Math Flashcards
0
Q
SQUARE ROOT OF 2
A
1.4
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
2
Q
SQUARE ROOT OF 3
A
1.7
3
Q
SQUARE ROOT OF 5
A
2.2
4
Q
SQUARE ROOT OF 7
A
2.6
5
Q
MEDIAN FORMULA
A
1rst + Last / 2
6
Q
NUMBER OF TERMS
A
Last - First / Spacing + 1
7
Q
AVERAGE SPEED
A
Total Distance / Total Time
8
Q
1/4
A
0.25
25%
9
Q
WHAT CAN YOU CANCEL IN A PROPORTION?
A
Just Vertical and Horizontal.
10
Q
WHAT CAN’T YOU CANCEL IN A PROPORTION?
A
It is illegal the Diagonal Cancellation.
11
Q
WHEN TRANSLATING FROM WORDS TO MATH, THE WORD “OF” MEANS:
A
Multiply
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
13
Q
How changing from Decimals to Percents?
A
Multiplying by 100. We move the decimal point two places to the right.
- 68 = 68%
- 3 = 230%
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
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
16
Q
Compounding periods that occur in a year?
A
Quarterly: n = 4
Monthly: n = 12
Daily: n = 365
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.
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
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”
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
21
Q
Compounding periods that occur in a year?
A
Quarterly: n = 4
Monthly: n = 12
Daily: n = 365
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.
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
24
How to estimate a compounded interest?
Estimate simple interest. The correct answer would be slightly more than it.
"Compound Interest > Simple Interest"
25
Number "ZERO" is:
Integer
| Even
26
FINDING "GCF" AND "LCM" using Venn Diagram
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
27
Calculate the GCF of 100, 140 and 250
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
28
Calculate the LCM of 100, 140 and 250
Take the highest power in any column. Multiply the factors.
| LCM = 2^2 x 5^3 x 7^1 = 3,500
29
(59)^3 (59)^2
59^3+2 = 59^5
30
150% of 48
72
The 100% is 48. I need 50% more of 48, which is 24. I add 48 (original) plus 24 = 72
31
How can I cancel (reduce) the following fraction in order to make it easier: 5/14 X 7/15
Cross cancelation:
5 and 15 = 1 and 3
7 and 14 = 1 and 2
New fraction: 1/2 X 1/3
32
Multiple Fractions - Addition and Subtraction Method
| 2/3 + 1/4 - 1/5
```
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
```
33
What are the prime divisors of 100?
2 x 2 x 5 x 5
| So the prime divisors are 2 and 5
34
What are the positive divisors of 372?
1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, and 372
35
The Prime Factors of 100:
2 x 2 x 5 x 5
36
The Factors of 100:
```
1 x 100
2 x 50
4 x 25
5 x 20
10 x 10
```
37
Prime Factors of 42
2 x 3 x 7
38
Factors of 42
```
All pairs of numbers when multiplied together result in 42:
1 x 42
2 x 21
3 x 14
6 x 7
```
39
(-3)^a / (-3)^2 =
(-3)^a-2
| The "3" maintains the negative sign.
40
Simplify, 7^5 x 5^3 =
Can't simplify, no common bases or exponents!
41
Simplify square root of 180 =
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
42
Simplify 8^3 x 2^6
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
2^0 =
1
44
What is the value of "y" in 6^y-3 = 36 ?
I have to equal exponents ones the bases are equal as well.
6^y-3 = 6^2
y-3 = 2
y = 5
45
If "J" is divisible by 12 and 10, is "J" divisible by 24?
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
If a/b has a reminder of 4, what is the smallest possible value of a + b?
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
While "zero" is neither positive nor negative, it is:
Even
48
8^4 (5^4) =
40^4
49
a^2 + a^4 = a^6 ?
NO
| Remember, you cannot combine exponential expressions linked by addition.
50
Whether it is positive or negative, any number raised to an even power is:
Positive
51
(1/2)^-y =
(2)^y
| The reciprocal of 1/2 is 2. We rise the the reciprocal to the positive version of the power (-y).
52
Which quantity is larger?
| a. (3^15) (2^8) or b. (3^12) (2^8)
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
Larger quantity?
A. Root of 150
B. 12
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
Formulas and order to find sum of terms in a set?
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
-2^2 =
-4
| Como no hay paréntesis (-2)^2, hay que elevar al cuadrado sólo al 2, no al signo negativo. El paréntesis IMPORTA
56
Divisibility rule for 4
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
In order to simplify the operation 48 x 75 / 3 can I first divide by 3?
Yes, but just one number, either 48 or 75.
58
What do I have to consider to simplify an inequality?
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
1/5?
20%
| .20
60
1/8?
12. 5%
| 0. 125
61
3/8?
37. 5%
| 0. 375
62
5/8?
62. 5%
| 0. 625
63
7/8
87. 5%
| 0. 875
64
x^2 - y^2 =
(x-y) (x+y)
65
(x-y)^2
x^2 - 2xy + y^2
66
Dividend Formula?
Dividend = Divisor x Quotient + Reminder
67
2 facts of Isosceles Triangle
1) Two equal sides, two equal angles.
| 2) With 1 angle I can find the other two.
68
Fact of Equilateral Triangle
3 equal sides, 3 equal angles: 60, 60, 60
69
2 Fact of Right Triangles 45 : 45 : 90
1) Triangle with one angle of 90.
| 2) Hired in squares.
70
Sides and measurements of Right Triangle
Leg : Leg : Hypotenuse
1 1 square root of 2
x x x and square root of 2
71
Pythagorean Triples
3 : 4 : 5
8 : 15 : 17
5 : 12 : 13
7 : 24 : 25
72
Special Right Triangle 30 : 60 : 90
1) One 90 degrees angle
2) Hidden in Equilateral Triangles
3) Leg : Leg : Hypotenuse
1 Square root of 3. 2
73
Distance Formula
Rate x Time
74
Set "S" has 4 consecutive integers. If the second number is "n", what are the others:
(n-1), (n+1), (n+2)
| With (n-1) the first term in the sequence.
75
.3875 to Fraction =
3875/10000
Numbers in the numerator equal number of zeros in denominator after 1.
Then, simplify it.
76
Skill: Generating examples of specific reminders.
| "What numbers when divided by 12, have a reminder of 5?
Start by adding 12 + 5 = 17
| Then, any multiple of 12 (24, 36, 48) + 5, will yield that specific reminder when division.
77
What is the smallest possible integer that, when divided by 12, has a reminder of 5?
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
Simplify Square Root of 81 + 169:
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
How many multiples of 11 are between 100 and 1,000, inclusive?
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
Simplify (3^3)^2 (15)^3 / 5^3
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
Is this rationale correct?
| Square Root of 20 + Square Root of 80 = Square root of 20 + 80 Square Root of 100 = 10
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
3^(3)^2 =
3^9
| The power of 3 is in parenthesis so it must be squared.
83
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?
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
(-x)^2 = -x^2 ?
No, the parenthesis matters.
| (-x)^2 = x^2
85
What is
| (a-1) (a) (a+1) ?
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
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?
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
How can I know that two numbers are reciprocals?
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
Can I cancel terms in the expression:
6x + 6y
---------
3x + y
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
ROUNDING DECIMALS
82.743196
Name the positions at the right to de decimal point:
82. 7 4 3
Tens Hundreds Thousands (TH)
1 9 6
Ten-TH Hundred-TH Millions
90
Rewrite the number in standard notation:
4 x 10^-2
Move the decimal point to the left:
| 0.04
91
Rewrite the number in standard notation:
2.5 x 10^-3
Move the decimal point to the left.
0.0025
92
Subtract:
| 0.40) - (0.0025
Order decimal points and add "zeros" as necessary.
0. 4000
0. 0025
- --------
0. 3975
93
Subtract:
| 8 - 7.98
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
573
------- (Division)
10^-2
5,730
When we divide by 10 raised to a negative exponent, we move the decimal to the right.
95
0.573 x 10^5
57,300
| When we are multiplying by 10 raised to a positive exponent, move the decimal to the right.
96
600
------- (Division)
133
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
In mixture problems withe percentages involved, remember...
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
Convert decimal to a fraction:
| 2.45
1. Get a mixed fraction:
45 9
2 ---- = 2 ----
100 20
49
2. Get an improper fraction: ----
20
99
Convert 0.008 to a fraction:
8 1
---- = ----
1000 125
100
Convert the fraction to a percent:
1000
----
10
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
Fractions Comparison fact =
Numerator bigger than denominator, so the fraction is greater than 1.
102
Fractions Comparison fact 2 =
When denominator is bigger than the numerator, the fraction is less than 1.
103
COMPOUND INTEREST
| If $10,000 is invested at 10%, compounded semi-annually, how much will the investment be worth after 18 months?
Final Amount = 10,000 (1+ .10/2)^(2*1.5)
| Final Amount = $11,576.25
104
WORK PROBLEMS
| La Cantidad de Trabajo es igual al Producto del Número de Trabajadores por el Tiempo Necesario para Terminarlo.
Amount of Work =
| of Workers or Machines) x (The Time needed to do it
105
WORK PROBLEMS
| Si el Tiempo aumenta...
El Número de trabajadores disminuye.
106
WORK PROBLEMS
| Si el tiempo disminuye...
El Número de Trabajadores aumenta.
