Quant Essentials Flashcards

1
Q

if a,b,c,d are positive, is a/b = c/d?

1) ad = bc
2) b = d

A

Equivalent fractions means that ad = bc
e.g. 1/8 = 7/56 -> 1x56 = 8x7
1 is sufficient, 2 is not.

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2
Q

Is 7/9 larger than 6/8?

A

Yes. Use bowtie method for positive fractions:
for a/b and c/d
if ad>bc
then a/b > c/d

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3
Q

Is 18/23 > 27/32?

A

No Find a common numerator and compare the denominator.
Larger denom -> smaller fraction

18/23 x 3/3 = 54/69
27/32 x 2/2 = 54/64

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4
Q

What are the DS choices?

A
A = First alone
B = Second alone
C = Both together
D = Both alone
E = Neither together

Split into AD + BCE blocks

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5
Q

1/6 as a decimal?

A

0.167

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6
Q

1/7 as a decimal?

A

0.143

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7
Q

1/9 as a decimal?

A

0.111

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8
Q

0.833 as a fraction?

A

5/6

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9
Q

When is x^2 < x < sqrt(x)?

A

When 0 < x < 1.

When x is between 0 and 1, the sqrt(x) is > x, and x^2 is < x

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10
Q

Perfect squares never end in?

A

2, 3, 7, 8

Because:
1, 4, 9, 25, 36, 49, 64, 81, 100

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11
Q

10011 - 7328 = ?

A

10011 = 9999 + 12
9999
- 7328
=2671 + 12 = 2683

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12
Q

What is 1/6 as a decimal?

A

0.167

5/6 = 0.833

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13
Q

What is 0.833 as a fraction?

A

5/6

0.167 = 1/6

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14
Q

What is 0.143 as a fraction?

A

1/7

2/7 = 0.286
6/7 = 0.875
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15
Q

What is 6/7 as a decimal?

A

0.857

5/7 = 0.714

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16
Q

What is 0.571 as a fraction?

A

4/7

5/7 = 0.714

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17
Q

3/7 as a fraction?

A

0.429

5/7 = 0.571

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18
Q

What are the positive 1-digit integers?

A

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

REMEMBER 0!

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19
Q

0! = ?

20
Q

If x is not equal to y,

(x-y)/(y-x) =?

A
(x-y)/(y-x)       = -1
(x-y)/(-1)(x-y)   = -1
21
Q

What are the 25 prime numbers less than 100?

A
2, 3, 5, 7
11, 13, 17, 19
23, 29
31, 37
41, 43, 47
53, 59
61, 67
71, 73, 79
83, 89
97
22
Q

How many different positive integers are factors of 12,000?

A
  1. Find the prime factorization
    12,000 = 2^5 x 3^1 x 5^3
  2. Add 1 to the value of each exponent, then multiply.
    Total number of factors = (5+1)(1+1)(3+1) = 48
23
Q

If we know the LCM and GCF of two positive integers (x,y), we know their product.

A

x * y = LCM (x,y) x GCF (x,y)
e.g. If LCM = 24, GCF = 2,
then, xy = 24 * 2 = 48

24
Q

Divisibility rule for 3

A

If sum of all digits is divisible by 3

25
Divisibility rule for 4
If last 2 digits are divisible by 4
26
Divisibility rule for 6
Even number where digits sum to multiple of 3
27
Divisibility rule for 8
Last 3 digits divisible by 8
28
Divisibility rule for 11
If sum of odd-place digits (1, 100, 10,000 etc) - sum of even-place digits (10, 1,000 etc) are divisible by 11. e.g. 2,915 = (9+5)-(2+1) = 11, so 2,915 divisible by 11
29
Divisibility rule for 12
If divisible by 3 and 4
30
Product of n consecutive integers are divisible by
n! | e.g. product of 3 consecutive integers is divisible by 3!
31
Algebraic consecutive integers
1. n^2 - n = n(n-1) 2. n^2+n = n(n+1) 3. n^3-n = n(n-1)(n+1) 4. n^5-5n^5+4n = n(n+1)(n-1)(n+2)(n-2)
32
Consecutive even integers
If n is odd, both (n-1) and (n+1) are even. | (n-1)(n+1) = n^2 -1 = product of two consecutive even integers
33
Product of n consecutive even integers is divisible by
2^n x n! | e.g. 3 consecutive even integers visible by 2^3 x 3! = 8x6 = 48
34
Division, general formula
x/y = Q + r/y Q = quotient, r = remainder *manipulate the formula to solve for each of x, Q, r. eg. Q = (x-r)/y
35
Any factorial >= 5! will
Always have 0 as its last digit, because it contains at least one 5 x 2 pair
36
Leading 0's
are to the right of the decimal point (e.g. 0.02 has 1 leading zero, 0.78 has no leading zeroes)
37
If X is an integer with k digits and X is not a perfect power of 10, then 1/X will have how many leading zeros?
k - 1 | e.g. 1/5000, X = 5000 (4 digits, not perfect power of 10), so decimal form 1/5000 has 3 leading zeros = 0.0002
38
If X is an integer with k digits and X is a perfect power of 10, then 1/X will have how many leading zeros?
k-2 leading 0s e.g. 1/10 = 0.1 (0 leading 0s) 1/1000 = 0.001 (2 leading 0s)
39
Product of any set of n consecutive integers is divisible by...
all integers between 1 and n inclusive. or, n! e.g 5x6 is divisible by 2!
40
To determine the largest number of a prime number x that divides into y!
1. divide y by x^1,...x^k, stopping when y/x^k produces a quotient of 0 2. Sum quotients = number of prime number x in y!
41
What is the largest integer n such that 30!/4^n is an integer?
1. 30!/4^n = 30!/2^2n 2. use factorial divisibility shortcut, e.g. there are 2^15+7+3+1 = 26, 2^26 in 30! 3. 2n <= 26, n=13 Largest value for n is 13
42
Reduced fractions with prime factorization with ONLY 2s, 5s or both will have a decimal that terminates.
Else, the fraction does not terminate. e.g. 7/80 = 2^4 x 5^1, so it terminates 5/60 = 1/12 = 2^2 x 4^1, does not terminate
43
Units digit for powers for 2 and 8
2^n has pattern 2,4,6,8 | 8^n has pattern 8,4,2,6
44
Units digit for powers for 3 and 7
3^n has pattern 3,9,7,1 | 7^n has pattern 7,9,3,1
45
Which prime factors do two consecutive integers share?
None. Therefore, GCF(n, n+1) = 1