
= 10x + y + 5x + y/2
= 15x + 3/2y
= 3(5x + 0.5y) —> sum is a multiple of 3 (eliminates 137 and 89)
we know that 0.5y is an integer, so 5x + 0.5*y is an int.
Ans: 129
Check: 129/3 = 43 = 5x + 0.5y
40 = 5*x
x = 8
If positive even integer p has a positive units digit and the units digit of (p^3 – p^2) is equal to 0, what is the units digit of the quantity p + 3?
p is an even integer, such that p cubed - p squared = 0
> since the units digit must be positive, it cannot be 0.
Units digit of p cubed and p squared must be equal!
Therefore p must be a multiple of 6 (always ends with 6)
6 + 3 has a units digit = 9
A gardener is planning a garden layout. There are two rectangular beds, A and B, that will each contain a total of 5 types of shrubs or flowers. For each bed, the gardener can choose from among 6 types of annual flowers, 4 types of perennial flowers, and 7 types of shrubs. Bed A must contain exactly 1 type of shrub and exactly 2 types of annual flower. Bed B must contain exactly 2 types of shrub and at least 1 type of annual flower. No flower or shrub will used more than once in each bed.
Identify the number of possible combinations of shrubs and flowers for bed A and the number of possible combinations of shrubs and flowers for bed B. Make only two selections, one in each column.
Strategy –> look at each bed separately AND each flower separately
Bed A: S A A P P
# of ways to pick 1 shrub: 7!/(1!6!) = 7
# of ways to pick 2 Annual Flowers: 6!/(2!4!) = 15
# of ways to pick 2 perennial flowers: 4!/(2!2!) = 6
Total number of possible beds (order doesn’t matter in the bed!) = 715*6 = 630
Bed B:
SSAPP or SSAAP or SSAAA
= 7!/(2!5!) * [6!/(1!5!)4!/(2!2!) + 6!/(2!4!)4!/(1!3!) + 6!/(3!3!)] —> use prev calculations from Bed A
= 2436
Solving equations with absolute values
x^2 - 8x + 21 = | x - 4 | + 5
Solve by using BRANCHES (positive, negative, and ZERO case for what’s inside the absolute value signs)
> Same approach as solving square root of a square
IDENTIFY “root” –> x = 4
Positive and Equal case: x - 4 >= 0 or x >= 4
x^2 - 8x + 21 = x - 4 + 5
x^2 - 9x + 20 = 0
(x - 4)(x - 5) = 0
x = 4 or x = 5 —> x = 4, or 5
Negative case: x - 4 < 0 or -(x-4) or x < 4
x^2 - 8x + 21 = -(x - 4) + 5
x^2 - 7x + 12 = 0
(x - 3)(x - 4) = 0
x = 3 or x = 4 —> x = 3
A certain college party is attended by both male and female students. The ratio of male-to-female students is 3 to 5. If 5 of the male students were to leave the party, the ratio would change to 1 to 2. How many total students are at the party?
24
30
48
80
90
Algebraic way:
M F T
3x + 5x = 8x (actual of students)
M new / Total = 1/3
(3x - 5)/(8x - 5) = 1/3
x = 10
Total original = 80
OR WORK BACKWARDS
> integer constraint –> total (8x) must be a multiple of 8 (eliminate 30, 90)
> Test 8x = 24 –> x = 3
M = 9
F = 15
Ratio M/F = 9/15 = 3/5
M - 5 = 4
M - 5 / F = 4/15 (WRONG)
> Test 8x = 48 —> x = 6
M = 18
F = 30
M - 5 = 13
M - 5 / F = 13/30 (Wrong)
Ans 80
If n is a positive integer, what must be true of n^3 - n?
SIMPLIFY and REWRITE n^3 - n
= n(n^2 - 1)
= n(n + 1)(n - 1) —-> Recognize this as the product of three consecutive integers!!
= (n - 1)(n)(n + 1)
Rule for product of 3 consecutive integers: Divisible by 3! = 6
Is z an even integer?
(2) 3z is an even integer
test FRACTIONAL cases
3*fraction = even (e.g., 2)
3*z = 2
z = 2/3
3*(2/3) = 2 —> z is not an even integer
NS
T is a set of y integers, where 0 < y < 7. If the average of Set T is the positive integer x, which of the following could NOT be the median of Set T ?
0
x
-x
1/3 y
2/7 y
Method 1) Create sets that conform to the facts and eliminate answers
> Problem with this method - Takes TOO LONG
Method 2) Pay attention to fractions and whether it makes sense. Also pay attention to Median rules
> Median is either (1) int in the set or (2) fraction with a DENOMINATOR = 2
e.g., y must be 1, 2, 3, 4, 5 or 6
2/7 * y will yield a fraction without a 2 in the denominator
> 2/7 y is wrong
Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?
Work problem
Fraction of the whole job done by Peter = [(P’s rate)*(Total Time worked)/1
Hourly Rates for each person:
T = 1/6 jobs per hour
P = 1/3 jobs per hour
J = 1/2 jobs per hour
First hour - T completes 1/6 of the job. 5/6 remains.
Second hour - T and P complete 1/6 + 1/3 = 3/6 of the job.
Total completed: 4/6. 2/6 or 1/3 remains.
Last Time Segment (WE DON’T KNOW YET):
> Hourly Rate of 1/6 + 1/3 + 1/2 = 1 job per hour
Time to Finish 1/3 of a job: 1 * t = 1/3
t = 1/3.
P’s fraction = 1/3 + 1/3*1/3 = 4/9
What percentage of the current fourth graders at Liberation Elementary School dressed in costume for Halloween for the past two years in a row (both this year and last year)?
(A) 60% of the current fourth graders at Liberation Elementary School dressed in costume for Halloween this year.
(B) Of the current fourth graders at Liberation Elementary School who did not dress in costume for Halloween this year, 80% did not dress in costume last year.
THIS IS AN OVERLAPPING SET / DOUBLE MATRIX QUESTION
Both statements are insufficient
Provide an example of a set in which the MEAN is SMALLER than the MEDIAN
Must the standard deviation of such a set be greater than a set in which Median = Mean?
Recall: Median splits the set into half. If Median > Mean, the numbers in the bottom half of the set must be farther away from the Median than the top half.
e.g.,
{3, 50, 70}
Mean = 123/3 = 41 (lower than 50)
{2, 6, 7}
Mean = 15/3 = 5 (lower than 6)
No - Median = Mean just means that the elements in both the bottom and upper half of the set are equally close/far from the median. But the elements can be spread however far apart.
e.g., {100, 200, 300}
e.g., {1, 2, 3}
Composite set (combination of two sets)
What are the properties of its mean?
What are the properties of its median?
MEAN: When two sets are combined to form a composite set, the mean of the composite set must EITHER be BETWEEN the means of the individual sets or be EQUAL to the mean of BOTH of the individual sets.
e.g., A = [3], B = [5]
A + B = [3, 5] –> mean is 4 (greater than A and less than B)
A = [5], B = [3]
A + B = [3, 5] –> Mean is still 4 (this time, it is greater than B and less than A)
MEDIAN:
when two sets are combined to form a composite set, the median of the composite set must EITHER be BETWEEN the medians of the individual sets or be EQUAL to the median of one or both of the individual sets.
> Median can equal or be greater than or less than its mean
e.g., A = [3], B = [5]
A + B = [3, 5] –> median is 4, which equals mean
Data sufficiency and inequalities - things to remember?
- One inequality can IMPLY another inequality
e.g., x > 5 —> x > 0 (positive) - Watch out for positive and negative unknowns and how it affects the direction of the inequality
- Inequalities can combine to yield a single answer
e.g. 0 < x < 2 AND x is an integer —> x = 1 - Many inequalities are actually disguised as positive/negative questions
How do you solve for inequalities in factored form?
(e.g., common statements for DS)
e.g., a(a - 2)(a + 1) < 0
Sewing approach:
First, solve as if it were an EQUATION to get the “roots”:
a(a - 2)(a + 1) = 0
a = 0, or a = 2, or a = -1
Second, draw a number line using the roots and do sewing approach, switching signs at the roots (unless the exponent on the factor is even)
Therefore a(a - 2)(a + 1) < 0 when a < -1 and 0 < a < 2.
Weighted Average Problems (especially hidden ones):
Can you find the average of something by simply taking the average OF an average?
e.g., A certain bank has ten branches. What is the total amount of assets under management at the bank?
(1) There is an average (arithmetic mean) of 400 customers per branch. When each branch’s average (arithmetic mean) assets under management per customer is computed, these values are added together and this sum is divided by 10. The result is $400,000 per customer.
(2) When the total assets per branch are added up, each branch is found to manage an average (arithmetic mean) of 160 million dollars in assets.
No - due to weighted average principles
> cannot use average of an average to get a number
e.g., Average Rate =/ average of rates
(1) NOT SUFFICIENT
We want to know AUM = AUM/branch * 10 = average AUM per branch * 10
We are given:
Customers/10 = 400 –> Total Customers = 4000 customers
Each branch’s Average AUM per customer = Sum AUM / N customers
When you take Sum of Each Branch’s Average Aum per customer / 10
= 400k per customer
You cannot do 400k per customer * 400 –> this assumes that each branch has 400 customers (equal weight), which is doesn’t!!
400 = customers/10 —> some branches have more than others
Reserve tank 1 is capable of holding z gallons of water. Water is pumped into tank 1, which starts off empty, at a rate of x gallons per minute. Tank 1 simultaneously leaks water at a rate of y gallons per minute (where x > y). The water that leaks out of tank 1 drips into tank 2, which also starts out empty. If the total capacity of tank 2 is twice the number of gallons of water actually existing in tank 1 after one minute, does tank 1 fill up before tank 2 ?
(1) zy < 2x^2 – 4xy + 2y^2
(2) The total capacity of tank 2 is less than one-half that of tank 1.
Manipulate statement such that z/(x - y) versus 2(x - y)/y
=> formula for time!!!
Statement 1 is sufficient
Ten years ago, scientists predicted that the animal z would become extinct in t years. What is t?
(1) Animal z became extinct 4 years ago.
(2) If the scientists had extended their extinction prediction for animal z by 3 years, their prediction would have been incorrect by 2 years.
Key #1 –> t is a PREDICTION, not actual year the animal became extinct.
Key #2 –> “incorrect by 2 years” means + or minus 2
E
What are regular figures in geometry? What are examples?
Regular figures are those for which you only need ONE MEASUREMENT to KNOW EVERY MEASUREMENT
> also have equal sides and angles
Include:
- Circles (radius = diameter = area = circumference = Sufficient)
- Square (side = area = perimeter = diagonal)
- 45-45-90 degree triangle (any side = all sides = area)
- 30-60-90 degree triangle (any side = all sides = area)
- Equilateral triangle (any side = area)
How would you express the following sequence as a formula?
10 17 24 31 …
Arithmetic Sequence:
Difference between terms = 7 (so each term is some multiple of 7 plus or minus something)
An = a1 + (n - 1)d
= 10 + (n - 1)7
= 7n + 3
n >= 1
What is the remainder of any integer divided by 10?
The integer’s UNITS DIGIT
e.g., 25/10 = 2 remainder 5 = 2.5
How many integers between 1 and 300 are divisible by 3?
300 integers in the set / 3 = 100
In other words, count the # of multiples of 3 between 1 and 300.
(a + b)^2 - (a - b)^2 = ?
4ab
a - b = sqrt(a) - sqrt(b)
What is a in terms of b?
If a does not equal b.
RECOGNIZE Special Factoring = Difference of Squares (any EVEN POWERS that are the SAME in both terms)
a - b = (sqrt(a) + sqrt(b))*(sqrt(a) - sqrt(b))
(sqrt(a) + sqrt(b))*(sqrt(a) - sqrt(b)) = sqrt(a) - sqrt(b)
sqrt(a) + sqrt(b) = 1
sqrt(a) = 1 - sqrt(b)
a = 1 - 2sqrt(b) + b
Is x + y > 0?
(1) (x + y)(x - y) > 0
Need to know signs
Statement 1: Means that x + y and x - y have the SAME SIGN (++ or –)
**IMMEDIATLEY recognize this is INSUFFICIENT to determine if it is one of the three cases above
(the two factors just have to be the same sign) !
Case 1) x + y > 0 and x - y > 0 —> Yes
Case 2) x + y < 0 and x - y < 0 —> No
pqp = p
Simplify
pqp - p = 0
p(pq - 1) = 0 —-> p = 0 or pq = 1
OR
pq = 1 IF p does not equal 0
Is r^2 / | r | < 1?
How would you simplify this?
r^2 and | r | are both positive
r^2 < | r | —-> only possible if -1 < r < 1 (and r does not equal 0)
OR
r^2 = | r | * | r | , so is | r | < 1 —->
is -1 < r < 1?
Calculate the Sum of Squares = a^2 + b^2
THIS IS NOT the same as a DIFFERENCE OF SQUARES
Recall cool trick when you ADD Square of a Sum and Square of a Difference:
(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)
Also a^2 + b^2 = (a + b)^2 - 2ab
How do you solve for the value of ab, given the following:
(a + b)^2 = # and (a - b)^2 = #
Recall cool trick when you SUBTRACT Square of a Difference FROM Square of a Sum:
(a + b)^2 - (a - b)^2 = 4ab
Summary of Quadratic Templates
3x quadratic templates
2x special manipulations
2x special applications
Square of a Sum: a^2 + 2ab + b^2 = (a + b)^2
Square of a Difference: a^2 - 2ab + b^2 = (a - b)^2
Difference of Squares: a^2 - b^2 = (a + b)(a - b)
(any EVEN POWERS that are the SAME in both terms, or 1)
e.g., a^4 - b^4; a^8 - b^8; a - b
**if given square of a sum or a difference equals a perfect square, you can calculate value of the inside!
e.g., (x - 1)^2 = 16
x - 1 = 4 (if x > 1)
1 - x = 4 (if x < 1)
Special Manipulations:
Addition of Square of a Sum and Square of a Difference: (a + b)^2 + (a - b)^2 = 2(a^2 + b^2)
Subtraction: (a + b)^2 - (a - b)^2 = 4ab
Disguised Quadratic Templates:
Multiplication (e.g., 198*202 = (200 - 2)(200 + 2) = 200^2 - 2^2)
a - b = (sqrt(a) + sqrt(b))(sqrt(a) - sqrt(b))
1/a^2 + a^2 = (1/a + a^2) - 2
Right Triangles and Area –> just need to know the hypotenuse and SUM of the two shorter sides or DIFFERENCE of the two shorter sides to calculate AREA!!
(a + b)^2 = c^2 + 4area
(a - b)^2 = c^2 - 4area
In the sequence A, A1 = 1, A2 = 100, and the value of An is strictly between the values of An-1 and An-2 for all n >= 3. Which of the following must be true?
A) A100 < A200 < A300 < A400
….
(Which order is correct?)
Since the terms in the answer choices are quite large, we have to find a PATTERN in the sequence.
A1 = 1
A2 = 100
A3 = between 1 and 100 (e.g., 50)
A4 = between 100 and 50 (e.g., 75)
A5 = between 75 and 50 (e.g., 60)
etc.
Strategy: DRAW A NUMBER LINE if the terms alternate from increasing to decreasing (+/-)
Pattern Observed: Even terms are larger than Odd terms (All the answer choices are EVEN)
> Largest Term = Smallest Even term (A100)
> Smallest Term = Largest Even Term (A400)
A400 < A300 < A200 < A100
A100
Which quadrilaterals have:
Diagonals that bisect one another
Parallelograms: Rectangles, Rhombuses, Squares
Which quadrilaterals have:
Diagonals that are perpendicular bisectors
Square and Rhombus –> due to equal sides
> every square is a rhombus (perpendicular bisectors, four equal sides)
Which quadrilaterals have:
Diagonals that bisect one another can are equal in length
Square, Rectangle —> due to 90 degree angles
Isosceles right triangle
MUST BE 45-45-90 degree triangle with sides in ratio x-x-xsqrt(2)
Right triangle with height drawn, creating two mini triangles –> what are the special properties of this triangle?
3 triangles (two smaller, one larger) are SIMILAR
Match up sides based on the angle
> Larger triangle: 90 degrees, angle a, angle b
> each mini triangle has 90 degrees and either angle a or angle b. So the third angle must be either angle b or angle a
Review the proportions
If h = height:
h/x = y/h
If 1/a^2 + a^2 represents the diameter of circle O and 1/a + a = 3, which of the following best approximates the circumference of circle O?
Reciprocals
Recognize this is related to square of a sum!
(a + b)^2 = a^2 + 2ab + b^2
(1/a + a) = 3 –> square both sides
(1/a + a)^2 = 9
1/a^2 + 2 + a^2 = 9
1/a^2 + a^2 = 7 = diameter
Circumference = 2pir = pid
= pi7
= 3.14*7
= 22
What is the area of the following right triangle:
> Hypotenuse is 8
> Sum of the legs is 12
Disguised quadratic template - you can solve for area if you know (1) hypotenuse and (2) Sum or Difference of the sides
(a + b)^2 = a^2 + 4(ab/2) + b^2
(a + b)^2 = c^2 + 4*area
(12)^2 = 8^2 + 4area
144 - 64 = 4area
area = 20
If xy > 0, is x/y > 2?
(1) x > 2y
xy > 0 means xy have the SAME SIGN (++, –)
(1) x > 2y —> alarm bells should be ringing (inequality division of an unknown sign)
If x and y are positive:
x/y > 2
e.g., 3 > 2(1)
3 > 2
If x and y are negative:
x/y < 2
e.g., -3 > 2(-4)
3/4 < 2
NOT SUFFICIENT
A rectangular wooden dowel measures 4 in by 1 in by 1 in. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?
Goal: # painted cube faces / total number of cube faces
DRAW OUT dimensions to help you figure out # of cubes
1) Determine the total number of cubes –> layer approach
>Length of 4 in can have 8 cubes
> Width of 1 in can have 2 cubes
—-> base can have lw = 16 cubes
> Height of 1 in can have 2 cubes (2 rows)
> Total # of Cubes = 82*2 = 32
2) Determine total number of faces = 6*32 = 192
3) Determine total number of painted cube faces –> Do it FACE BY FACE (surface area of the rectangular wooden dowel)
= 2(16 cubes) + 2(4 cubes) + 2(16 cubes)
= 2(16 + 4 + 16)
= 2(36)
= 72
4) Find the fraction
= 72/ 192 = 3/8
What’s a faster way to calculate the simple average of non consecutive integers?
e.g., 1954, 1942, 1980, 1999
Baseline approach –> focuses on differences from a chosen “baseline” (smallest number, e.g., 1942, median term, largest term, a round number near the range of values etc.)
> keep signs
1942 - 1942 = 0
1954 - 1942 = +12
1980 - 1942 = +38
1999 - 1942 = +57
Then, COMPUTE THE AVERAGE of the differences
= (0 + 12 + 38 + 57) / 4
= 107/4
= 26.7
Finally, ADD the average difference to the Baseline
= Average = Baseline + Avg Difference
= 1942 + 26.7
= 1968.7
Sometimes you want to find the TERM of a list, given the mean. Set mean as the baseline. The number above and below the mean should yield an average distance = 0.
Mean/Median Q: Maximizing or Minimizing one term strategy
e.g., in a certain lottery drawing, five balls are selected from a tumbler in which each ball is printed with a different two-digit positive integer. If the average (arithmetic mean) of the five numbers drawn is 56 and the median is 60, what is the greatest value that the lowest number selected could be?
43, 48, 51, 53, 56
Focus on the term you want to maximize or minimize
Maximize Term –> minimize other terms –> draw out a number line detailing what each value can equal
Minimize Term –> maximize other terms –> draw out a number line detailing what each value can equal
_ _ 60 _ _ (median is 60, since it is an odd number of balls, 60 is a part of the list).
a < b < 60 < c < d
Maximize lowest number (a) by MINIMIZING others given constraints
c minimum value = 61
d minimum value = 62
b minimum value depends on a –> one more than a –> b = 1 + a
Also mean < median, which indicates that the numbers above the median are CLOSER to 60 than the numbers below the median.
> Minimum distance from median of the values greater than 60 = 61, 62
56 = (a + 1 + a + 60 + 61 + 62)/5
2a = 96
a = 48
How do you tell if an integer is a PERFECT square?
- In factored form –> ALL even exponents (divisible by 2) sqrt = 1/2
- Odd number of factors (remember factors are always distinct)
> One of the factor pairs is a repeat
e.g. 4 –> factors: 1, 4, 2
e.g., 9 –> factors: 1, 9, 3
Palindromes - how would you solve?
Slot method - multiply the # of possible numbers for each position
e.g., Number of odd numbered four digit palindromes
= (5)(10)(1)*(1)
= 50
> last two spots is 1 because once you have chosen the first two spots, the last two must be fixed.
a^2 is equivalent to what?
| a | is equivalent to what?
a^2 = | a |^2
| a | = sqrt(a^2)
All are POSITIVE numbers
> can freely divide both sides of an inequality without changing the direction of the sign
> can freely square both sides of an inequality without changing the direction of the sign
Use these interchangeably!
Square inscribed in another square properties
1) Largest area of the inscribed square?
2) The four triangles (shaded)
1) Largest area is 1/2 of the larger square (proven by using 45-45-90 degree triangle ratios)
2) Each of the four triangles are CONGRUENT triangles
Properties of remainders
X/a = Z + R/a —-> anything that is a fraction after division is a remainder
Z is the integer, but could also be in OTHER forms as long as they are integers (n^2 + n)
What if X/a has a remainder equal to X?
How many equations do you need to solve for values of X?
What if you know X/a –> R1 and X/b –> R2?
What about negative remainders?
Special remainder formulas
“What is the remainder of X/a” data sufficiency questions
Even/Odd properties with remainders
Decimals and unknown divisors
1) Patterns in the numbers (X) that share the SAME REMAINDER (R) and DIVISOR (a)
> X goes up by a
e.g., x/5 = z + 1/3 —> x = 5z + 1
x = 1, 11, 16, 21 (go up by 5 or a)
2) Possible remainders for a divisor, a
= [0, 1, … up to a - 1]
> a (divisor) must always be LARGER than the remainder
> R/a is always LESS THAN 1
> start off every remainder question by writing out a > R
3) X = aZ + R
(X - R) = aZ —> a and Z are factors of X - R —> list them out!
> combined with a > R, you should be able to figure out properties of a
4) If X/a has a remainder of X –> means that X < a, z = 0, X/a is the remainder
e.g., 40/80 = 0 + 40/80
5) Just need one equation to come up with possible values for X.
BUT if you know the remainders when X is divided by TWO DIFFERENT DIVISORS, you can come up with a COMBINED EQUATION FOR X.
X = Least Common Multiple of a and b * Z + Smallest Value of X
> LCM of DIVISORS
6) If you get X/a = int - R/a —> you must convert the remainder into a positive one
e.g., (48^2 - 1)/8 = int - 1/8
= 8*int - 1 —> one less than a multiple of 8
Therefore, R = 7
7) special remainder formulas:
(ab)/c –> Remainder = (Ra*Rb)/c
(a + b)/c –> Remainder = (Ra + Rb)/c
(a - b)/c –> Remainder = (Ra - Rb)/c
> applies to real numbers too
8) Finding the remainder of an unknown integer with a known divisor (data sufficiency)
e.g., X/8 has a remainder = ?
> Set up the equation of the integer in the form X = a*Z + R
> Divide both sides by the divisor
> Recall the remainder of (a + b)/c = (Ra + Rb)/c
> If a is a MULTIPLE of the divisor => Certain, fixed remainder
> Otherwise if a is NOT a multiple of the divisor => UNCERTAIN remainder
9) Harder remainder problems are combined with even/odd property questions
e.g., Even = even + even —> remainder must be even
10) Divisor * decimal = Integer Remainder
If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?
(1) The remainder when a is divided by 40 is 0
(2) The remainder when 40 is divided by a is 40.
SET UP:
a > 0 int
81/a = z INTEGER + 1/a
81 = az + 1
80 = az —> factors of 80, list possible values of a
1 80
2 40
4 20
5 16
… etc.
(1) a/40 = int + 0
a is a MULTIPLE OF 40 –> a = 40, 80 (NS)
(2) 40/a = int + 40/a
40/a = 0 + 40/a
Meaning a > 40—-> a = 80 (S)
If X divided by 3 has a remainder of 2 and X divided by 5 has a remainder of 2, what is the remainder of X/15?
Also 2
CONCEPT:
A/m –> R
A/n –> R
Then A/least common multiple –> same R
If n is a positive integer, what is the number that separates the factors of n into half?
sqrt(n)
e.g., If n is a perfect square (odd # of factors), then half of the factors are above sqrt(n), and half of the factors are below sqrt(n)
e.g., if n is not a perfect square (even # of factors), then half of the factors are above sqrt(n), and half of the factors are below sqrt(n)
if n = 20, then half of the factors are above sqrt(20) = 2sqrt(5), and half of the factors are below sqrt(20).
n = 2^2 * 5 ==> 6 factors (1, 2, 4, 5, 10, 20)
If a divided by c has a remainder of R1, and b divided by c has a remainder of R2, what is the remainder of:
(a + b)/c?
(R1 + R2)/c
LONG WAY: a/c + b/c => remainder equals (R1 + R2)/c
e.g., 10/3 = 3 + 1/3 –> R1 = 1
13/3 = 4 + 1/3 –> R2 = 1
(10 + 13)/3 = 23/3 —> 7 R = 2
Formula - Remainder = (1 + 1)/3 = 2/3 –> R = 2
If a divided by c has a remainder of R1, and b divided by c has a remainder of R2, what is the remainder of:
(a - b)/c?
(R1 - R2)/c
e.g., 10/7 = 1 + 3/7 –> R1 = 3
8/7 = 1 + 1/7 –> R2 = 1
(10 - 8)/7 = (2)/7 = 0 —> remainder 2
Formula Remainder = (3 - 1)/7 = 2/7 –> remainder 2
(Must be positive differences)
If a divided by c has a remainder of R1, and b divided by c has a remainder of R2, what is the remainder of:
(ab)/c?
(R1 * R2)/c
e.g., (47*49)/8
e.g., (7^50)/8 —> manipulate exponents so that you can get an integer in the numerator (part of it has remainder = 0)
LONG WAY:
(ab)/c = (a/c)b
—> remainder equal to (R1/C)b = R1(b/C)
= R1(R2/c)
= (R1*R2)/c
You have n books on a bookshelf, where 20 < n < 40.
If you place the books on five different columns, there will be three books leftover. If you place the books on six different columns, there will be two books leftover.
How many books are leftover when you place them on four different columns?
REMAINDER PROBLEM
1) Come up with GENERAL EQUATION for n
We know that:
n = 5z + 3
n = 6y + 2
n = LCM of divisors * m + smallest value of n
> LCM –> list out multiples of 5 and 6, find the least common multiple.
> From eq’n 1, n = 3, 8, 13, 15
> From eq’n 2, n = 2, 8, 14
> Therefore the smallest value of n = 8
n = 30m + 8
2) List out some values of n that meet the constraint
n = 8, 38, 68 —> n = 38
3) Find the remainder
38/4 = 9 + 2/4 —> R = 2
** For remainder questions, DO NOT SIMPLIFY the division (e.g., 38/4 keep it as is, do not simplify further)
DS statement:
y < 1/y
One variable inequality - sewing approach
Move terms to one side so that one side equals 0
y - 1/y < 0 —> combine and factor
(y + 1)(y - 1)/y < 0
In other words, (y)(y + 1)(y - 1) < 0
Roots = 0, -1, 1
y < - 1
0 < y < 1
x^2 > 1
What is the range of x?
x > sqrt(1)
or
x < - sqrt(1)
If K is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following is less than k?
CONCEPT - consecutive integers and EXTREME values
Compute the range of what K can be.
n terms = 60 - 41 + 1 = 20
MAX value of K –> all 20 terms are 1/41
= 1/41 * 20 = 20/41 (less than 50%)
MIN value of K –> all 20 terms are 1/60
= 1/60 * 20 = 1/3
Therefore, 1/3 < K < 20/41
Triplets Adam, Bruce and Charlie enter a triathlon. There are nine competitors in the triathlon. If every competitor has an equal chance of winning, and three medals will be awarded, what is the probability that at least two of the triplets will win a medal?
___ ____ ____
9 competitors = 3 are ABC + 6 others
P(at least two win) = P(2 win) + P(3 win)
CONCEPT –> When you see P(ABC) –> think DEPENDENT EVENTS
= P(A) * P(B|A) * P(C|AB)
1) P(2 win) = (P(ABx) + P(ACx) + P(BCx))6 ways to arrange each one
= (1/9)(1/8)*(6/7) * 3 * 6
= 3/14
2) P(3 win) = P(ABC) * 6 ways to arrange
= (1/91/81/7)*6
= 1/84
3) SUM = 19/84
OR
(1) P(2 win) = T T O
= (3/9 * 2/8 * 6/7) * (3!/2! ways to arrange)
= 3/14
(2) P(3 win) = T T T
= (3/9 * 2/8 * 1/7)
= 1/84
Which of the following values for b makes x closest to zero?
x = 2^b - (8^8 + 8^6)
First get all terms into base 2
x = 2^b - (2^24 + 2^18)
We want x to be closest to 0:
0 = 2^b - (2^24 + 2^18)
2^24 + 2^18 = 2^b
Factor and evaluate so that 2^value = 2^b:
2^18(2^6 + 1) = 2^b
2^18(65) = 2^b
Approx 2^18(2^6) = 2^b
so 2^24 = 2^b
b = 24
Properties of isosceles triangles
1) An isosceles right triangle is always a 45-45-90 degree triangle.
2) The altitude of an isosceles triangle is a PERPENDICULAR BISECTOR and splits the triangle into two CONGRUENT triangles (and bisects the ANGLE)
> not necessarily special angle triangles
How do you correctly match the side dimensions to a polygon?
e.g., ABC is an isosceles triangle.
The length of AB = 9, BC = 4.
Properties of polygons:
> Triangle: The sum of two sides must be GREATER than the third side
For all positive integers n, the sequence An, is defined by the following relationship:
An = (n - 1)/n!
What is the sum of all the terms in the sequence from A1 through A10, inclusive?
Sequences: Try to find the pattern (NOT just in the individual terms, but also the SUM)
Ans: (10! - 1)/10!
Valid triangles?
- Angle sizes must match side lengths
> hypotenuse has the LARGEST length - Sum of any two sides is larger than the third side.
- Difference of any two sides is smaller than the third side.
What is the remainder of 2^50/7?
Pattern in the remainder:
2^1/7 –> R = 2
2^2/7 –> R = 4
2^3/7 –> R = 1
2^4/7 –> R = 2
Remainder cycle [2,4,1] repeats every three terms.
# of cycles = 50/3 = 16 R2
so Remainder of 2^50/7 = 4
Alternatively, Concept: Remainder of (ab)/c = (r1*r2)/c
SIMPLIFY FIRST, get multiples of 7 in the numerator as a SUM of something, preferably with 1(which have a remainder = 0):
= (2^3)^16 * 2^2 / 7
= (8)^16 * 2^2 / 7
= (7 + 1)^16 * 2^2 / 7
The remainder of (7 + 1)^16/7 = 1
> exponent is just a long multiplication!
> (7 + 1) / 7 has a remainder of 1
> So (7 + 1)^16 / 7 has a remainder of (111*1….1)/7
The remainder of 2^2/7 = 4
So the overall remainder = (1*4)/7 = 4/7 –> 4
Pigeonhole principle
e.g., What is the LEAST number of cards that must be drawn from a standard deck of cards to guarantee at least four cards of the same suit?
if n items are put into m containers, with n>m, then at least one container must contain more than one item.
CONCEPT to solve: think of WORST CASE SCENARIOS
E.g.,
Worst case scenario is that you draw 4 cards with different suits in a row, three times (a total of 12 cards drawn). Therefore, the 13th card must complete the set of one of the suits (4 of a suit).
Is xy + xy < xy?
(1) x^2/y < 0
(2) x^3*y^3 < (xy)^2`
When you see an inequality with the same terms on both sides –> rearrange first!
Is 2xy < xy?
Is 2xy - xy <0?
Is xy < 0?
(1) y < 0, however, we don’t know the sign of x
(2) Can divide both sides by (xy)^2 (since it is a positive value).
xy < 1 –> NS either
(3) together –> still don’t know the sign or value of x
E
Grass Problem
Characteristics:
> Given some unknown rate of inflow and outflow
> Given # of outflow “terminals” (e.g., cows, stations, tubes), each having an equal outflow rate.
> Given at least two “equations”
> Some “fixed” starting capacity
> Asked for some variable
e.g., 10 cows can finish the entire grass in 20 minutes. 15 cows can finish the entire grass in 10 minutes.
How many minutes does it take 5 cows to finish the entire grass?
e.g., There is a large tank being filled up with water. At a certain volume, water leaves the tank. When there are 2 tubes, it takes 8 minutes to empty the tank. When there are 3 tubes, it takes 5 minutes to empty.
How long should the tank be filled up before opening the tubes?
Strategy: Set up a system of equations such that
In = Out
> inflow rate, r
> outflow rate, x
> SAME time
> # of terminals = n
Solve for an expression via elimination!
Cow Grass Problem:
Starting Grass + rt = nx*t
–> asked for solve for t when there are 5 cows.
Tank Problem:
Starting Tank + rt = nx*t
–> asked to solve for how LONG it takes to reach STARTING TANK amount
e.g., L0 = 40*r —> t = 40
What is the arithmetic mean of these numbers: 12, 13, 14, 510, 520, 530, 1115, 1120, 1125?
Long way - add up the numbers and divide by 9.
Recognize that the numbers are three sets of EVENLY SPACED integers (average = median)
Find the formula:
average = [(a1 + b1 + c1)/3 + (a2 + b2 + c2)/3 + (a3 + b3 + c3)/3]*3 then divide by 9
= (average1 + average2 + average 3)/3
= 551
Do the sets have equal standard deviation?
M: {1,2,3,4,5,6,7}
N: {3,4,5}
What about Z: {41, 42, 43, 44, 45, 46, 47}
No - N has a smaller standard deviation
> M has values that are farther away/more spread out from the mean than set N
Z has the same standard deviation as M.
Concept: Two sets have the same standard deviation IF:
> They have the SAME # of terms AND
> The terms have the SAME GAP from the mean in each set (e.g., -3, -2, -1, 0, +1, +2, +3)
Sequences formula for calculating the sum of geometric sequences
e.g., the term grows by 2
A1 * [(1 - r^n)/(1 - r)]
where r is the shared factor
where n starts at 1 (A1 = position 1)
When will | x + y | = | x | + | y |?
What is the general inequality?
Equal when x and y have SAME SIGNS
e.g., | - 5 - 3 | = |-5| + |-3|
Not equal when x and y have DIFFERENT SIGNS
e.g., |5 - 3| =/ |5| + |-3|
x | - | y | <= | x + y | <= | x | + | y |
What is the formula for the surface area and volume of a CONE
SA = pir^2 + pirlength
> derived from calculating the area of the sector
= (2pir/2piL) * (piL^2)
V = 1/3 * volume of cylinder = 1/3 * (pi*r^2 * h)
What is the formula for the surface area and volume of a SPHERE
SA: 4pir^2
V: 4/3 pir^3
Set S is the set of all prime integers between 0 and 20. If three numbers are chosen randomly from set S, and no number is chosen more than once, what is the probability that the sum of all the three numbers is Odd?
of prime integers in set: 2,3,5,7,11,13,17,19 = 8
Probability Q:
How to get odd sum? = odd + even
> Recall Odd + Odd = Even
> therefore P(three odds)
= (7/8)(6/7)(5/6)
= 5/8 —-> there is NO NEED to multiply by the number of arranges (OOO has no arrangements).
**Strategy: when multiplying probabilities dealing with a characteristic (e.g., odds)–> usually no need to multiply by cases
e.g., P(draw no pairs in a row)
= (1/1 * different number/total # * different number/total #)
What is the greatest prime factor of (2^10)(5^4) - (2^13)(5^2) + 2^14?
Prime Factor problem: NEED TO BE IN A PRODUCT, NOT sum or difference!!
First, start by factoring out the greatest common factor, 2^10, from each term
Then, COMBINE the remaining terms in the bracket and re-do the prime factorization
You will realize that 3 and 7 appear in the remaining output –> 7 is the greatest prime factor
Come up with different sets that have a mean equal to 50 (6 terms)
CONCEPT: Use the average gap method
mean = mean + avg gap of 0
50 50 50 50 50 50
0 0 50 50 100 100
47 48 49 51 52 53
etc.
** question did not specify that the terms have to be different!
Bill has a set of 6 black cards and a set of 6 red cards. Each card has a number from 1 through 6, such that each of the numbers 1 through 6 appears on 1 black card and 1 red card. Bill likes to play a game in which he shuffles all 12 cards, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?
Probability Q:
P(at least one pair) = 1 - P(no pairs)
= 1 - (1/110/118/10*6/9) —-> NO NEED to multiply by # of cases because we did GENERAL approach
= 1 - 16/33
= 17/33
Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address?
Probability Q:
P(only 1 correct) –> C W W W (arrange 4!/3! ways)
= (1/4 * 2/3 * 1/2 * 1) * 4 –> 1/4 chance to get it correct * 2/3 chance to get second one wrong * 1/2 chance to get the third one wrong * 100% chance to get the last one wrong * 4 ways to arrange C W W W
= 1/3
Exponential growth
General formula
Starting Value * (growth factor)^# of periods
> amount grows by a factor of X each period OR decays by a factor X each period
Periods increase by Y
Periods relate to TIME –> # of periods = t/y
e.g., Periods go up by 2, while population increases by 2x
Formula: A0 * (2)^(t/y)
You have an n-sided die, each numbered from 1 to n.
What is the probability that you role
A) Two different numbers
B) Two equal numbers
A) Two different numbers
= (1/1 on first role) * (n-1)/(n) on second role
OR (1/n on first role)*(1/n on second role) * # of unique orders (n * n-1)
B) Two equal numbers
= (1/1 on first role) * (1/n) on second role
OR (1/n on first role)*(1/n on second role) * # of unique orders (n)
At a birthday party, x children are seated at two tables. At the table with the birthday cake, there will be exactly y children seated, including the birthday girl.
How many different groups of children may be seated at the birthday cake table?
Combination Q
> order doesn’t matter (groups of children)
> remove 1 from x because Sally is guarantee to sit at the table
n! / (duplicates! * duplicates!)
= n! / [# chosen! * (n - # chosen)!]
= (x - 1)! / [(y - 1)!(x - 1 - y + 1)!]
= (x - 1)! / [(y-1)!(x-y)!]
Set A consists of all the integers between 10 and 21, inclusive. Set B consists of all the integers between 10 and 50, inclusive. If X is a number chosen randomly from set A, y is a number chosen randomly from set B, and y has no factor z such that 1 < z < y, what is the probability that the product xy is divisible by 3?
of multiples of 3 between 10 and 21 —> [12, 21] = (21 - 12)/3 + 1 = 4
Key info:
> y is a prime number between 10 and 50: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
> x must be a multiple of 3
# of integers between 10 and 21 –> (21 - 10 + 1) = 12
# of primes between 10 and 50 = 11
# of total products = 12 * 11 = 132
# of multiples of 3 = 4 * 11 = 44
P(xy is divisible by 3) = 44/132 = 11/33 = 1/3
Combinations versus permutations (with and without duplicates)
A) # of ways to arrange unique items
B) # of ways to arrange items with duplicates
C) # of ways to create different groups
Combination questions –> “Choose” r from a total of n
> order doesn’t matter
Permutations –> arranging all the elements of n, taking into account duplicates
ASK YOURSELF:
1) Does order matter?
Y - permutation
N - combination
2) Choose your method (slot or anagram)
A - permutation - slot method or formula (m!/(m-n)!)
B - permutation w repetition - anagram method
C - combination - anagram method (Y/N)
