Mathematics Sin, Cosine, Percentages, Year 8 mv im

Factorization of a² - b²

(a+b)(a-b) [45]

Factorize x² - 11

(x + sqrt(11)) (x - sqrt(11))

10^3 =

1,000 (exponent is the number of zeros after the 1)

cos of 60 degrees

1/2

10^2 =

100 (exponent is the number of zeros after the 1)

Scientific notation: 263.57

2.6357 x 10²

Factorize 3x² + 6x - 12

3(x + 6)(x - 4) [50] Note: start by looking for a common factor: 3(x² + 2x - 4), then factorize...

Scientific notation: 0,041 x 10-2

4.1 x 10-4

(a - b)² =

a² - 2ab + b²

order of operations

"Please Excuse My Dear Aunt Sally" is common. It stands for Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction. - source: http://en.wikipedia.org/wiki/Order_of_operations

What does % (per cent) mean?

"per 100"; so 5% means 5 per 100 or 5/100. It's the same thing.

Factorize 4x² - 25

(2x + 5)(2x - 5) because this fits the pattern a² - b² where a = 2x and b = 5 [45]

Factorize x² - 7x + 12

(x - 3)(x - 4) [49] note: if the sum is negative, but the product is positive then both numbers must be negative.

Factorize (x-1)(x+2)+3(x-1)

(x-1)(x+5) [43]

-1^7 =

-1

cos of 90 degrees

0

sin of 0 degrees

0

tan of 0 degrees

0

What is the 1/4, represented as a decimal?

0.25 = Think there are 4 quarters in a dollar and each quarter has 25 cents; 25 PER cent that is, which is .25.

What is the 1/2, represented as a decimal?

0.5.

cos of 0 degrees =

1

sin of 90 degrees

1

tan of 45 degrees

1

the hypotenuse of a unit circle equals ?

1

(5/2)^(-2) =

1 / (5/2)² = (2/5)²

a^(-x) =

1 / (a^x), e.g., 2^(-3) = 1 / (2^3)= 1/8

10^6 =

1,000,000 = 1 million

10^9 =

1,000,000,000 = 1 billion = 1 with 9 zeros after it

Scientific notation: 18,17 x 10^6

1.817 x 10^7

sin of 30 degrees

1/2

If I have 3 chocolate donuts and 9 vanilla donuts, what fraction of the donuts are chocolate?

1/4 => You get this answer by dividing the number of chocolate donuts (3) by the TOTAL number of donuts (3 + 9 = 12). 3 divided by 12 = 3/12 = 1/4. You get 1/4 by dividing both top and bottom by 3.

What is 10% of $100? What is 20% of $100?

10 percent means 10 per 100 = 10/100 = 0.1. To get the percent "of" something this means multiply, so 10% of $100 = 0.1 x $100. 0.1 times anything just moves the decimal point one place to the left, so $100 becomes $10. 10% of $100 is $10.00. 20% of $100 is $20 because a.) 20% is twice 10%, so since 10% = $10, 20% must equal $20. The other way is to translate the English into math: 20% of $100 = 20 per 100 times $20 = 20/100 x $100 = .2 x $100 = $20.

What is 10% of $20? What is 20% of $20.

10 percent means 10 per 100 = 10/100 = 0.1. To get the percent "of" something this means multiply, so 10% of $20 = 0.1 x $20. 0.1 times anything just moves the decimal point one place to the left, so $20 becomes $2.0. 10% of $20 is $2.00. 20% of $20 is $4 because a.) 20% is twice 10%, so since 10% = $2, 20% must equal $4. The other way is to translate the English into math: 20% of $20 = 20 per 100 times $20 = 20/100 x $20 = .2 x $20 = $4.

What are two other ways of writing 10%?

10/100 or 1/10 or 0.1. These are all the same.

If something doubles in size, what percent does it increase?

100% - this you have to know. This is because if something doubles in size, it is increasing by 1 whole unit (whatever its original size was) which equals 1, and 1 = 100%.

10^1

10; any number to the power of 1 is itself

(10^-5)^(-8)

10^40

100/10^(-5)

10^5

Isabella bought exactly twice as many green sweaters as pink sweaters. What could be the total number of sweaters she bought? a.) 10, b.) 13, c.) 16, d.) 18, e.) 25.

12 (D.) This is actually a much easier question than it seems. Let's say that the number of pink sweaters you buy is designated by the letter p. Therefore, the number of green sweaters must be 2 x p (twice as many). So the TOTAL NUMBER of sweaters must be p + 2p = 3p. Therefore, since you can't have fractions of sweaters, the correct answer must be divisible by 3. 18 is the only choice that falls in this category.

10^0

1; any number to the power of 0 (except 0) is 1

Scientific notation: 2050000

2.05 x 10^6

19 x 20 x 21 =

203 - 20 because this is the product of (a-1) x a x (a+1) = a (a+1)(a-1) = a (a² - 1) = a³ - 1

If 1/4 N = 12 then 1/2 N = ?

24. The easiest way is to solve for N, then plug N into the second equation (1/2 N). So if 1/4 N = 12, you try to get everything except N on the right side of the equal sign. To do this you have to get rid of the ugly fraction. How do you do this? To get rid of 1/4, multiply times 4, to make it 1 x N and remember you don't write the 1 x part so just end up with N. Problem: you can't just multiply times one side of the equation - you have to do the same thing to the other side to make them equal. So N = 12 x 4, or N = 48. If N = 48 then 1/2 N = 1/2 x 48 = 24. If you're really stumped, try to think it through: Whatever N is (unless it's a negative number), 1/2 N must be more than 1/4 N, right? so since 1/4 N is 12, any answer less than or equal to 12 must be wrong. Eliminate 3 and 6 in this case. Now, if you're really smart, you can figure out that 1/2 of something is twice as much as 1/4 of something, for the same reason that a half dollar is twice as much as a quarter. So whatever N is, 1/2 N has to be twice as much as 1/4 N. If 1/4 N is 12, then 12 x 2 = 24, and this must the answer to 1/2 N. (Note that we have not solved for N in this case, saving ourself some time, but if this gets you confused, don't do it.)

9 + 6 x (8 - 5)

27 = 9 + 6 x 3 = 9 + 18 // you have to do parentheses first

16/25 expressed in powers of 2, 3, or 5 =

2^4 x 5^(-2)

What is 2(x+3)?

2x + 6 => you must multiply 2 times x plus 2 times 3

40 expressed in powers of 2, 3, or 5 =

2³ x 5

Simplifier et donner les résultats en notation scientifique: (8x10³)(4x10-²)

3.2 x 10² because 32 x 10(3-2) = 32 x 10 = 320; notation scientifique = 3.2 x 10²

(3^2)^-4 =

3^(-8)= 1 / (3^8) = 1 / 6,561

81 expressed in powers of 2, 3, or 5 =

3^4

Factorize 4a + 12

4(a + 3) [42]

What is 5x - x?

4x since 5x - x is really 5x + (-1)x = (5-1)x = 4x

Simplifier et donner les résultats en notation scientifique: (3x10^5) / (6x10-²)

5 x 10^6 because (3 x 10^7) / 6 = .5 x 10^7 = 5 x 10^6

If something is on sale for half price, what percent reduction is this?

50% = 1/2.

5^(-9) / 5^(-2) =

5^(-7)

(2x + 1)(3x - 2)

6x² - x - 2 [34]

(1/7)^(-1) =

7 = 1/ (1/7)

Srijana has sold 23 more boxes of girl scout cookies than Isabella. If there are 3 weeks remaining before they have to turn in their orders, how many more boxes per week must Isabella sell to beat Srijana?

8 boxes per week. This is a little tricky, but think of it this way. Isabella is behind by 23 boxes. She has 3 weeks to close the gap. So 23 boxes divided by 3 weeks is the number Isabella must sell just to tie. 23 divided by 3 works out to 7 and 2/3. But remember, this would only tie, and the question asks how much must Isabella sell to win. Also, boxes can't be divided up, so 8 (the next whole number up from 7 and 2/3) is the right answer. In other words, if Isabella sells 8 more boxes on average per week more than Srijana over the next 3 weeks, she will sell 3 x 8 or 24 boxes. Hint: if you're really pressed for time, use the "brute force" method. Guess an answer, see if it works, then go higher or lower. For example, you might have guessed 2 boxes per week, but 2 x 3 = 6 which is less than 23, so that would not close the gap. The answer must be higher than 2 boxes per week. how about 4? Well, 4 x 3 = 12 which is still less than 23. Hmmm. How about 8? 8 x 3 is 24 and - aha - that's the right answer.

Find the highest common factor of 8a and 12b

8a = 2 x 2 x 2 x a, 12b = 2 x 2 x 3 x b, HCF = 2 x 2 = 4 [41]

Scientific notation: 0.965 x 1011

9.65 x 1010

What is an average?

An average is a total divided by the number of things we are adding up. For example, the average of 1, 2, and 4 is (1 + 2 + 4) / 3 = 7 /3 = 2 1/3.

What is 15% of $8.95 closest to? a.) 1.95, b.) 1.75, c.) 1.50, d.) 1.35, e.) 1.00.

D $1.35. Since they are saying "closest to", you might get away with rounding off $8.95 to the $9 to make the math easier then seeing which of the answers is close. Now 15% is a common number because we use it to give tips. It's easy once you learn this trick. 15% of something is 10% of something but half of whatever that number is. In this case, 10% of 9.00 (rounded off) is 10 per 100 x 9 = 10/100 x 9 = .9. (Hint: just move the decimal point left one place). Now half of .9 = .9 / 2 = .45. So .9 plus .45 = 1.35. Voila! This is answer D.

True or false: pi ∈ Q.

False, because pi cannot be expressed as one integer over another so is therefore not a rational number (Q); it is an irrational number.

True or false: -2 ∉R.

False, since -2 can be plotted on a number line so is a real number.

Factorise 3x² + 17x + 10.

Find the ac = 3x10 = 30, then find the factors of 30 that add up to b (2 + 15), replace bx by these factors (2x + 15x), complete the factorization: 3x² + 2x + 15x +10 = x(3x + 2) + 5(3x + 2) = (x + 5)(3x + 2) [53]

The first thing you must do to solve a problem such as x + 12 = 16 - x is ...

Get everything with x on 1 side of the equal sign and everything that does not have x on the other side.

Natural numbers

N to represent the set of all natural or counting numbers {0, 1, 2, 3 ...}

Write a set, P, that includes all prime numbers less than 10.

P = {2, 3, 5, 7}, where {} = "the set of"

Rational numbers

Q represents the set of all rational numbers which have the form p/q, where p and q are integers and q <> 0

Real numbers

R represents the set of all real numbers, which are numbers which can be placed on a number line.

What is an easy way to remember what percent means using money?

There are 100 cents in a dollar. Therefore, a cent is 1 per cent of a dollar! This is because the word roots are the same (cent = 100, per cent = per 100). So if you get stumped on a percentage problem think of it in terms of cents per dollar.

Betsy rakes leaves from 9:30 am to 11 am and finished 1/4 of the yard. She wants to finish raking by 6:30 pm. If she plans to rake at the same rate, what is the latest time she can start raking the leaves again?

This is a very hard problem and requires several steps, but each of these steps is relatively simple, so if you break it down, you can solve it. First, we must calculate the time it took her to do the work she did already. If you start at 9:30 and end at 11 am, that's 1.5 hours. She did only 1/4 of the yard in this time, though, so we know she has 3/4 of her work ahead of her. So, if she could do 1/4 in 1.5 hours, how many hours would it take her to do 3/4? Since 3/4 is 3 times 1/4, all we have to do is multiply 1.5 hours time 3 to get 4.5 hours. But we're not done. They asked us the latest time she could start to be done by 6:30 pm? So we have to subtract 4.5 hours from 6:30 pm. 6:30 - 4.5 hours is 2:00 pm., choice C. That is a hard one!

True or false: 2.5 ∈ Q

True, because Q = set of all rational numbers, having form p / q where p and q are integers. And 2.5 = 5/2

True or false: 2 ∈ Z

True, because Z = set of all integers and 2 is an integer

Is .363636 (repeating) a rational number?

Yes, because if x = .3636..., 100x = 36.3636.. = 36 + x, so 99x = 36, so x = 36/99 = 4/11

Integers

Z represents the set of all integers {0, ±1, 12. ±3, ±4, ±5, ±6

Z+ represents

Z+ represents all positive integers {1, 2, 3, 4, 5. 6....

a^x * a^y =

a ^ (x+y) Example: 3^2 x 3^4 = 3^6

.49 x 81 is closest to a.) 1/2 of 80, b.) 1/2 of 90, c.) 1/4 of 80, d) 1/4 of 90, e.) 4 x 80

a.) 1/2 of 80. The point here is estimation of "about" which is very important in the real world. You should have a feel for what these numbers mean. .49 is about .5 which is about a half. (Think of 49 cents is about equal to 2 quarters or half a dollar.) 81 is close to 80. So, of the answers given, .49 x 81 is probably closest the .5 x 80, or 1/2 x 80.

If 5 x (P + Q) = 30 and P is greater than zero, then Q could NOT be: a.) 7, b.) 4, c.) 3 1/2, d.) 0, e.) -2

a.) 7. This looks like a much harder problem than it is. All you have to do is to substitute the numbers for P one at a time to see what they're getting at. Start with the biggest and smallest number. If you were lucky enough to start with seven, you would realize that even if Q were zero (the smallest number that Q could be), 5 x (7 + 0) = 30, but 5 x 7 = 35 which is greater than 35. This tells you that P could not be seven - By the way, in a problem like this once you find any number that P could not be, you're done. Unless you made a mistake, you do not need to waste time on the other possible answers.

a(b + c) =

ab + ac => Distributive law [32]

(a + b)(c + d) =

ac + ad + bc + bd => FOIL: First, Outer, Inner, Last [33]

Quadratic trinomial

algebraic expression of the form ax² + bx + a where x is a variable and a, b, and c are constants, a <> 0. [49]

In the equation y = mx + b, what is the slope and the y intercept and what sort of equation is this?

an equation in the form y = mx + b, where m is the slope and b is the y intercept is known as a linear equation

What is a linear equation?

an equation in the form y = mx + b, where m is the slope and b is the y intercept.

(a + b)² =

a² + 2ab + b²

(a + b)(a - b) =

a² - b² [35]

formula for cos

cos = adjacent / hypotenuse

term

either a single number or variable, or the product of several numbers and/or variables separated from another term by a + or - sign in an overall expression

∈

element of e.g., 2 ∈ {1, 2, 3} is true; opposite is ∉

Power equation

equation of the form xn = k where n = 2, 3, 4... ; note: these are NOT linear equations.

linear equation

equation which contains a variable which is not raised to any power other than 1

The opposite of factorization is [1]

expansion [42]

∉

not element of e.g., Isabella ∉{all boys at La Chat}

Prime numbers

numbers with exactly 2 factors (versus composite numbers)

Composite numbers

numbers with more than 2 factors (versus prime numbers which have only 2 factors)

Algebraic product =

products which contain variables, such as 6c or 4x²y [40]

slope of a line = [1]

rise over run, or how many units y increases for every unit x increases; it is the steepness of a ski slope - if it rises 4 feet for every 1 foot you go forward (horizontally) it is much steeper than if it rises 1 foot for every foot you go forward

formula for sin

sin = opposite / hypotenuse

cos of 45 degrees

sqrt(2) / 2

sin of 45 degrees

sqrt(2) / 2

tan of 60 degrees

sqrt(3)

cos of 30 degrees

sqrt(3) / 2

sin of 60 degrees

sqrt(3) / 2

tan of 30 degrees

sqrt(3) / 3

formula for tan

tan = opposite / adjacent

(5t^(-2))^(-1) =

t²/5 because 5t-2 = 5/t², and this expression raised to the power of -1 equals 1 over this expression, so 1/ (5/t²) = t²/5

tan of 90 degrees

undefined

Translate into an equation: "The sum of 2 consecutive even integers is 34"

x + (x + 2) = 34 = the equation [84]

John has x dollars. Ann has $5 more than John. If Ann gives John $10, then, in terms of x, how many dollars will Ann have?

x + 5. It might help to make a table here. We start with John having x dollars. We don't know what x is and don't need to know. We know that Ann starts with $5 more than John, however much that is, so we can express this as x + 5 dollars. Now, Ann gives John $10. What does this do to our table. Well, Ann loses (minus) 10 and John gains (+) 10. So we add these to what they started with. John started with x and now has $10 more so whatever x is, he now has x plus 10. Ann started with x + 5 dollars and gave away $10 so she must now have x + 5 - 10 or x -5 dollars. This is the correct answer, C.

Solve x - 1 = 2 (x + 3)

x - 1 = 2x + 6; -x - 1 = 6; -x = 7; x = -7

Solve for x: (4x + 3) / 5 = -2

x = - 3 1/4 [78]

Solve for x: x/3 + 2 = -2

x = -12 [example 3, p. 77]

Solve for x: 5x + 2 = 3x - 5

x = -3 1/4 [79]

if -x = 7, x = ?

x = -7 => if -x = y then x = -y

Solve for x: (2x + 3) / 4 = (x - 2) / 3

x = -8 1/2 [81]

Solve for x: (3x + 1) / (x - 1) = -2

x = 1/5 [82]

Solve 3 - 2x < 7

x >= -2 [90]

xy^-3 =

x^(-3)y^(-3) = 1 / (x^3 * y^3)