Maths! :D

PAGE 1!

Question 1 is BEDMAS (that's what I was taught it was anyway xD) Order of Operations.
So you solve the equation by dividing it into parts and solving from B to S
B - Brackets
E - Exponents
D - Division
M - Multiplication
A - Addition
S - Subtraction

So for 7 * 19 + 6
We solve the Multiplication part first, so it becomes
133 + 6
Then solve the Addition which gives us
139

Question 2 is to be solved with substitution.
For each of the letters there is a value which you subtitute into the equation and then solve same as Question 1.
So when we substitute a = 3, b = 6, and c = 8 into ab - c (when the letters are right beside each other it means they are multiplied)
We get the following
3 * 6 - 8
And solving with BEDMAS we get
18 - 8
and then
10! :D

Question 3 is to be solved with fraction logic (not sure if it has a proper name xD).
So with fractions, when you want to add or subtract, you first make then both have the same divisor. (The top part is the numerator and the bottom part is the denominator/divisor).
You do this by finding the LCM (Lowest Common Multiple, which is the smallest number that is divisible by both) of the 2 denominators. (Or you can just multiply them together, which sometimes is the LCM (it is in this situation! :D)) and multiplying both fractions so that they both have the same denominator.
For denominators 3 and 4 the LCM is 12, so we multiply the 7/3 by 4 and 9/4 by 3 so they are both 12 denominators.
So 7/3 + 9/4 becomes
(7*4)/(3*4) + (9*3)/(4*3)
which gives us
28/12 + 27/12
Now we can add them together to get the result (which we will turn into a Mixed number like the answers are in from the improper fraction from adding them together)
So the improper fraction is 55/12
and convert that to a Mixed number by extracting whole numbers out of the fraction until the numerator is less than the denominator.
So 55/12 would take out 4 whole numbers (48 = 12 * 4, you can just keep taking 12 out of the top number until it's less than 12)
so in the end we get 4 + 7/12

Where's question 4?! XD

For Question 5 it's some more fraction logic (still haven't found a proper name for it :P)
For multiplying fractions together, you multiply the 2 numerators together and the 2 denominators together to get the result.
So 3/8 * 4/5 would give us
12 / 40
Since this doesn't match one of the answers (and normally I think you would do so anyway) we need to simplify the fraction.
To simplify fractions you find the highest factor (factors are numbers that a number can be divisible by) of the top and bottom and divide both the top and the bottom by the highest factor.
So for 12 and 40 the highest factor is 4, (12 / 3 = 4, 40 / 10 = 4).
So we divide both of these numbers by 4 and the result is then
3 / 10

Question 6 is similar to question 2 in that it requires substitution, but the resulting equation from the substitution is a bit more complex with exponents. We still will solve this in the same was with BEDMAS.
So to begin with we will substitute the values into the equation.
So 3 * x^2 + 9^x = ? by substituting x = 3 into this we get
3 * 3^2 + 9^3
So now we do BEDMAS. Since there is no brackets, we solve the exponents first, which gives us
3 * 9 + 729
Then we do the Multiplication of 3 * 9, which gives us
27 + 729
And the final addition gives us
756 :D

Question 7 i've no idea how to explain it easily, but I do remember the answer from geometry class :P
The answer is 2 * pi * r, which is the same as saying pi * diameter (the distance across the circle through the center)
pi * r^2 is the area of the circle also :P
The other 2 answers aren't anything that I know of xD
Crazy genius mathematicians from years back found a relationship between the diameter of the circle and it's circumference that they called pi. You can see it in action in this gif (sorry it's a little fast xD) http://i.imgur.com/WeBLRRM.gif

PAGE 2!

Question 1 is the opposite of substitution, in this one we have to find the value of the letter (X) by solving the equation to give us X = (the value).
To solve this kind of equation you need to make the left (or right depends on how you want to do it) equal to just X.
Let's do X on the left.
So to make X alone on the left we have to move the other stuff to the other side of the equals.
Firstly we will move the -23 to the right side (since we can't move the ^2 with it there).
To do so we have to do the same operation to both sides of the equals so that the -23 is negated.
So we will add 23 to both sides of the equation.
Adding 23 to -23 will give us 0 so we can just remove that.
So from that we get
X^2 = 2 + 23
And solve the addition gives us
X^2 = 25
Now we have to move the ^2 over to the right side of the equals.
The negating operation of ^2 would be 2nd root (or square root as it's commonly known, for ^3 it would be the 3rd root, etc, complicated stuff xD)
so we apply the square root to both sides which gives us
X = square root(25)
When we solve the square root of 25 we get our answer.
The square root of 25 is 5
So X = 5!

For Question 2 it is another BEDMAS question, the brackets are just to indicate the ^2 is applied to the whole thing rather than an operation.
X^2 is another way of saying X*X
So we can show the equation as such to make it easier.
2/3 * 2/3
Then we can apply the fraction multiply logic from Page 1 question 5
This will give us
(2*2)/(3*3)
And solving the multiplications we get
4/9

Question 3 I explained in the previous question xD
Another way to write (x-y)^2 would be
(x-y)*(x-y)

Question 4 is a combination of the substitution and then the opposite to find the value of the unknown A
We would start by substituting the known values into the equation first.
So A=2(DX)/V^2 with D=7; X=9; V=2
We get (the lack of brackets mean that the 2 is applied only to the top and not the whole equation)
A = (2 * (7 * 9)) / (2 * 2)
We can make things easier for ourselves here and do some more fraction logic (like we did at the end of question 4 by simplifying the fraction by dividing by a common value). We can see there is a common value of 2 on both the top and the bottom so we can just take those away to give us
A = (7 * 9) / 2
Solving the multiplication first (BEDMAS says otherwise but there are actually brackets even if not explicit on the contents of the top and bottoms of fractions like this so we solve that first).
A = 63 / 2
Which results in 31.5

Question 5 is about trying to solve for the unknown A but since there are no values to substitute we just have to rearrange the equation so the A is alone on one of the sides (lets do left again)
So A / B = C
We would apply the same operation to both sides to negate the effect of the / B so that A will be alone.
So opposite of division is multiplication so we multiply both sides by B
So that gives us
A = BC (rearranged the letters so it's in Alphabetical order xD and the same as the answer :D)

MAGIC! :D