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GMAT > Algebra > Flashcards

Algebra Flashcards

1
Q

(x+y+z)^2

A

x^2 + y^2 + z^2 + 2(xy + xz + yz)

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2
Q

x^3 + y^3
x^3 - y^3

A

(x+y)(x^2 + y^2 - xy)
(x-y)(x^2 +y^2 +xy)

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3
Q

T or F. a^n - b^n is always divisible by a- b

A

T - as proved in x^2 - y^2 and x^3 - y^3

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4
Q

a^n - b^n is divisible by a+b if
a^n + b^n is divisible by a+b if

A

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

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5
Q

How do you express negative exponents ?

A

reciprocal of the inside base and turn negative to positive exponent

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6
Q

T or F. if 0^x = 0 = 0^y , x is equal to y

A

False. 0^4 = 0 = 0^3 but 4 ≠ 3

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7
Q

T or F. In GMAT, you should always subtract or divide two inequalities

A

F. You should never

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8
Q

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)

A

G^n is less than original G
G^n is larger than original G

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9
Q

If x < y, then how does 1/x compare to 1/y when x and y are both positive or both negative?

A

1/x > 1/y (flip the sign)

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10
Q

T or F. If you don’t know the sign of x or y, you can still take the reciprocals.

A

False.

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11
Q

if |x| > a then:
if |x| < a then:

A

x > a or x < -a
-a < x < a

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12
Q

If |A| = |B| then

A 
A = B or 
A = - B
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13
Q

What is the equation expression for directional proportionality?

A

y(1)/ x(1) = y(2)/ x(2)

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14
Q

What is the equation expression for inverse proportionality?

A

y(1) . x(1) = y(2) . x(2)

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15
Q

√2
√3
√5

A

about 1.4
about 1.7
around 2.25

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16
Q

√121
√144
√169

A

11
12
13

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17
Q

13^2

A

169

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18
Q

√196
√225
√625

A

14
15
25

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19
Q

√256
√361

A

16
19

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20
Q

How do you translate “The retailer has less than twice as many radios as clocks in inventory” in math expression?

A

r < 2c

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21
Q

The trick to translate fractional exponent of a number to a root

A

denominator -> OUT
numerator -> IN

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22
Q

The trick to translate a root to fraction exponent

A

in NUMERATOR
out DENOMINATOR

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23
Q

Can absolute value result in negative number?
Can square root of a negative number result in value?

A

NO. its results are always greater than 0
NO. square root always involves with positive numbers

24
Q

x^0 = ?
0^x = ?
what is the value of 0^0 ?

A

1
0
undefined

25
Q

proper fraction

A

a fraction between 0 and 1

26
Q

What does x^2 - x < 0 imply?

A

x^2 < x, so 0 < x < 1

27
Q

√324
√289

A

18
17

28
Q

What is the vertex (x,y) of equation: a(x-h)^2 + k = 0

A

(h,k)

29
Q

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)

A

True.

30
Q

T or F. In x-y plane, the quadrants is upper shaded area if the inequality is equal to or greater (>=)

A

True.

31
Q

T or F. In x-y plane, the quadrants is below shaded area if the inequality is equal to or less (=

A

True.

32
Q

When I see, I’ll think/do

Which of the following must be ____?
(A) variables
(B) variables

A

Pick values that fit the constraints and plug in/eliminate

33
Q

T or F: We can add two inequalities but can’t subtract them

A

T

34
Q

What is xy?

-5< X< 4
-12<Y<2

A

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

35
Q

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?

A

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

36
Q

Is xy< 6. and what is the minimum of xy?

If 1/2 < x < 2/3
-8< y < 8

A

Max: xy < 8(2/3) ~ 5 + 1/3 <6, so yes

Min: (-8) x (2/3) < xy -> -16/3 < xy

37
Q

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

A

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

38
Q

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.

A

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

39
Q

What does this problem test?

If n is an integer, is (0.1)^n greater than (10)^n ?
(1) n > –10
(2) n < 10

A

1) Negative exponent = reciprocal of base
2) Exhaustive test cases

40
Q

When do you change the sign of the inequalities?

A

1) when divide 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)

41
Q

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?

A

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

42
Q

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

A

Recognize whether you can simplify further by cross-multiply: with known sign

43
Q

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

A

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

44
Q

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

A

You can set up: (x+r)(x+s)

45
Q

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

A

No since there exist a case of fraction (i.e: 4 > 3 & 1>0.5 but 4/1 < 3/0.5 = 6)

46
Q

What does it mean “define” here?

If a and be are constants, is the expression x+b / √x+a defined for x = -2?

A

defined = there are possible values for the expression
undefined = no possible values or when denominator is 0

47
Q

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.

A

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

48
Q

Spot the potential trap here

Is x = y?
1) 2x/3 – y/3 = 1/3

A

2x = y AND x= y = 0 then YES
x= 1/2 & y = 1, since X # Y then NO

49
Q

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|

A

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)

50
Q

When you see a number line marked into smaller ratio, to find distance among them you will….

A

find the LCM to have a nice round number - 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

51
Q

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

A

1) Notice that a = 5 only when b cancels out, in which both statements is required
2) Test cases

52
Q

What else is this statement equivalent to if it is true?

T or F: difference of x and y is equal to x - y

A

y subtracted from x
x decreased by y

53
Q

T or F: There is only a single estimate value for this equation

x = (2.32^2 - 2.536)/ (2.68^2 + 2.79)

A

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

54
Q

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?

A

Key Words: “MUST BE TRUE” -> pick number for test cases

55
Q

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?

A

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!

56
Q
A

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