Each day’s work starts with a section title and ends with a FOR YOU part. The learner is meant to do that part as part of that work. (Solutions are at the back of the book.) Some days may be devoted to the games for the sake of review: Vap (multiplication), …. Feel free to skip around if you know the earlier topics – they are always there if you need them. 

CHAPTER ONE: MULTIPLES AND DIVIDENDS

DIVISION
	One number divides another if the first number times a whole number equals the second number. For example, 
5 divides 15 because 5 x 3 = 15
But 5 does not divide 16. Every number divides itself, so 7 divides 7 and 30 divides 30. The number 1 divides every number. For example,
1 divides 17 because 1 x 17 = 17
Practice these problems to make sure you understand division. 
	FOR YOU:
		1) Does 8 divide 40?
		2) Does 2 divide 10456739?
		3) Does 17 divide 357?

FACTORING
	You may remember that a prime number is one that can be divided only by 1 and itself. Seven is a prime. Fifteen is not, since both 3 and 5 divide 15. 
	Completely factoring a number means listing the prime number(s) whose product(s) equal that number. For example, completely factoring 
15 yields 3,5. 
(Or 5,3 - the order does not matter, so we will always list from smallest to largest by convention). Completely factoring 
30 yields 2, 3, 5. 
60 yields 2, 2, 3, 5. 
A remarkable fact about numbers is that every complete factorization of a number yields the same list. For example, we may factor 
140 into 2, 70, then 2, 7, 10, then (here is the complete factorization) 2, 2, 5, 7.
If we started with 140, but broke the number down in a different way, the end result is not affected. Let us take a look: 
140, then 7, 20, then 4, 5, 7, then 2, 2, 5, 7. 
This fact about complete factorizations is known as the Fundamental Theorem of Arithmetic. 
	FOR YOU:
		1) Completely factor 25.
		2) Completely factor 60.
		3) Find the smallest number having five different positive factors (not including 1).

COMMON MULTIPLES
	If I give you two numbers, say 4 and 6, and ask you to find a third number such that both 4 and 6 divide that third number, which number would you choose? Well, you could not go wrong with 6 x 4 = 24 since both 6 and 4 divide 24. We call 24 a common multiple of 6 and 4. 
	Most often though, we are interested in the least common multiple of two numbers, which is the smallest number that both numbers divide. To find that number, we completely factor the two numbers: 
4 has factors 2, 2 
6 has factors 2, 3 
The least common multiple has the smallest collection of factors that contains both lists. 
In this case: 2,2,3
We multiply these numbers together and get
2 x 2 x 3 = 12
We know we have found the desired number since 4 divides 12 (because 4 x 3 = 12) and 6 divides 12 (because 6 x 2 = 12). 
	FOR YOU:
		1) Find the least common multiple of 10 and 20.
		2) Find the least common multiple of 60 and 80.
		3) Completely factor 228 and 316.

GREATEST COMMON DIVISOR
	The greatest common divisor of two numbers, say 15 and 35, is the largest number that divides both. In this case, that number is 5 because 5 divides 15 and 35. No larger number does this.
	We can find the greatest common divisor by factoring. 
Factoring 15 yields 3,5
Factoring 35 yields 5,7
The greatest common divisor is whatever is in both lists, in this case 5. Let us see what happens if we take larger numbers such as 60 and 75: 
60 factors into 2,2,3,5
75 factors into 3,5,5
Common elements are 3,5, so the greatest common divisor is 15, or 3 x 5. 
	FOR YOU:
		1) Find the greatest common divisor of 20 and 25.
		2) Find the greatest common divisor of 18 and 25.
		3) Find the greatest common divisor of 400 and 350?

FRACTIONS
	Fractions are used to express portions of things. A half (1/2) of a pizza is a portion of a pizza, as is an eighth (1/8). The basic things you need to learn about fractions: how to multiply them, how to add them, how to reduce them, and how to convert them to decimals and percentages and ratios.
	MULTIPLYING FRACTIONS
	Multiplying fractions is a piece of cake. (As it happens slices of cake can be combined using fraction mathematics!) Consider:
(4/5) x (3/8)
Just multiply the tops, called the numerators, and multiply the bottoms, called the denominators: 
4 x 3 = 12
5 x 8 = 40
So the result is (12/40)	
Since multiplying any number by 1 yields that number and since any fraction with the same numerator and denominator equals 1, we can always multiply a fraction by any form of 1. Examples include these fractions: (4/4), (16/16), (19/19), ((1/2)/(1/2)), ((1/8)/(1/8)). 
	As an example of multiplying by 1, take (5/4) and multiply it by (3/3) to yield ((5/4) x (3/3)) = (15/12). Such multiplications will be useful when we want to convert a fraction with one denominator to a fraction with another. For example, if you want to convert (6/3) to fifteenths (which means 15 is in the denominator), you can use this method. Since you know:
5 x 3=15
then ((6/3) x (5/5)) = (30/15).
	YOU TRY:
		1) Convert 14/6 to have a denominator of 12.
		2) Convert 4/6 to have a denominator of 3.
		3) Convert 60/24 to have a denominator of 2.

	ADDING FRACTIONS
	Remember that the top number in a fraction is called the numerator and the bottom is the denominator. For example, the numerator of (2/7) is 2, and the denominator is 7. To read a fraction, you say, “Numerator denominator-ths,” with a few exceptions. 
	So (2/7) is two sevenths,  (3/10) is three tenths, (4/3) is four thirds, (15/2) is fifteen halves (an exception). 
	It is good to know how to read fractions because their names suggests the basic way to add and subtract them. For example, you would expect that adding 3 apples to 10 apples would give you 13 apples. Similarly, 
(3/17) + (10/17) = (13/17)
(Read it “three seventeenths plus ten seventeenths equals thirteen seventeenths”). Also, (3/2) + (10/2) = (13/2)
	FOR YOU:
		1) (6/8) + (1/8)
		2) (14/17) – (11/17)
		3) ((14/13) + ( 7/13)) - (5/13)

ADDING FRACTIONS WITH DIFFERENT DENOMINATORS
	Adding (or subtracting) fractions with the same denominator is like adding (or subtracting) apples.. When the fractions have different denominators, we make them have the same one by using least common multiples. Consider (5/4) + (1/6). Factoring the first denominator 4 we get 2, 2. Factoring 6 we get 2, 3. So the least common multiple of 4 and 6 has factors 2, 2, 3. Therefore the least common multiple is 2 x 2 x 3 = 12. We notice that 4 x 3 = 12, so we want to convert (5/4) to some number of twelfths, by multiplying the denominator by 3. But we can’t do that all by itself. We have to multiply the numerator by 3 as well so that the net effect is to multiply (5/4) by 1.	This gives
(3/3) x (5/4) = (15/12)
	By similar reasoning, observe that 6 x 2 = 12, so multiply both numerator and denominator of (1/6) by 2. 
(1/6) x (2/2) = (2/12)
This now tells us that 
(5/4) + (1/6) = (15/12) + (2/12)
And we know how to do that last part since they have the same denominator:
(15/12) + (2/12) = (17/12) 
What were the steps?
		a. Factor the denominators and find the least common multiple of the denominators. In this case twelfths. 
		b. For each fraction, determine what to multiply numerator and denominator by to get the least common multiple. For (5/4), multiply top and bottom by 3, yielding (15/12). For (1/6), multiply top and bottom by 2, yielding (2/12). 
		c. Add the fractions the way you would add apples: (15/12) + (2/12) = (17/12). 
	Let’s do one more example before I leave it to you:
a.	You want to compute (5/8) + (3/10). 
b.	Factor 8, yielding 2,2,2. Factor 10, yielding 2,5. Find the factors of the least common multiple: 2,2,2,5. Find the product of the factors: 2 x 2 x 2 x 5 = 40.
c. Convert (5/8) to fortieths: (5/8) x (5/5) = (25/40). Convert (3/10) to fortieths: (3/10) x (4/4) = (12/40). 
d. Add (25/40) + (12/40) = (37/40). 
	FOR YOU:
		1) (1/2) + (1/4)
		2) (7/8) + (5/6)
		3) (15/12) - (8/9)
REDUCING A FRACTION TO LOWEST TERMS
            There are many ways to express the same quantity with fractions. For example, if you have eight slices of cake, then four slices make up half the cake.  That is 4/8 = ½. How can we tell for sure? Well, (4/4) x (1/2) = 4/8. So, why should we prefer writing the fraction as ½ rather than 4/8?  It’s hard to tell, but ½ is reduced to lowest terms (that means that the numerator and denominator are as small as they can get while keeping the same quantity) and 4/8 are not.  
        To reduce a number to its lowest terms, completely factor the numerator and the denominator and remove the common factors. Let’s try this with 8/12. Completely factoring 8, we get 2, 2, 2. Completely factoring 12, we get 2, 2, 3. The common factors are 2,  2. So, if we remove these from 8, we get 2, which becomes the numerator. If we remove these from 12 we get  a denominator of 3.  So, the net result is 2/3.
        Let’s try a new example: 60/72. Completely factoring 60 gives 2, 2, 3, 5. Completely factoring 72 gives 2, 2, 2, 3, 3. So, the common factors are 2, 2, 3. Removing these from 60 leaves 5. Removing these from 72 leaves 6, so the result is 5/6.
       FOR YOU:
1.	Reduce 11/22 to lowest terms.
2.	Reduce 6/9 to lowest terms.
3.	Reduce 60/90 to lowest terms.

DECIMALS AND FRACTIONS
	A decimal number is a way to indicate a certain number of 10ths, 100ths, 1000ths, and so on. For example, 0.5 is read “zero point 5” but is equal to 5/10. On the other hand, 0.43 is 43/100.  And 0.375 is 375/1000. The nice thing about decimals is that they can be added, subtracted, multiplied, and divided just like whole numbers.
	For example, 0.5 + 0.2 = 0.7 (because 5/10 + 2/10 = 7/10). 
Similarly, 0.53 + 0.2 = 0.73 as we can see by converting this to a fractions problem:
	0.53 + 0.2 
= (53/100) + (2/10)
	= (53/100) + (10/10) x (2/10)
	= (53/100) + (20/100)
	= 73/100
The general technique for adding decimals is to add zeros to the right of the shorter decimal number and then add as if you were adding whole numbers:
	0.53 + 0.2 
	= 0.53 + 0.20
	= 0.73
FOR YOU:
1)	0.23 + 0.385
2)	0.43 – 0.394
3)	6.24 – 4.321

DECIMAL MULTIPLICATION
Multiplication works like multiplication for whole numbers except that the result has as many decimal places as the number of decimal places in the first number plus the number in the second number. For example, to calculate
	0.26 x 0.2, note that 25 x 2 = 52 and that there are two positions to the right of the decimal point in 0.26 and one position to the right in 0.2. This gives a total of three positions to the right of the decimal point. So, the result is 0.052. 
	Why should this work this way? Express this in fractions:
	0.26 x 0.2 
	= (26/100) x (2/10)
	= 52/1000
	= 0.052
	Here’s another example: 0.432 x 3.4. First we do 432 x 34 = 14688. Then we count the number of digits beyond the decimal point: three for 0.432 and one for 3.4. This gives a total of four. So, the result is 1.4688.	
FOR YOU:
1)	0.54 x 0.77
2)	4.2 x 5.12
3)	3.33 x 1.6
CONVERTING FRACTIONS TO DECIMALS
	To convert a fraction to a decimal, simply do the division the fraction represents and keep going until you arrive at enough digits. For example, to convert 5/8 to a decimal, divide 8 into 5.0000. Since 8 > 5, you start with a decimal point and then divide 8 into 50, getting 0.6, with a remainder of 2, then you divide 8 into 20 and get 2 (and therefore 0.62)  with a remainder of 4 and then you divide 8 into 40 to get 5 (and therefore 0.625).
	In that example, we came out with an exact division at the end. This doesn’t always happen. For example, if you try to convert 2/3 to a decimal, you write down a decimal point and then divide 20 by 3 to get 6 (and therefore 0.6) with a remainder of 2,  then divide 20 by 3 to get 6 (and therefore 0.66) with a remainder of 2, and so it goes forever.  One way to write this is 0.6 OVERBAR.

FOR YOU: 
     Convert the following to decimals:
1)	1/5
2)	5/6
3)	13/7.
RATIOS
	Ratios are closely related to fractions. For example, if the ratio of Jack’s age to Katie’s age is 2:3 (read this “two to three”), then Jack is (2/3) the age of Katie. So, if Katie is 15 years old, then Jack is (2/3) x 15 = (2/3) x (15/1) = (30/3). Reducing this to lowest terms yields 10/1 = 10. 
	Ratios can be used to measure tall buildings with the help of a little geometry. Take a measuring stick outside and let it dangle directly downward to get an exact vertical. Then line it up so the top of the stick is even with the top of a building. 
DIAGRAM 1

Let us try this out with some examples: DIAGRAM 2

The ratio of horizontal to vertical is 4:1 at the top of the stick (assuming the stick is of height 1). Therefore, it must also be so for the height of building h. The total distance is 
	4 + 10 = 14
So the ratio becomes
	14 : h = 4 : 1
Therefore 
	(14/h) = (4/1)
Therefore
	h = (14/4) = 3.5
	FOR YOU:
		1) What is the height of the tower in this case?
DIAGRAM 3
SOLUTIONS:
RATIOS
1) Jack’s age is 10 and is (2/3) the age of Jill. So Jill must be 
(3/2) x 10 = (30/2) = 15

2) Jack is (2/3) x 45 = (90/3) = 30

3) The ratio is 35:40. This is equivalent to the fraction (35/40). Reducing this to lowest terms gives (7/8), so the ratio is 7:8. 

POST RATIOS
	Ratios are calculated by dividing. The ratio of 3 to 2 is (3/2). 
	FOR YOU:
		1) What is the ratio of 20 to 15? Reduce to the greatest common factor.
		2) If A to B is 6 to 5, and if B to C is 5 to 3, what is A to C? 
POST RATIO SOLUTIONS:
	1) 4:3
	2) A = 6, B = 5, C = 3. So 6 to 3 equals (6/3) = 2

SQUARE ROOT ESTIMATION
	FOR YOU:
		1) What is the square root of 90? Well, the square root of 81 is 9 and the square root of 100 is 10, so 9< (90)^(1/2) <10. Figure out what it is to one decimal level.
		2) What is the square root of 150? 
		3) What is the square root of 1017?
	
CUBE ROOTS
	The cube root of 81 is 3 because (3)(3)(3) = 37 so (27)^(1/3) = 3.
	FOR YOU:
		1) Estimate (100)^(1/3) 
SQUARE ROOT SOLUTIONS
1) 9.5

2) 12.2

3) 31.9

	CUBE ROOTS
1) 4.6
PERCENTAGE
	Percentages are easily interchanged with decimals: just take the decimal number and multiply it by 100. So, 1 (one)  becomes 100% (read “100 per cent”). Similarly, 0.25 becomes 25%. This means moving the decimal point two digits to the left. When going from percentages to decimals, move the decimal point to digits to the right. Thus 8% becomes 0.08.
	In everyday life, percentages are often used to denote interest payments.  For example, if Mike puts $200 in the bank at 5% interest, then after 1 year Mike will have 5% more than he put in. Translating 5% to a decimal we get 0.05. So, Mike will receive 
$200 x 0.05 = $10
in interest and his total bank account will have the value $200 + $10 = $210. Another way to look at this is that his bank account should read 105% of $200. Translating 105% into decimals gives 1.05, so his bank account will have the value: 
200 x 1.05 = $210.
	
FOR YOU:
		1) If Mike put in $300 at 6% interest, how much would he have after one year?
		2) How much after two years?
		3) If Ellen gets a 10% discount on train tickets and the normal price of a ticket is $5.20, how much does she pay? Hint: Find 10% of the price and subtract it.

PERCENTAGE SOLUTIONS
1) 300 x 1.06 = 318

2) 318 x 1.06 = 337.08

3) 10% of 5.20 is 0.52, so the result is 5.20 - 0.52 = 4.68.

FINDING PERCENTAGES

Taxi drivers commonly receive a tip.  In some cities, their parting words are closely related to their happiness with the tip: If the tip is under 15%, they grunt at you. If it’s over 20%, they may even say something nice. How do they calculate the percentage so fast?
Let’s try an example. Suppose the cab fare is $8 and you leave a $1 tip. What percentage is the tip? Expressed as a fraction the tip is 1/8 of the fare. Convert this to decimals by dividing 1.00 by 8. This gives 0.125. Finally, convert this to percentages by moving the decimal point over two places, giving 12.5%.  You get a grunt.
So the basic method is to express the tip as a fraction of the price, convert the fraction to a decimal and the decimal to a percentage.

	FOR YOU:
1)	If Jillian gives a tip of $2.00 on a fare of $14.00, what percentage would she have given?
2)	If a class of 20 students has 13 boys, then what is the percentage of boys?
3)	If the price of gasoline rises from $1.50 per gallon to $2.00 per gallon, how much of a percentage increase is that?

FINDING PERCENTAGE SOLUTIONS
1)	$2 on $14 is 2/14. Expressed as a decimal, this is 0.143 approximately. In percentage, this is 14.3%.
2)	13 out of 20 is 13/20. Expressed as a decimal, this is 0.65, so the class is 65% boys.
3)	The price has increased by $0.50 from $1.50. Expressed as a fraction this is 0.50/1.50, which is the same as 50/150. Expressed as a decimal, this is 0.333 approximately. So, the increase is 33.3%.

HOW TO THINK ABOUT ALGEBRA
	There are two steps to most mathematical problems: 
		a. Write down the right equations
		b. Solve them
	Algebra is especially useful when there is too much to keep track of to do a problem in your head. Even though algebra relies on symbols (usually letters), which you may not have used before in math problems, once you learn algebra, you will likely find that it is one of the simplest, most straightforward methods to solving problems.
	FOR YOU (AND ME):
		a. Given: Mary is five times the age of Bill. Mary is 10. Find: Bill’s age.
		b. The first step – if you are willing to be patient – is to represent the various quantities by symbols: 
M = Mary’s age
B = Bill’s age. 
		c. Next you must identify equations relating these symbols to one another and to regular numbers. This gives: 
M = 10
M = 5B. 
Note that we write 5B to mean “five times B.” We can take out the multiplication sign so that we don’t mistake it for the letter “x” and to be concise. 
		d. Now we use the Rule of Substitution, which says that it is always okay to substitute variables by their equals. In other words, if the same variable equals two values, then those two values equal one another. (Variables are symbols; it is simple to remember because the value of symbols can vary, depending on the problem, while the value of numbers always remains the same.) Using the rule, we can rewrite 
M = 5B as 10 = 5B. 
		e. Next, we use the Rule of Equal Treatment, which states that it is always okay to do the same thing to both sides of an equation. Multiply both sides of this last equation by (1/5). Why (1/5)? Because (1/5) times 5 is 1, and we want to find out the age of a single Bill.
(1/5)(10) = (1/5)(5B) 
by the Rule of Equal Treatment. Solving this, we get 
(1/5)(5B) = (1/5)(5B) = 1B = B 
because multiplication is  associative (so we can move the parentheses around) and 1 times anything is anything. Since 
(1/5)(10) = 2
we get the result that 
2 = B
Since B is Bill’s age, we know that Bill is 2 years old.
	FOR YOU: 
		1) Mary is three times older than Bob. Mary is 24. How old is Bob?
		2) Mary has 5 times as much money as Bob. Mary has $77. How much does Bob have?
		3) Mary has 2.5 times as much money as Bob. Mary has $63. How much does Bob have?
	The Rule of Equal Treatment applies not only to multiplication, but also to addition and subtraction. Consider: Hector is two years more than three times Luke’s age. Hector is 29. How old is Luke? Setting up the equations as before
H = Hector’s age
L = Luke’s age
H = 29
3L+2=H. 
We use the Rule of Substitution to get 3L + 2 = 29. Now, since our goal is to find an equation of the form L = .... we want to get rid of the pesky plus 2 part. So we write 
(3L+2) - 2 = 29-2
by the Rule of Equal Treatment. Performing the arithmetic on each side leads to 
3L=27
(1/3)3L = (1/3)27 or L = 9 
     FOR YOU:
		1) Mary is three years less than five times the age of her grandson Harry. If Mary is 62, how old is Harry?
		2) The Empire State Building is 488 feet taller than the Eiffel Tower. If the Eiffel Tower is 984 feet, how tall is the Empire State Building? What is the ratio of their heights?
		3) Heather has 4.2 times as much gasoline in her tank as Jennifer. If Heather has 11 gallons, how much does Jennifer have?
	The rules extend and become helpful when solving different problems. For example, suppose that Alice hears from a friend that first class concert tickets cost twice as much as regular ones, and then she hears from another friend that they are $41 more. How much are the two kinds of tickets? F = first class ticket price in dollars. R = regular ticket price in dollars. 
F = 2R
F=R+41
Now the Rule of substitution allows us to replace one side of an equality by the other side. We can change 
F = 2R to R + 41 = 2R since F = R+41
By the Rule of Equal Treatment, we can subtract R from both sides. (Parent: This step may confuse your child: “How can you subtract a letter?” You must emphasize the fact that this letter represents a number. We just don’t know yet which number.) This gives 
(R+41) - R = 2R - R or 41 = R
Now we can use the discovered value of R to determine the value of F. Since 
F = 2R
F = 2(41), so F = 82 
	FOR YOU:
		1) Suppose first class tickets cost 3 times as much as regular tickets and also $55 more. How much do these tickets cost? 
		2) First class tickets cost 4.3 times as much as regular ones and $38 more. How much do these tickets cost?
		3) An adult ticket plus two children tickets cost $100. Adult tickets cost $15 more than children's tickets. How much does each kind of ticket cost?

WHICH IS BETTER? 
	Sometimes algebra goes hand-in-hand with estimation. For example, suppose you can pay for two kinds of telephone services: 
    Plan A: $20 fee per month + $0.03 per minute. 
    Plan B: $10 fee per month + $0.05 per minute. 
One question is: If I know how many minutes I will call, which plan is better? Let us try Plan A first. Suppose I’m going to call 800 minutes in a month. Then Plan A gives me 
20 + (800 x 0.03) = 20 + 24 = 44 dollars
Plan B gives me 
10 + (800 x 0.05) = 10 + 40 = 50
So Plan A would be more economical. You could spend the extra $6 on ice cream and gumballs! 
	But sometimes you may not know for how many minutes you will end up talking. In that case, you will want to know the answer to a different question: When is one plan better than another? 
	Observe first that until you talk for a certain number of minutes, Plan B will be better. So we will calculate the Breakeven Point, or the number of minutes when the cost of Plan A equals the cost of Plan B. Above that number, Plan A is better. Below that number, plan B is better.
	FOR YOU (AND ME):
1)	N = number of minutes that will be the Breakeven Point. 
20 + 0.03N = 10 + 0.05N
We want N alone. 
2)	We use the Rule of Equal Treatment to get  
(20 + 0.03N - 10 = (10 + 0.05N) - 10.
Then 10 + 0.03N = 0.05N
Then we use that same rule again: 
(10+0.03N)-0.03N=(0.05N)-0.03N or 10=0.02N. 
3)	Now we must get rid of 0.02. How? We want to multiply it by some number to get 1. So let us make it a fraction. 
0.02 = (2/100)
Reducing this, we get (1/50). So the equation now reads 10 = (1/50)N. We now use the Rule of Equal Treatment: 
(10)50 = (1/50)50N or N = 50
So if you call 500 or more minutes per month, you should use plan A. 
SQUARE LAWS
	One of the most fundamental concepts in physics is the Law of Gravitation. In the version applied to earth, it says that the distance traveled downwards by an object initially at rest is 4.9(t^2) meters, where t is time in seconds.
	Where does the 4.9 come from? It comes from measurement. The number would be larger for a more massive planet like Jupiter and less for a less massive object like the moon or a comet.
	Stated as an equation: Distance traveled = D = 4.9t^2. Now you might say, “Wait a minute. Feathers don’t fall that fast. What about air resistance?”
	You would be right. This equation ignores air resistance. It applies pretty well to lead balls, falling rocks and other dense, massive objects. And for all such objects it applies very well no matter what the weight of the object.
	So let us use the gravitation equation to learn some physics while doing algebra. The beauty of math is that it actually helps us both in real life situations and in understanding science.
	How long would it take to fall 100 meters? 
		D = 100
		D = 4.9t^2
		By the Rule of Substitution: 100 = 4.9t^2
		By the Rule of Equal Treatment: 
(1/4.9)100 = (1/4.9)4.9t^2
		100/4.9 = t^2
	Now what do we do? Well, we have to get rid of the exponent 2. How do we do that? We use the Rule of Equal Treatment and take the square root of both sides:
		(100/4.9)^(1/2) = (t^2)^(1/2) = t
	Now there is the matter of computing (1///4.9)^(1/2). You know how to do that (first divide, then take the square root). The answer should be somewhere between 4 and 5 seconds. 
	FOR YOU:
1)	How long does it take for a lead ball to drop 200 meters?
2)	If a lead ball, B1, is dropped from a height of 200 meters at time 1:00 PM and another ball, B2, is dropped two seconds later from a height of 100 meters, which will reach the ground first?
3)	If ball B1 is at 300 meters and ball B2 is dropped at 600 meters, how many seconds later must B1 be dropped for the two balls to hit the ground at the same time? 
SOLUTIONS
MULTIPLES AND DIVISORS
	
4)	Yes, 8 x 5 = 40

5)	No, 2 x 5,228,369.5 = 10456739. If a decimal is required, that means there will be a remainder so something will be left over.

6)	Yes, 17 x 21 = 357.

FACTORING
1)	25 yields 5, 5

2)	60 yields 2, 30 
then 2, 2, 15
then 2, 2, 3, 5

3)	2,310 because 2, 3, 5, 7, 11 are the smallest prime factors in the number system. 
The product of these factors is 2,310.

COMMON MULTIPLES
4)	Factor 10: 2, 5
Factor 20: 2, 2, 5.
So the factors of the least common multiple are: 2, 2, 5
The resulting product is 20.

5)	Factor 60: 2, 2, 3, 5
Factor 80: 2, 2, 2, 2, 5
So a list of factors that contains both of these is:
2, 2, 2, 2, 3, 5
The product of these factors is 240.

6)	228 yields 12, 19
then 2, 6, 19
then 2, 2, 3, 19
316 yields 4, 79. Since 79 is a prime number and cannot be broken down any further, that is the final factorization.

	GREATEST COMMON DIVISOR
1)	20 yields 2, 2, 5
25 yields 5, 5
So the largest number which both lists have in common is 5.

2)	18 yields 2, 3, 3
25 yields 5, 5
So, since the lists have no numbers in common, only 1 divides them both.

3)	400 yields 2, 2, 2, 2, 5, 5
350 yields 2, 5, 5, 7
Since 2, 5, 5 are the numbers in common, we must find the product, which is 50.

	MULTIPLYING FRACTIONS
1)	(14/6) x (2/2) = (28/12)

2)	(4/6) x ((1/2) / (1/2)) = ((4/2) / (6/2)) = (2/3)

3)	(60/24) x ((1/12) / (1/12)) = (5/2)

ADDING FRACTIONS
1) 6 + 1 = 7, so (6/8) + (1/8) = (7/8)

2) 14 – 11 = 3, so (14/17) – (11/17) = (3/17)

3) The order of operations is important here. Remember that you should always calculate the problem within a parentheses first. In other words, work your way out. 
14 + 7 = 21, so ((14/13) + (7/13)) = (21/13)
21 – 5 = 16, so ((21/13) - (5/13)) = (16/13)

	ADDING FRACTIONS PART II
1) The factors of 2 are 2, 1. (So we can ignore the 1, of course). The factors of 4 are 2, 2. Therefore, the least common multiple has factors 2, 2. The product, 2 x 2 = 4, tells us that 4 should be the denominator used to make the addition simpler. 
(1/2) x (2/2) = (2/4)
(1/4) x (1/1) = (1/4)
(2/4) + (1/4) = (3/4)

2) Factor 8: 2, 2, 2. Factor 6: 2, 3. So the common factors are 2, 2, 2, 3. The product of these factors is 24.
(7/8) x (3/3) = (21/24)
(5/6) x (4/4) = (20/24)
(21/24) + (20/24) = (41/24)

3) Factor 12: 2, 2, 3. Factor 9: 3, 3. So the least common factors are 2, 2, 3, 3. The product of these factors is 36.
(15/12) x (3/3) = (45/36)
(8/9) x (4/4) = (32/36)
(45/36) - (32/36) = (13/36)
	HOW TO THINK ABOUT ALGEBRA
1) M = 24 and 3B = M
Since (M/3) = (24/3), Bob is 8

2) M = 77 and 5B = M
Since (M/5) = (77/5), Bob has $15.40

3) M = 63 and 2.5B = M
Since (M/2.5) = (63/2.5), Bob has $25.20
	PART II
1) M = 62
M + 3 = 5H
(62 + 3) = 65
(65/5) = 13, so H = 13

2) ET = 984
ET + 488 = EB
488 + 984 = 1472 = EB

3) H = 11
(4.2)J = H
J = H/(4.2)
J = (11)/(4.2)
J = 2.6
	PART III
1) F = 3R
F = R + 55
3R = R + 55
2R = 55
R = (55/2) = 27.5

2) F = (4.3)R
F = R + 38
(4.3)R = R + 38
(4.3)R - R = 38
(3.3)R = 38
R = ((38)/(3.3))
R = 11.5

3) A + 2C = 100
A = C + 15
A = 100 - 2C
C + 15 = 100 - 2C
3C + 15 = 100
3C = 85
C = 28.3
A + 2(28.3) = 100
A + 56.6 = 100
A = 100 - 56.6
A = 43.3

POST ALGEBRA
	FOR YOU:
		1) (4/9) + x = 5
		2) (5/3)(13/8) = x
		3) (5/3) y = 5
		4) x + 2y = 30
		2x + y = 21
		5) How many rotations of A will result form one rotation of B?
DIAGRAM 1

POST ALGEBRA SOLUTIONS
1) x = 5 - (4/9)
x = 4 + (5/9)
x = (41/9)

2) (65/24)

3) y = 5/(5/3)
y = (15/5)
y = 3

4) (2x + 4y = 60) - (2x + y = 21) = (0x + 3y = 39)
y = 13
2x = (21-13) 
2x = 8
x = 4

5) 2.5

PROBABLY STOP HERE ------------
SETS
Intersection-elements in common:
{3, 5, 7, 5} ^ {5, 3, 3, 4, 5} = {3, 5, 5}
Union elements in either one:
	{3, 5, 7, 5} u {5, 3, 3, 4, 5} = {3, 5, 7, 5, 5, 3, 3, 4, 5}
Minimum common superset – all elements of set one are in T. All elements in set two are in T.
	Set 1 = {3, 5, 7, 5}
	Set 2 = {5, 3, 3, 4, 5}
	T = {5, 5, 3, 3, 4, 7}
	FOR YOU:
		1) Find union, intersection, and minimum common superset if
	Set 1 = {7, 11, 13, 11, 13}
	Set 2 = {3, 11, 13, 13}
SETS SOLUTIONS
1) Union: {7, 11, 13, 11, 13, 3, 11, 13, 13}
Intersection: {11, 13, 13}
Minimum common superset: {7, 11, 13, 11, 13}