Math Flashcards
SQUARE ROOT OF 2
1.4
Slope of the line with equation 3x + 7y = 9
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
SQUARE ROOT OF 3
1.7
SQUARE ROOT OF 5
2.2
SQUARE ROOT OF 7
2.6
MEDIAN FORMULA
1rst + Last / 2
NUMBER OF TERMS
Last - First / Spacing + 1
AVERAGE SPEED
Total Distance / Total Time
1/4
0.25
25%
WHAT CAN YOU CANCEL IN A PROPORTION?
Just Vertical and Horizontal.
WHAT CAN’T YOU CANCEL IN A PROPORTION?
It is illegal the Diagonal Cancellation.
WHEN TRANSLATING FROM WORDS TO MATH, THE WORD “OF” MEANS:
Multiply
CHANGING PERCENTS TO DECIMALS
Simply dividing by 100, so we move the decimal point two places to the left.
42.5% = 0.425
How changing from Decimals to Percents?
Multiplying by 100. We move the decimal point two places to the right.
- 68 = 68%
- 3 = 230%
How changing from Percents to Fractions?
We just put the percent over 100. After that, simplify.
20% = 20/100 = 1/5
0.02% = 0.02/100 = 2/10000 = 1/5000
Sheila invests $4000 in an account that yields 6% compounding annually for 8 years. What is the total amount after 8 years?
Multiplier for a 6% increase = 1.06
The account is multiplied by this multiplier 8 times.
Math:
A = 4000 (1.06)^8
Compounding periods that occur in a year?
Quarterly: n = 4
Monthly: n = 12
Daily: n = 365
IF A BANK PAYS 5% ANNUAL INTEREST, COMPOUNDING QUARTERLY, HOW MUCH DOES THE BANK PAYS US?
5%/4 = 1.25% each quarter.
We divide the annual percent of interest by “n” to get the percent of each individual compounding period.
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?
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
How to estimate a compounded interest?
Estimate simple interest. The correct answer would be slightly more than it.
“Compound Interest > Simple Interest”
Sheila invests $4000 in an account that yields 6% compounding annually for 8 years. What is the total amount after 8 years?
Multiplier for a 6% increase = 1.06
The account is multiplied by this multiplier 8 times.
Math:
A = 4000 (1.06)^8
Compounding periods that occur in a year?
Quarterly: n = 4
Monthly: n = 12
Daily: n = 365
IF A BANK PAYS 5% ANNUAL INTEREST, COMPOUNDING QUARTERLY, HOW MUCH DOES THE BANK PAYS US?
5%/4 = 1.25% each quarter.
We divide the annual percent of interest by “n” to get the percent of each individual compounding period.
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?
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
How to estimate a compounded interest?
Estimate simple interest. The correct answer would be slightly more than it.
“Compound Interest > Simple Interest”
Number “ZERO” is:
Integer
Even
FINDING “GCF” AND “LCM” using Venn Diagram
- Prime Factorization:
30 = 2x3x5
24 = 2x2x2x3 - Create a Venn Diagram
- Place each shared factor into the shared area of the diagram. 30 and 24 share one 2 and one 3.
- Place the remaining (non-shared) factors into the non-shared areas.
- The GCF is the product of the primes in the shared region: 2x3 = 6
- The LCM is the product of all primes in the diagram:
5x2x3x2x2 = 120
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
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
(59)^3 (59)^2
59^3+2 = 59^5
150% of 48
72
The 100% is 48. I need 50% more of 48, which is 24. I add 48 (original) plus 24 = 72
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
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
What are the prime divisors of 100?
2 x 2 x 5 x 5
So the prime divisors are 2 and 5
What are the positive divisors of 372?
1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, and 372
The Prime Factors of 100:
2 x 2 x 5 x 5
The Factors of 100:
1 x 100 2 x 50 4 x 25 5 x 20 10 x 10
Prime Factors of 42
2 x 3 x 7
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
(-3)^a / (-3)^2 =
(-3)^a-2
The “3” maintains the negative sign.
Simplify, 7^5 x 5^3 =
Can’t simplify, no common bases or exponents!
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
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
2^0 =
1
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
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”.
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
While “zero” is neither positive nor negative, it is:
Even
8^4 (5^4) =
40^4
a^2 + a^4 = a^6 ?
NO
Remember, you cannot combine exponential expressions linked by addition.
Whether it is positive or negative, any number raised to an even power is:
Positive
(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).
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
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
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)
-2^2 =
-4
Como no hay paréntesis (-2)^2, hay que elevar al cuadrado sólo al 2, no al signo negativo. El paréntesis IMPORTA
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.
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.
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
1/5?
20%
.20
1/8?
- 5%
0. 125
3/8?
- 5%
0. 375
5/8?
- 5%
0. 625
7/8
- 5%
0. 875
x^2 - y^2 =
(x-y) (x+y)
(x-y)^2
x^2 - 2xy + y^2
Dividend Formula?
Dividend = Divisor x Quotient + Reminder
2 facts of Isosceles Triangle
1) Two equal sides, two equal angles.
2) With 1 angle I can find the other two.
Fact of Equilateral Triangle
3 equal sides, 3 equal angles: 60, 60, 60
2 Fact of Right Triangles 45 : 45 : 90
1) Triangle with one angle of 90.
2) Hired in squares.
Sides and measurements of Right Triangle
Leg : Leg : Hypotenuse
1 1 square root of 2
x x x and square root of 2
Pythagorean Triples
3 : 4 : 5
8 : 15 : 17
5 : 12 : 13
7 : 24 : 25
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
Distance Formula
Rate x Time
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.
.3875 to Fraction =
3875/10000
Numbers in the numerator equal number of zeros in denominator after 1.
Then, simplify it.
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.
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.
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
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
Simplify (3^3)^2 (15)^3 / 5^3
- (3^3)^2 = 3^6
- 15^3 can be written as (3^3) (5^3) -> 3 x 5 = 15 and exponents can be distributed.
- New numbers and powers: (3^6) (3^3) (5^3)
- Exponents with the same base are combined: (3^6) (3^3) = 3^9
- New Numerator : (3^9) (5^3)
- Same Denominator : (5^3)
- Both (5^3) cancel
- Answer : 3^9
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.
3^(3)^2 =
3^9
The power of 3 is in parenthesis so it must be squared.
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
(-x)^2 = -x^2 ?
No, the parenthesis matters.
(-x)^2 = x^2
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.
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.
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.
Can I cancel terms in the expression:
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.
ROUNDING DECIMALS
82.743196
Name the positions at the right to de decimal point:
- 7 4 3
Tens Hundreds Thousands (TH)
1 9 6
Ten-TH Hundred-TH Millions
Rewrite the number in standard notation:
4 x 10^-2
Move the decimal point to the left:
0.04
Rewrite the number in standard notation:
2.5 x 10^-3
Move the decimal point to the left.
0.0025
Subtract:
0.40) - (0.0025
Order decimal points and add “zeros” as necessary.
- 4000
- 0025
- ——– - 3975
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.
573
——- (Division)
10^-2
5,730
When we divide by 10 raised to a negative exponent, we move the decimal to the right.
0.573 x 10^5
57,300
When we are multiplying by 10 raised to a positive exponent, move the decimal to the right.
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…
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).
Convert decimal to a fraction:
2.45
- Get a mixed fraction:45 9
2 —- = 2 —-
100 20
49 - Get an improper fraction: —-
20
Convert 0.008 to a fraction:
8 1
—- = —-
1000 125
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.
Fractions Comparison fact =
Numerator bigger than denominator, so the fraction is greater than 1.
Fractions Comparison fact 2 =
When denominator is bigger than the numerator, the fraction is less than 1.
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
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
WORK PROBLEMS
Si el Tiempo aumenta…
El Número de trabajadores disminuye.
WORK PROBLEMS
Si el tiempo disminuye…
El Número de Trabajadores aumenta.
