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0
Q

SQUARE ROOT OF 2

A

1.4

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1
Q

Slope of the line with equation 3x + 7y = 9

A

Remember the Slope Formula: y = mx + b (“b” is the “y” intercept) (“m” is the Slope).
Process: all goes into the Slope Formula.
7y = -3x + 9
y = -3/7x + 9/7

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2
Q

SQUARE ROOT OF 3

A

1.7

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3
Q

SQUARE ROOT OF 5

A

2.2

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4
Q

SQUARE ROOT OF 7

A

2.6

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5
Q

MEDIAN FORMULA

A

1rst + Last / 2

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6
Q

NUMBER OF TERMS

A

Last - First / Spacing + 1

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7
Q

AVERAGE SPEED

A

Total Distance / Total Time

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8
Q

1/4

A

0.25

25%

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9
Q

WHAT CAN YOU CANCEL IN A PROPORTION?

A

Just Vertical and Horizontal.

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10
Q

WHAT CAN’T YOU CANCEL IN A PROPORTION?

A

It is illegal the Diagonal Cancellation.

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11
Q

WHEN TRANSLATING FROM WORDS TO MATH, THE WORD “OF” MEANS:

A

Multiply

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12
Q

CHANGING PERCENTS TO DECIMALS

A

Simply dividing by 100, so we move the decimal point two places to the left.
42.5% = 0.425

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13
Q

How changing from Decimals to Percents?

A

Multiplying by 100. We move the decimal point two places to the right.

  1. 68 = 68%
  2. 3 = 230%
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14
Q

How changing from Percents to Fractions?

A

We just put the percent over 100. After that, simplify.
20% = 20/100 = 1/5
0.02% = 0.02/100 = 2/10000 = 1/5000

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15
Q

Sheila invests $4000 in an account that yields 6% compounding annually for 8 years. What is the total amount after 8 years?

A

Multiplier for a 6% increase = 1.06
The account is multiplied by this multiplier 8 times.
Math:
A = 4000 (1.06)^8

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16
Q

Compounding periods that occur in a year?

A

Quarterly: n = 4
Monthly: n = 12
Daily: n = 365

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17
Q

IF A BANK PAYS 5% ANNUAL INTEREST, COMPOUNDING QUARTERLY, HOW MUCH DOES THE BANK PAYS US?

A

5%/4 = 1.25% each quarter.

We divide the annual percent of interest by “n” to get the percent of each individual compounding period.

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18
Q

Build the correct math: If Susan invests $1000 in an account that yields 5% annual, compounding quarterly, then how much does she have after six years?

A

Quarterly percent: 5/4 = 1.25%
Multiplier = 1.0125
Percent increase: 4 times each year, or 24 times in 6 years.
-Math- A = 1000 (1.0125)^24

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19
Q

How to estimate a compounded interest?

A

Estimate simple interest. The correct answer would be slightly more than it.
“Compound Interest > Simple Interest”

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20
Q

Sheila invests $4000 in an account that yields 6% compounding annually for 8 years. What is the total amount after 8 years?

A

Multiplier for a 6% increase = 1.06
The account is multiplied by this multiplier 8 times.
Math:
A = 4000 (1.06)^8

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21
Q

Compounding periods that occur in a year?

A

Quarterly: n = 4
Monthly: n = 12
Daily: n = 365

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22
Q

IF A BANK PAYS 5% ANNUAL INTEREST, COMPOUNDING QUARTERLY, HOW MUCH DOES THE BANK PAYS US?

A

5%/4 = 1.25% each quarter.

We divide the annual percent of interest by “n” to get the percent of each individual compounding period.

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23
Q

Build the correct math: If Susan invests $1000 in an account that yields 5% annual, compounding quarterly, then how much does she have after six years?

A

Quarterly percent: 5/4 = 1.25%
Multiplier = 1.0125
Percent increase: 4 times each year, or 24 times in 6 years.
-Math- A = 1000 (1.0125)^24

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24
Q

How to estimate a compounded interest?

A

Estimate simple interest. The correct answer would be slightly more than it.
“Compound Interest > Simple Interest”

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25
Q

Number “ZERO” is:

A

Integer

Even

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26
Q

FINDING “GCF” AND “LCM” using Venn Diagram

A
  1. Prime Factorization:
    30 = 2x3x5
    24 = 2x2x2x3
  2. Create a Venn Diagram
  3. Place each shared factor into the shared area of the diagram. 30 and 24 share one 2 and one 3.
  4. Place the remaining (non-shared) factors into the non-shared areas.
  5. The GCF is the product of the primes in the shared region: 2x3 = 6
  6. The LCM is the product of all primes in the diagram:
    5x2x3x2x2 = 120
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27
Q

Calculate the GCF of 100, 140 and 250

A

Make a column listing 100, 140 and 250. Besides, a column named as each of the prime numbers that appear in the factorization of 100, 140 and 250. Take the lowest power in any column. GCF is 2x5 = 10

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28
Q

Calculate the LCM of 100, 140 and 250

A

Take the highest power in any column. Multiply the factors.

LCM = 2^2 x 5^3 x 7^1 = 3,500

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29
Q

(59)^3 (59)^2

A

59^3+2 = 59^5

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30
Q

150% of 48

A

72

The 100% is 48. I need 50% more of 48, which is 24. I add 48 (original) plus 24 = 72

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31
Q

How can I cancel (reduce) the following fraction in order to make it easier: 5/14 X 7/15

A

Cross cancelation:
5 and 15 = 1 and 3
7 and 14 = 1 and 2
New fraction: 1/2 X 1/3

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32
Q

Multiple Fractions - Addition and Subtraction Method

2/3 + 1/4 - 1/5

A 
Each numerator must be multiplied by the other two denominators.
2/3 + 1/4 - 1/5
a) 2 x 4 x 5
b) 1 x 3 x 5
c) 1 x 3 x 4
New fraction:
40 + 15 - 12 / 60
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33
Q

What are the prime divisors of 100?

A

2 x 2 x 5 x 5

So the prime divisors are 2 and 5

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34
Q

What are the positive divisors of 372?

A

1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, and 372

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35
Q

The Prime Factors of 100:

A

2 x 2 x 5 x 5

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36
Q

The Factors of 100:

A 
1 x 100
2 x 50
4 x 25
5 x 20
10 x 10
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37
Q

Prime Factors of 42

A

2 x 3 x 7

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38
Q

Factors of 42

A 
All pairs of numbers when multiplied together result in 42:
1 x 42
2 x 21
3 x 14
6 x 7
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39
Q

(-3)^a / (-3)^2 =

A

(-3)^a-2

The “3” maintains the negative sign.

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40
Q

Simplify, 7^5 x 5^3 =

A

Can’t simplify, no common bases or exponents!

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41
Q

Simplify square root of 180 =

A

Square root of 2 x 2 x 3 x 3 x 5
(Square root of 2 x 2) (square root of 3 x 3) (square root of 5)
2 x 3 x (square root of 5)
6 times Square root of 5

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42
Q

Simplify 8^3 x 2^6

A

I have to equal 8 to 2.
8 is 2 x 2 x 2, so: (2 x 2 x 2)^3 = (2^3)^3 x 2^6
2^9 x 2^6
= 2^15

43
Q

2^0 =

A

1

44
Q

What is the value of “y” in 6^y-3 = 36 ?

A

I have to equal exponents ones the bases are equal as well.
6^y-3 = 6^2
y-3 = 2
y = 5

45
Q

If “J” is divisible by 12 and 10, is “J” divisible by 24?

A

CANNOT BE DETERMINED
Prime Factors of 12 = 2 x 2 x 3
Prime Factors of 10 = 2 x 5
(There are only TWO 2’s that are definitely in the prime factorization (PF) of “J”, because the 2 in the PF of 10 may be REDUNDANT -it may be the same 2 as one of the 2’s in the PF of 12-.
Prime Factors of 24 = 2 x 2 x 2 x 3
Since there are only TWO 2’s in the prime box of “J”, and 24 requires THREE 2’s, 24 is not necessarily a factor of “J”.

46
Q

If a/b has a reminder of 4, what is the smallest possible value of a + b?

A

9
Since a/b has a reminder of 4, “b” must be at least 5 (remember, the reminder must always be smaller than the divisor). The smallest possible value for “a” is 4 (it could also be 9, 14, 19).
Thus, the smallest possible value of a + b = 9

47
Q

While “zero” is neither positive nor negative, it is:

A

Even

48
Q

8^4 (5^4) =

A

40^4

49
Q

a^2 + a^4 = a^6 ?

A

NO

Remember, you cannot combine exponential expressions linked by addition.

50
Q

Whether it is positive or negative, any number raised to an even power is:

A

Positive

51
Q

(1/2)^-y =

A

(2)^y

The reciprocal of 1/2 is 2. We rise the the reciprocal to the positive version of the power (-y).

52
Q

Which quantity is larger?

a. (3^15) (2^8) or b. (3^12) (2^8)

A

a.
We divide away the original quantities by the common terms. Both quantities contain the product (3^12) (2^8). The smaller powers of each number.
a. 27 b.4

53
Q

Larger quantity?
A. Root of 150
B. 12

A

A is larger.
Square both quantities:
A. Root of 150^2 becomes just 150 (cancel out the root with the 2).
B. 12^2 = 144

54
Q

Formulas and order to find sum of terms in a set?

A

In order:

A) Get the Median: adding up the first and the last term. And then, divide by 2.

B) Find the number of terms: subtract last term “minus” first term; and then divide by the spacing or interval, +1

  The sum of terms will be the (Median) x (No. of terms)
55
Q

-2^2 =

A

-4

Como no hay paréntesis (-2)^2, hay que elevar al cuadrado sólo al 2, no al signo negativo. El paréntesis IMPORTA

56
Q

Divisibility rule for 4

A

Look at the last two digits: the tens place and the ones place. If the las two digits form a two-digit number divisible by 4, then the entire number is divisible by 4.

57
Q

In order to simplify the operation 48 x 75 / 3 can I first divide by 3?

A

Yes, but just one number, either 48 or 75.

58
Q

What do I have to consider to simplify an inequality?

A

I can add a cero on one side as necessary.
Example:

b-c+a > b+c+a
-c+a > c+a
-c > c
0 > 2c
c < 0
59
Q

1/5?

A

20%

.20

60
Q

1/8?

A
  1. 5%

0. 125

61
Q

3/8?

A
  1. 5%

0. 375

62
Q

5/8?

A
  1. 5%

0. 625

63
Q

7/8

A
  1. 5%

0. 875

64
Q

x^2 - y^2 =

A

(x-y) (x+y)

65
Q

(x-y)^2

A

x^2 - 2xy + y^2

66
Q

Dividend Formula?

A

Dividend = Divisor x Quotient + Reminder

67
Q

2 facts of Isosceles Triangle

A

1) Two equal sides, two equal angles.

2) With 1 angle I can find the other two.

68
Q

Fact of Equilateral Triangle

A

3 equal sides, 3 equal angles: 60, 60, 60

69
Q

2 Fact of Right Triangles 45 : 45 : 90

A

1) Triangle with one angle of 90.

2) Hired in squares.

70
Q

Sides and measurements of Right Triangle

A

Leg : Leg : Hypotenuse
1 1 square root of 2
x x x and square root of 2

71
Q

Pythagorean Triples

A

3 : 4 : 5
8 : 15 : 17
5 : 12 : 13
7 : 24 : 25

72
Q

Special Right Triangle 30 : 60 : 90

A

1) One 90 degrees angle
2) Hidden in Equilateral Triangles
3) Leg : Leg : Hypotenuse
1 Square root of 3. 2

73
Q

Distance Formula

A

Rate x Time

74
Q

Set “S” has 4 consecutive integers. If the second number is “n”, what are the others:

A

(n-1), (n+1), (n+2)

With (n-1) the first term in the sequence.

75
Q

.3875 to Fraction =

A

3875/10000
Numbers in the numerator equal number of zeros in denominator after 1.
Then, simplify it.

76
Q

Skill: Generating examples of specific reminders.

“What numbers when divided by 12, have a reminder of 5?

A

Start by adding 12 + 5 = 17

Then, any multiple of 12 (24, 36, 48) + 5, will yield that specific reminder when division.

77
Q

What is the smallest possible integer that, when divided by 12, has a reminder of 5?

A

5
12 is bigger than 5, so if we divide 5/12, 12 goes into it zero times, an integer quotient of zero, and the reminder is 5.

78
Q

Simplify Square Root of 81 + 169:

A

1) Because there is addition in the expression I add first.
Square Root of 81 + 169 = Square Root of 250
2) Prime Factorization of 250 under the radical: 2 x 5 x 5 x 5.
3) The expression can be rewritten as:
(Square Root of 2 x 5) (Square Root of 5 x 5) =
5 times Square Root of 10

79
Q

How many multiples of 11 are between 100 and 1,000, inclusive?

A

81
1) Determine the smallest and largest multiples of 11 (of the interval).
10 x 11 = 110 90 x 11 = 990
2) Subtract the 990 - 110 and divide by 11 (equals 80), then add 1 = 81

80
Q

Simplify (3^3)^2 (15)^3 / 5^3

A
  1. (3^3)^2 = 3^6
  2. 15^3 can be written as (3^3) (5^3) -> 3 x 5 = 15 and exponents can be distributed.
  3. New numbers and powers: (3^6) (3^3) (5^3)
  4. Exponents with the same base are combined: (3^6) (3^3) = 3^9
  5. New Numerator : (3^9) (5^3)
  6. Same Denominator : (5^3)
  7. Both (5^3) cancel
  8. Answer : 3^9
81
Q

Is this rationale correct?

Square Root of 20 + Square Root of 80 = Square root of 20 + 80 Square Root of 100 = 10

A

NO
A sum of two Roots CANNOT be rewritten as a single Square Root.
Instead, I should simplify as much as possible by factoring out a perfect square for inside each original root.
Square Root of 20 = Square Root of 4 x 5 = 2 times Square Root of 5.
Square Root of 80 = Square Root of 16 x 5 = 4 times Square Root of 5.
Gran Total : 6 times Square Root of 5.

82
Q

3^(3)^2 =

A

3^9

The power of 3 is in parenthesis so it must be squared.

83
Q

What is the approach:
Sam has 40% more marbles than Emma. However, if he gives 45 of his marbles to Emma, then Emma will have 10% more marbles then Sam. How many marbles did Sam begin with?

A

245.
Draw a table to relate quantities.
Horizontal axis: BEFORE / AFTER
Vertical axis: Emma x x + 45
Sam. 1.40x 1.40x - 45
Emma starts with “x” marbles.
Sam starts with “x” + 40%
Emma will end with 10% more marbles than Sam:
x + 45 = 1.10 (1.4x - 45) —-> 45 + 49.5 = 1.54x - x —–> 94.5/0.54 = x
x = 175, therefore: Sam = 1.40 (175) = 245

84
Q

(-x)^2 = -x^2 ?

A

No, the parenthesis matters.

(-x)^2 = x^2

85
Q

What is

(a-1) (a) (a+1) ?

A

It is the product of 3 consecutive integers. That means it will always contain a multiple of 3 so it will always be divisible by 3.

86
Q

What is the best SMART number to pick?

Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food.
Carrie, who earns twice as much as Lisa, spends 1/4 on rent and 1/2 on food. What is the reminder of their combined salary?

A

Since there is no amount specified, for example the reminder of the salary, I can chose a SMART number. Since the denominators in the problem are 8,12,4, and 2, assign Lisa a monthly paycheck of $24, since 24 is Least Common Multiple of the denominators.

87
Q

How can I know that two numbers are reciprocals?

A

The product of a number and its reciprocal must equal 1. To test whether two numbers are reciprocals, multiply them. If the product is 1, they are reciprocals; if it is not, they are not.

88
Q

Can I cancel terms in the expression:

3x + y

A

No.
The most we can simplify the expression is:
6 (x + y)
———
3x + y
Remember: in this example the x’s and the y’s, cannot cancel out in order to leave the 6 and a 3 alone.

89
Q

ROUNDING DECIMALS
82.743196
Name the positions at the right to de decimal point:

A
  1. 7 4 3
    Tens Hundreds Thousands (TH)
    1 9 6
    Ten-TH Hundred-TH Millions
90
Q

Rewrite the number in standard notation:

4 x 10^-2

A

Move the decimal point to the left:

0.04

91
Q

Rewrite the number in standard notation:

2.5 x 10^-3

A

Move the decimal point to the left.

0.0025

92
Q

Subtract:

0.40) - (0.0025

A

Order decimal points and add “zeros” as necessary.

  1. 4000
  2. 0025
    - ——–
  3. 3975
93
Q

Subtract:

8 - 7.98

A

8.00
7.98
—–
0.02 IMPORTANT: 8 to 10 = 2 and I have “1” left to add up to 9.
9 plus 1 = 10, to 10 = 0 and I have a “1” left to add up to 7, which becomes 8.

94
Q

573
——- (Division)
10^-2

A

5,730

When we divide by 10 raised to a negative exponent, we move the decimal to the right.

95
Q

0.573 x 10^5

A

57,300

When we are multiplying by 10 raised to a positive exponent, move the decimal to the right.

96
Q

600
——- (Division)
133

A

133 x 4 = 532
600 - 532 = 068, then I need to add a decimal point to the quotient and add a 0 to 068 = 680
Continue…

97
Q

In mixture problems withe percentages involved, remember…

A

To follow the story of the total amount of liquids.
I don’t have to focus just in the percentages, but also in how this percentages impact the overall quantities of something (for example liquids).

98
Q

Convert decimal to a fraction:

2.45

A
  1. Get a mixed fraction:45 9
    2 —- = 2 —-
    100 20
    49
  2. Get an improper fraction: —-
    20
99
Q

Convert 0.008 to a fraction:

A

8 1
—- = —-
1000 125

100
Q

Convert the fraction to a percent:
1000
—-
10

A

Rewrite the fraction with a denominator of 100.
10,000
—-
100
Alternatively, I can convert the fraction to a decimal (divide), and shift the decimal point two places to the right; and add a percent symbol.

101
Q

Fractions Comparison fact =

A

Numerator bigger than denominator, so the fraction is greater than 1.

102
Q

Fractions Comparison fact 2 =

A

When denominator is bigger than the numerator, the fraction is less than 1.

103
Q

COMPOUND INTEREST

If $10,000 is invested at 10%, compounded semi-annually, how much will the investment be worth after 18 months?

A

Final Amount = 10,000 (1+ .10/2)^(2*1.5)

Final Amount = $11,576.25

104
Q

WORK PROBLEMS

La Cantidad de Trabajo es igual al Producto del Número de Trabajadores por el Tiempo Necesario para Terminarlo.

A

Amount of Work =

of Workers or Machines) x (The Time needed to do it

105
Q

WORK PROBLEMS

Si el Tiempo aumenta…

A

El Número de trabajadores disminuye.

106
Q

WORK PROBLEMS

Si el tiempo disminuye…

A

El Número de Trabajadores aumenta.

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