Algebra Flashcards
(x+y+z)^2
x^2 + y^2 + z^2 + 2(xy + xz + yz)
x^3 + y^3
x^3 - y^3
(x+y)(x^2 + y^2 - xy)
(x-y)(x^2 +y^2 +xy)
T or F. a^n - b^n is always divisible by a- b
T - as proved in x^2 - y^2 and x^3 - y^3
a^n - b^n is divisible by a+b if
a^n + b^n is divisible by a+b if
n is even - x^2 - y^2 always has (x-y) as factor
n is odd - x^3 + y^3 has (x+y) as factor
How do you express negative exponents ?
reciprocal of the inside base and turn negative to positive exponent
T or F. if 0^x = 0 = 0^y , x is equal to y
False. 0^4 = 0 = 0^3 but 4 ≠ 3
T or F. In GMAT, you should always subtract or divide two inequalities
F. You should never
If 0< G < 1 then how does G^n compare to G (if n is odd/even positive integer)
If 0< G < 1 then how does G^n compare to G (if n is odd/even negative integer)
G^n is less than original G
G^n is larger than original G
If x < y, then how does 1/x compare to 1/y when x and y are both positive or both negative?
1/x > 1/y (flip the sign)
T or F. If you don’t know the sign of x or y, you can still take the reciprocals.
False.
if |x| > a then:
if |x| < a then:
x > a or x < -a
-a < x < a
If |A| = |B| then
A = B or A = - B
What is the equation expression for directional proportionality?
y(1)/ x(1) = y(2)/ x(2)
What is the equation expression for inverse proportionality?
y(1) . x(1) = y(2) . x(2)
√2
√3
√5
about 1.4
about 1.7
around 2.25
√121
√144
√169
11
12
13
13^2
169
√196
√225
√625
14
15
25
√256
√361
16
19
How do you translate “The retailer has less than twice as many radios as clocks in inventory” in math expression?
r < 2c
The trick to translate fractional exponent of a number to a root
denominator -> OUT
numerator -> IN
The trick to translate a root to fraction exponent
in NUMERATOR
out DENOMINATOR
Can absolute value result in negative number?
Can square root of a negative number result in value?
NO. its results are always greater than 0
NO. square root always involves with positive numbers
x^0 = ?
0^x = ?
what is the value of 0^0 ?
1
0
undefined
proper fraction
a fraction between 0 and 1
What does x^2 - x < 0 imply?
x^2 < x, so 0 < x < 1
√324
√289
18
17
What is the vertex (x,y) of equation: a(x-h)^2 + k = 0
(h,k)
T or F. Any two points the same distance from the origin along perpendicular lines will have the opposite coordinates from one another - if one point is (a,b), the other will be (b,a)
True.
T or F. In x-y plane, the quadrants is upper shaded area if the inequality is equal to or greater (>=)
True.
T or F. In x-y plane, the quadrants is below shaded area if the inequality is equal to or less (=
True.
When I see, I’ll think/do
Which of the following must be ____?
(A) variables
(B) variables
Pick values that fit the constraints and plug in/eliminate
T or F: We can add two inequalities but can’t subtract them
T
What is xy?
-5< X< 4
-12<Y<2
It’s not: -60<xy<8 (you have to think about min/max value & notice both possible negative signs in x & y)
-48<xy<60
Hint: Practice common sense algebra
The population of a bacteria culture doubles every 2 minutes. Approximately how many minutes will it take for the population to grow from 1,000 to 500,000 bacteria?
What is the general formula for this exponential growth?
1st term: 2,000 -> 2nd: 4,000 -> 3rd: 8,000…-> 9th: 512,000 or
- 500,000 = 1,000 x 2^n –> 2^n = 500 or n= 9 -> 9 times x 2 min/time = 18 mins
y = x.k^t
- x = initial population
- k = constant multiplier (double, triple, quadruple…)
- t = number of multiplication
Is xy< 6. and what is the minimum of xy?
If 1/2 < x < 2/3
-8< y < 8
Max: xy < 8(2/3) ~ 5 + 1/3 <6, so yes
Min: (-8) x (2/3) < xy -> -16/3 < xy
What principle in equalities do we use to solve?
If x + y + z >0, is z >1 ?
(1) z > x + y +1
(2) x +y +1 < 0
Add two equalities, which have the common sign:
1) z- x- y -1 >0 => 2z > 1then z > 1/2 but can be less than 1 -> Not sufficient
2) -z - x - y < 0 => 1 - z < 0 -> z > 1-> sufficient
How do we know there is no constraint (one possible solution) in (2)?
A citrus fruit grower receives $15 for each crate of oranges shipped and $18 for each crate of grapefruit shipped. How many crates of oranges did the grower ship last week?
(2) Last week the grower received a total of $38,700 from the crates of oranges and grapefruit shipped.
We can find the extreme range of R & G to prove that there are two solutions in 15R + 18G = 38,700, i,e:
G= 0 then R = 2,580
R = 0 then G= 2150
In other words, because there is no restriction in the question stem that the grower must ship both, we can draw the case either say (2) is not sufficient
What does this problem test?
If n is an integer, is (0.1)^n greater than (10)^n ?
(1) n > –10
(2) n < 10
1) Negative exponent = reciprocal of base
2) Exhaustive test cases
When do you change the sign of the inequalities?
1) when divide/multiply by the negative number (i.e: -3x < -3y -> x > y)
2) reciprocal of a known sign inequalities (i.e: x>y>0 -> 1/x < 1/y)
Within 30 seconds before approaching problem, what you do?
3p - q = 9
|q| =< 14
For how many ordered pairs (p,q) that are solutions of the system above are p and q both integers?
1) Recognize the uniqueness/restriction in the number properties of the equation: q = 3p - 9 = 3 (p-3) so q must be a multiple of 3
2) Combine with the limited range in -14 < q < 14, we can how many q & hence p
Within 30 seconds before approaching problem, what you do?
If x and y are positive numbers, is x + 1/ y+1 > x / y
(1) x > 1
(2) x < y
Recognize whether you can simplify further by cross-multiply: with known sign
what concepts you need to succeed in solving this problem?
If a and b are positive integers, is a/b < 9/11?
(1) a /b < 0.818
(2) b /a > 1.223
1) repeating decimal digits pattern: 9/11 = 81/99 -> pattern: 0.818181
2) Rephrase the question if a/b < 9/11 then b/a > 11/9 = 121/99 -> pattern: 1 + 22/99= 1.222
When you see two roots…
If r and s are the roots of the equation x^2 + bx + c = 0, where b and c are constants, is rs < 0 ?
(1) b < 0
(2) c < 0
You can set up: (x-r)(x-s)= x^2 - x(r+s) +rs
Same sign division..
If a,b,c and d are positive number, with: c > a & d > b, can we conclude that c/d > a/b
No since there exist a case of fraction (i.e: 4 > 3 & 1>0.5 but 4/1 < 3/0.5 = 6)
What does it mean “define” here?
If a and be are constants, is the expression x+b / √x+a defined for x = -2?
defined = there are possible values for the expression
undefined = no possible values or when denominator is 0
Use smart number to quickly realize that MN is greater than MQ
Two points, N and Q (not shown), lie to the right of point M on line ℓ. What is the ratio of the length of QN to the length of MQ?
(1) Twice the length of MN is 3 times the length of MQ.
(2) Point Q is between points M and N.
If MQ=2 units, 2MN=3*2, MN=3 units, and hence Q must lie in between of M &N. Therefore, QN =1 QN:MQ=1:2 -> statement A sufficient
Spot the potential trap here
Is x = y?
1) 2x/3 – y/3 = 1/3
2x = y AND x= y = 0 then YES
x= 1/2 & y = 1, since X # Y then NO
Why is this statement sufficient?
If q, s, and t are all different numbers, is q < s < t ?
(1) t - q = |t - s| + |s - q|
1) Since sum of absolute value is greater than 0, t -q > 0 -> t > q
2) If we draw the number line, the sum of these absolute values implies that s in the middle. Therefore, t> s> q (either to the left or right of 0)
When you see a number line marked into smaller ratio, to find distance among them you will….
find the LCM to have a nice round common unit - avoid ratio subtraction
i.e: 1/5 & 1/7 marked between 0 & 1 -> LCM = 35 so we have distance among members: 5-7-10-14-15-21-20-28-25-30-35
Try this problem w/o simplifying the absolute value
If a and b are integers and a = |b + 2| + |3 – b|, does a = 5?
1) b < 3
2) b > –2
1) Notice that a = 5 only when b cancels out, in which both statements is required
2) Test cases
Translate this equation into english statement
x - y
y subtracted from x
x decreased by y
T or F: There is only a single estimate value for this equation
x = (2.32^2 - 2.536)/ (2.68^2 + 2.79)
F: We need to find the value x within the upper bound & lower bound of estimations, so:
(2^2 - 3)/ (3^2 +3) < X < (3^2-2)/ (2^2+2)
-> 1/12 < X < 7/6
What strategy to use?
Today Rose is twice as old as Sam and Sam is 3 years younger than Tina. If Rose, Sam, and Tina are all alive 4 years from today, which of the following must be true on that day?
Key Words: “MUST BE TRUE” -> pick number for test cases
What does this problem test?
One week a certain truck rental lot had a total of 20 trucks, all of which were on the lot Monday morning. If 50 percent of the trucks that were rented out during the week were returned to the lot on or before Saturday morning of that week, and if there were at least 12 trucks on the lot that Saturday morning, what is the greatest number of different trucks that could have been rented out during the week?
Logic w/ test cases + min/max:
- The # of trucks (not rented out) + The # of trucks return on Saturday will be at least 12
Since 1/2 were rented out, these number must be even:
- 18 rented x 1/2 = 9 (return) + 2 (left-over) = 11
- 16 rented x 1/2 = 8 (return) + 4 (left-over) = 12: BINGO!
Translate into the common sense inequality of max/min
At least 2/3 of the 40 members of a committee must vote in favor of a resolution for it to pass. What is the greatest number of members who could vote against the resolution and still have it pass?
If at least 2/3 of the member must vote in favor of a resolution, then this is the minimum threshold for the pass. Therefore:
- At max is the remaining 1/3 of the members can vote against, and the resolution still pass
Note: At max is also called “No more than”
Key lesson
A sales commission plan pays a sales representative $2,000 or 20% of monthly gross sales, whichever is lower. The company is now introducing a new plan that pays the sales representative $500 each month plus 10% of all monthly gross sales above $2,000. Which of the following is a monthly gross sales amount for which both sales commission plans pay out the same amount?
Critical thinking all scenarios since the other side of the equation can be $2,000 or 1/5 of the total monthly gross sales:
Look at the answer choices we see that 1/5 of the total sale is lower than $2,000; therefore, this is the preferred paid method
What is your strategic execution?
A casino uses chips in $5 and $7 denominations only. Which of the following amounts CANNOT be paid out using these chips?
Since we are asked about the sum of possible combinations, we MUST write the potential values of multiple 5 and 7
- This will help us see the adding more efficiently
