ABE Math Review Package

Transcription

1 P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the topics included here. Read the explanations carefully, and do as many of each type of question in each exercise until you have mastered the material. Section One: Check your answers in the answer key at the end of each topic in section one. Section Two: Check your answers in the answer key on pages - 6. Do not use a calculator, unless you are instructed to. Students wishing to take placement tests for math 00 should complete section one. Students wishing to take placement tests for math 0, math 06 and math 0 should complete section one and two. TABLE OF CONTENTS SECTION ONE page Whole Numbers.. Decimals... 6 Exponents and Roots. Fractions. 6 Percent SECTION TWO Signed Numbers. Add and Subtract signed numbers.. Multiply and Divide signed numbers. Exponents... 6 Order of Operations... Three Rules of Exponents.. Solving Equations add and subtract Solving Equations multiply and divide... 0 More solving equations.. Factoring common factor Factoring difference of squares... Factoring basic trinomials Answer Key for Section Two. Appendix Graphing a Line... VIU/CAP/\\capitan\redirects\h\hunterg\Desktop\math assessment\completemathreview.doc/jan, 00

2 P a g e Whole Numbers Review Before studying fractions, you should be familiar with all operations on whole numbers. This work will provide a quick review. Answers are at the end on p... ADDITION In addition, remember to begin with the right hand column, and work to the left. e.g. + Try these: (a) + (b) + (c) + Remember that when the total of any column is greater than, you must carry the left digit in that total to the column to the left. e.g Here is some practice: (d) + 6 (e) (f) + (g) SUBTRACTION Remember to begin at the right hand column and move column by column to the left. e.g. 6 Do these: (a) 6 (b) 6 (c) Remember that you should check the accuracy of your answer by adding it to what you took away. If it is correct, your result should be the number you began with. In the example above, + 6, so we can assume that it is correct.

3 P a g e Remember that when a digit in the bottom number is too big to subtract from the digit in the top number, you borrow from the next column in the top number first. e.g. You can not subtract from, so you borrow ten from the tens column first. Cross out the in the tens place, and replace it with.. Borrow 0 from to make in the ones column.. Subtract from. Subtract from. Try these: (d) 0 (e) 0 (f) 0 (g) (h) (i) 0. MULTIPLICATION Take the time to learn your multiplication tables. That will save you a lot of time later on! Multiply each digit in the top number by the bottom number. Write the answer from right to left, starting in the ones column. e.g. x x 6 Remember that any number multiplied by 0 is 0. Try these: (a) 0 x (b) 0 x (c) 0 x When you multiply by a two or three digit number, be sure to begin your answer under the ones column. Then continue on the line below the tens and then the hundreds. Always multiply from right to left. Leave a space under the bottom digit that has already been multiplied OR add a zero in that space to keep the columns lined up neatly and correctly. Carrying in multiplication is like carrying in addition. Multiply first and then add the number being carried. Line up the digits carefully under the correct column. e.g. 6 0 x x x _

4 P a g e Now try these: (d) x 6 (e) 06 x 0 When you multiply by 0, 00, 000, etc, add the same number of zeroes to the right of the number. e.g. x 0 0 x x Tricky!. DIVISION You will need your multiplication tables here too! Learn them well! Division is the opposite of multiplication. That helps you to check your division answer. Pay careful attention to lining up digits, as you did in multiplying large numbers, so that you can keep your working straight. Here are the words: 6 (the dividend) divided by (the divisor) is (the quotient) OR into 6 is. e.g. 6 Step : Divide the divisor into 6. Place the above the 6. - Step : Multiply x. Place the under the 6. Step : Subtract: 6. - Step : Bring down the next number to the right 0 Step : The new number is. Step 6: Divide into. Multiply by the divisor and the answer is. Step : Subtract from 0. The division is complete. Check: In division, it is easy to check the answer by multiplying your answer Division problems (the quotient) do not by always the number work out you with are an dividing even answer by (the as divisor). in the last example. x 6 so the answer is likely to be correct. Sometimes there is an amount left over, which is called the remainder. It is placed on the top line with the letter r for remainder and is part of the quotient. e.g. 6 r Try these: (a) 6 (b) 06 (c) 00

5 P a g e When you check your answer for a division problem with a remainder, multiply your answer (the quotient) by the number you are dividing by (the divisor) and add the remainder. When you divide by a double or triple digit number, you need to use another different skill. You need to estimate, which is a process of thoughtful guessing. This takes time and practice. e.g Step : Think of about how many times goes into 6. To do that, round to 0 ( 0) and divide that into 6. The answer is approximately or close to. Step : Place above the last digit of 6. Step : x (the divisor) 6. Then Step : Bring down. Estimate how many times, 0, divides into 6. Place on the top line. Step : x Step 6: Bring down. Estimate how many times, 0, divides into. Sometimes your estimate is not quite right, but it should only be more or less than the correct number, so the estimate gives a good starting point. Here is an example with a remainder: e.g. r Step : Round 0. Divide 0 into the first part of the number. Step : 0 goes into about times. Place above the last digit of. Step : x. Subtract from. Step : Bring down. Estimate how many times 0 goes into. Step : Multiply x. from. Step 6: Bring down. Estimate how many times 0 goes into. x. Step :. Divide into. It cannot divide, so is the remainder. Try these: (d) (e) 6 60 (f) 6 6 (g) 6 ANSWERS Addition: (a) (b) 6 (c) 06 (d) (e) 0 (f) 0 (g) Subtraction: (a) (b) (c) 6 (d) (e) (f) 6 (g) 66 (h) 6 (i) 0 Multiplication: (a) 0 (b) (c) 0 (d) 6 (e) 06 Division: (a) 0 r (b) r (c) r (d) 6 (e) r (f) 0 (g) r

6 P a g e 6 DECIMAL REVIEW A. INTRODUCTION TO THE DECIMAL SYSTEM The Decimal System is another way of expressing a part of a whole number. A decimal is simply a fraction with a denominator of 0, 00, 000 or etc. The number of decimal places refers to how many zeros will be in the denominator. Note that the number.6 is read as five point six two. The first decimal place refers to tenths The second decimal place refers to hundredths The third decimal place refers to thousandths Similarly, six decimal places would be a fraction with a denominator of (millionths). The most common usage of decimals is in our monetary system where 00 cents ( decimal places) make up one dollar. For example, $. is really two dollars and forty-one hundredths of a dollar. ( ) 00 Examples:. Change.0 to a fraction Notice that.0 is the same as. In fact, etc.. Change 0. to a fraction Notice that 0.. The zero in front of the decimal place is not needed.. Change.00 to a fraction Notice that the zeros in this example are important.. Simplify Notice that zero at the end or zero as a whole number (to the left of the decimal) is not needed

7 P a g e B. OPERATIONS WITH DECIMALS. Addition: Add $.0 and $. Place numbers in columns, so that the decimal places are in line. Place the decimal point in the same line for the answer. Now, add as if adding whole numbers. Add. and. and. Add zeros to fill in columns. This will not change the value of the number (see examples on previous page.) $.0 + $. $ Subtraction: Subtract $.0 from $. Set up columns as in addition, and subtract as if subtracting whole numbers. Calculate minus. Fill in columns with zeros, as in addition. Note the difference in wording in these two examples. $. - $.0 $ Multiplication: Multiply. by. At first, ignore the decimals, and multiply as if calculating x. Now, add up the decimal places in both numbers and your answer will have that total number of decimal places. Product of 0.0 and.0 Multiply. Add enough zeros to show the correct number of decimal places. ( places) x. ( place) 0.0 ( places).0 ( places) x.0 ( places).00 (total 6)

8 P a g e Multiply.6 by 000 Notice.6 x , so multiplying x 000 (0 decimal places) is the same as moving the decimal place places (since 000 has zeros) to the right. Similarly,. x x x..0. x Division: Divide. by. The divisor (.) must be changed to a whole number () by moving the decimal point places to the right, in both numbers. In the answer, place the decimal point directly above the numbers. Note that zeros must be added to complete the division ( places) x 000 (0 places) (total ) or 6. Divide.00 into Firstly, notice the difference in wording in these two division questions. Zero must be added to the number in order to move the decimal places to the right Divide. by 000 Notice So, dividing by 000 is the same as moving the decimal places to the left (since 000 has zeros).

9 P a g e Similarly,

10 P a g e 0 EXERCISE : Decimal Operations Do not use a calculator. a) Change to fractions: (remember to reduce) ). ).00 ) 0.0 ) 0.0 ).0 b) Addition: ). +. ) 0.0 plus.00 ). and and.6 ) ).6 more than.0 c) Subtraction: ).06 from. ). from ) 60. minus. ) 0.00 minus 0.0 ).6 decreased by. d) Multiplication: ). by.0 ). x. ). times.6 ) Product of.00 and.00 ). of 6). by 000 ). by.00 e) Division: ) divided by 0. ). divided by.0 ) 0. into. ) 0.0 divided by.0 ).00 into. 6).6 divided by 000 ).6 divided by.0

11 P a g e C. CONVERSION AND ROUNDING OFF. Converting Decimals to Fractions: When changing decimals to fractions, simply create a fraction with 0, 00, 000 etc. in the denominator. The number of zeros in the denominator is the same as the number of decimal places Converting Fractions to Decimals: When changing a fraction to a decimal, simply divide the denominator into the numerator. Every time a fraction is changed to a decimal, the division will either stop (as in 0.) or the division will go on forever by repeating (as in. 6 ). Notice that a repeating decimal is shown by a dot over the number (if only one number repeats) or as a bar (if more than one number repeats, as in. ) Repeating Decimals Following is a list of some repeating decimals: or.. or.. or.. or.. or.. or.. Rounding off: If we want to divide $.00 into equal parts, we would want our answer to be to the nearest cent (or nearest hundredth). Since our answer is closer to 6 cents than 66 cents, we would round off our answer to $ When rounding to the nearest thousandth, we.6 to nearest thousandth.6

12 P a g e want our answer to have decimal places. If the fourth decimal place has a or greater, round up. If less than a, do not round up..6 to nearest hundredth.6. to nearest tenth.0 EXERCISE : Conversion and Rounding Off a) Change to Fractions: (Remember to reduce whenever possible) b) Change to Decimals: (Do not round off) c) Round Off:..6 to nearest hundredth..6 to nearest thousandth.. to nearest tenth..0 to nearest tenth.. to nearest hundredth 6.. to nearest whole number.. to nearest thousandth.. to nearest hundredth to nearest thousandth

13 P a g e ANSWERS EXERCISE : Decimal Operations ) a) ) 0 ) 0 ) 0 00 ) 0 b) ). ).0 ) 6. ). ). c) ). ). ) 0. ).00 ). d) ).00 ).06 ). ).00 ) 6). ).00 e) ) 0 ) 0 ) 0. ).06 ) 0 6).6 ) 6 EXERCISE : Conversion and Rounding Off a) ) ) ) 0 ) b) ). ). ). ).6 ). 6). 6 ). ). 6 ). 0). c) ).6 ).6 ). ). ).0 6) ). ).6 ).

14 P a g e EXPONENTS AND ROOTS A. EXPONENTS In math, many symbols have been developed to simplify certain types of number expressions. One of these symbols is the exponent. Rule: An exponent indicates how many times a base number is used as a factor. Example: 0 0 x 0 x 0 x 0 or (0) (0) (0) (0) x x x x x (x) (x) The exponent is written smaller and is placed above the base number (the number to be multiplied). The first example can be read ten exponent four or ten to the power of four. Second and third power have special names: second power is usually called squared, and third power is usually called cubed. Example: is five squared is five cubed A simple way to work out exponents is to write the base digit the same number of times as the value of the exponent, and put a multiplication sign between each digit. Example: x x x x (the exponent is, so write five times) 0 PRACTICE A: Write as an exponent or power (also called exponential notation ):.).).) 0 0.) Evaluate the following:.) 6.).).).) 0 0.).).).).) 0 +

15 P a g e B. SQUARE ROOTS Example: (the square root of ) because () (the square root of ) because () The square of a number can be positive (+) or negative ( ). This is because two positive numbers multiplied together or two negative numbers multiplied together make a positive number. However, the symbol tells us we should write only the positive root. Example:, but remember, (+) or ( ), but either (+) or ( ) PRACTICE B: Rule: Taking the square root of a number is the reverse of squaring the number. The symbol for square root is Evaluate the following:.).).).).) 00 6.) 0000.) 6 6.).) 0.) Practice A (exponents): ANSWERS.).).) 0.) 6.) x x x 6 6.) x x.) x x x.) x x x x.) 0 x ) x x x x x x.) x 6.) x 6 Practice B (square roots).) (0 x 0 x 0) + ( x ) 06.) x 6.).).).).) ) 00.).) 6.) 00 0.) 0

16 P a g e 6 FRACTION REVIEW A. INTRODUCTION. What is a fraction? A fraction consists of a numerator (part) on top of a denominator (total) separated by a horizontal line. For example, the fraction of the circle which is shaded is: Fraction numerator denominator (parts shaded) (total parts) In the square on the right, the fraction shaded is and the fraction unshaded is. Equivalent Fractions Multiplying The three circles on the right each have equal parts shaded, yet are represented by different but equal fractions. These fractions, because they are equal, are called equivalent fractions. Any fraction can be changed into an equivalent fraction by multiplying both the numerator and denominator by the same number x x or x x so Similarly x 0 x or x x so 0 You can see from the above examples that each fraction has an infinite number of fractions that are equivalent to it.

17 P a g e. Equivalent Fractions Dividing (Reducing) Equivalent fractions can also be created if both the numerator and denominator can be divided by the same number (a factor) evenly. This process is called reducing a fraction by dividing a common factor (a number which divides into both the numerator and denominator evenly) Simplifying a Fraction (Reducing to its Lowest Terms) It is usual to reduce a fraction until it can t be reduced any further. A simplified fraction has no common factors which will divide into both numerator and denominator. Notice that, since and have a common factor of, we find that is an equivalent fraction. But this fraction has a factor of common to both numerator and denominator. So, we must reduce this fraction again. It is difficult to see, but if we had known that was a factor (divides into both parts of the fraction evenly), we could have arrived at the answer in one step e.g. 60

18 P a g e. EXERCISE : Introduction to Fractions a) Find the missing part of these equivalent fractions x ) 6 ) ) 0 Example: 0 x ) ) 6 6.) Since x 0, multiply the numerator by, also. ) ) 0 00 So, 6 0 b) Find the missing part of these equivalent fractions. ) 6 ) Example: 0 ) 6 ) 0 Since 0 divide the numerator by, also. ) 0 6) So, 0 c) Simplify the following fractions (reduce to lowest terms). ) ) ) 0 66 ) 6) 0) 0 ) ) 6 6 ) ) 0 6

19 P a g e B. TYPES OF FRACTIONS. Common Fractions A common fraction is one in which the numerator is less than the denominator (or a fraction which is less than the number ). A common fraction can also be called a proper fraction. e.g.,,, are all common fractions.. Fractions that are Whole Numbers Some fractions, when reduced, are really whole numbers (,,, etc). Whole numbers occur if the denominator divides into the numerator evenly. e.g. is the same as or 0 is the same as 0 6 or 6 So, the fraction 0 is really the whole number 6. Notice that a whole number can always be written as a fraction with a denominator of. e.g Mixed Numbers A mixed number is a combination of a whole number and a common fraction. e.g. (two and three-fifths) (twenty-seven and two-ninths) 6 (always reduce fractions)

20 P a g e 0. Improper Fractions An improper fraction is one in which the numerator is larger than the denominator. From the circles on the right, we see that (mixed number) is the same as (improper fraction). An improper fraction, like, can be changed to a mixed number by dividing the denominator into the numerator and expressing the remainder () as the numerator. e.g. 6 A mixed number can be changed to an improper fraction by changing the whole number to a fraction with the same denominator as the common fraction. 0 and 0 0 and A simple way to do this is to multiply the whole number by the denominator, and then add the numerator. e.g. x 0 0 x 6 0. Simplifying fractions All types of fractions must always be simplified (reduced to lowest terms). e.g. 6,, Note that many fractions can not be reduced since they have no common factors. e.g.,,

21 P a g e 6. EXERCISE : Types of Fractions a) Which of the following are common fractions (C), whole numbers (W), mixed numbers (M) or improper fractions (I)? ) ) ) ) ) 6) ) ) ) 0) b) Change the following to mixed numbers: ) ) ) 0 6 ) ) 00 6) c) Change the following to improper fractions: ) ) 6 ) ) ) 6) d) Simplify the following fractions: ) 0 ) 0 0 ) ) ) 6) 0

22 P a g e C. COMPARING FRACTIONS In the diagram on the right, it is easy to see that is larger than (since is larger than ). However, it is not as easy to tell that is larger than 6. In order to compare fractions, we must have the same (common) denominators. This process is called Finding the Least Common Denominator and is usually abbreviated as finding the LCD or LCM (lowest common multiple). Which is larger: or 6? In order to compare these fractions, we must change both fractions to equivalent fractions with a common denominator. To do this, take the largest denominator () and examine multiples of it, until the other denominator (6) divides into it. Notice that, when we multiply x, we get, which 6 divides into. Now change the fractions to th s. When we change these fractions to equivalent fractions with an LCD of, we can easily see that is larger than since is greater than 0. 6 x (6 doesn t divide into ) x 6 (6 doesn t divide into 6) x LCD x x x x 0 6

23 P a g e Which is larger: or? Examine multiples of the larger denominator () until the smaller denominator divides into it. This tells us that the LCD is 6. Now, we change each fraction to equivalent fractions with the LCD of 6. x x x 6 (LCD) LCD is 6. x 6 6 x x 6 x So, is larger than Which is larger: or or? Find the LCD by examining multiples of. Notice that, when we multiply x, we find that 60 is the number that all denominators divide into. x x 0 x x 60 (LCD) x x x x x x So, is the largest fraction.

24 P a g e Which is larger: or? Notice that one denominator () divides into the other denominator (). This means that the LCD and we only have to change one fraction to an equivalent fraction. x x So, is larger than. EXERCISE : Comparing Fractions Which is the largest fraction? (find LCD first) ) or 6 ) or 0 0 ) or 0 ) or ) or 6) or 6 or ) or ) or or ) or 6

25 P a g e D. ADDING FRACTIONS There are four main operations that we can do with numbers: addition ( + ), subtraction ( ), multiplication ( x ), and division ( ). In order to add or subtract, fractions must have common denominators. This is not required for multiplication or division.. Adding with Common Denominators To add fractions, if the denominators are the same, we simply add the numerators and keep the same denominators. + e.g. Add and Since the denominators are common, simply add the numerators. + 6 Notice that we must reduce the answer, if possible.. Adding When One Denominator is a Multiple of the Other Add and x 6 x Notice that the denominators are not common. Also notice that is a multiple of (since x ). This means that the LCD (see the last example in Comparing Fractions )

26 P a g e 6. Adding Any Fraction Add and We must find a common denominator by examining multiples of the largest denominator. We find that the LCD Add 6 and When adding mixed numbers, add the whole numbers and the fractions separately. Find common denominators and add If an improper fraction occurs in the answer, change it to a common fraction by doing the following. total equals +. The Language of Addition + CAN BE WORDED plus and total of and sum of and addition of and combined with more than (or greater than) Note: All of these can be worded with the fractions in reverse order: e.g. plus is the same as plus

27 P a g e. EXERCISE : Adding Fractions a) Add the following: ) + ) + ) + ) + + b) Find the sum of: ) + ) + ) + 6 ) + c) Add the following: ) + ) + 6 ) ) ) ) 6 6) 6 ) d) Evaluate the following: ) and ) total of 6 and ) plus ) combined with ) greater than 6) sum of 6 and

28 P a g e E. SUBTRACTING FRACTIONS. Common Fractions As in addition, we must have common denominators in order to subtract. Find the LCD; change the fractions to equivalent fraction with the LCD as the denominator. Then subtract the numerators, but keep the same denominator. - - or 6 -. Mixed Numbers When subtracting whole numbers, subtract the whole numbers, and then subtract the fractions separately. - However, if the common fraction we are subtracting is smaller than the other common fraction, we must borrow the number from the large whole number. i.e. To subtract + +, or from 6, first change the common fractions to equivalent fractions with the LCD. Since is smaller than, borrow from

29 P a g e. The Language of Subtraction minus 6 from 6. EXERCISE : Subtracting Fractions a) Subtract the following: (NOT minus) subtracted from 6 decreased by or lowered by 6 - CAN BE WORDED 6 less than 6 the difference of 6 and NOTE: Unlike addition, we can not reword the above with the fractions in reverse order: i.e. - is NOT the same as - ) - ) - 6 ) - 6 ) - ) - 6 6) 6 - ) - ) - b) Subtract the following: ) 6 - ) - ) - 6 ) - ) 6-6) - 6 c) Find the following: ) What is minus? ) decreased by is what? 6 ) What is less than? ) What is from? 6

30 P a g e 0 F. MULTIPLYING FRACTIONS. Common Fractions When multiplying fractions, a common denominator is not needed. Simply multiply the numerators and multiply the denominators separately. x 0 Sometimes, we can reduce the fractions before multiplying. x Any common factor in either numerator can cancel with the same factor in the denominator. Multiply after cancelling (reducing). Note that any whole number (6) has the number understood in its denominator. x 6 x If more than two fractions are multiplied, the same principles apply. x x Mixed Numbers Mixed numbers must be changed to improper fractions before multiplying. Remember that a mixed number (like ) can be changed to an improper fraction by multiplying the whole number () by the denominator () and then adding the numerator. See page 6 for instructions on changing a mixed number to an improper fraction. x x x or x

31 P a g e. The Language of Multiplication multiplied by of x CAN BE WORDED. EXERCISE 6: Multiplying Fractions a) Multiply (cancel first, when possible): by the product of and NOTE: When multiplying, it doesn t matter which fraction is first. i.e. x is the same as x ) x ) x 6 ) x ) ) 0) x 6 x x 0 x 0 x 6 x x x x ) ) ) x x x 6) ) ) 6 x x 6 x x 6 b) Find the following: ) What is of? ) by is what number? ) What is times? 0 ) of is what number? ) 6 of is what number? 6

32 P a g e G. DIVIDING FRACTIONS To divide fractions, we invert (take the reciprocal of) the fraction that we are dividing by, then cancel (reduce), and then multiply. Taking the reciprocal of a fraction involves flipping the fraction so that the numerator and denominator switch places. Note that a whole number is really a fraction (e.g. ). reciprocal reciprocal reciprocal. Common Fractions Simply invert (take the reciprocal of) the fractions that we are dividing by ( ). Then cancel and multiply. Note: you can only cancel after the division is changed to a multiplication Mixed Numbers As in multiplication, mixed numbers must be changed to improper fractions.

33 P a g e. The Language of Division CAN BE WORDED divided by into NOTE: In multiplication, the order of the fractions was not important. i.e. x is the same as x divide by In division, this is not the case. The order of the fractions is important. Consider the following: but. EXERCISE : Dividing Fractions a) Divide: ) ) ) 6 ) ) 6) 0 0 ) 0) ) 0 ) b) Find the following: ) divided by ) divided by ) into

34 P a g e ) Divide by ) into 6) Divide by

35 P a g e FRACTION REVIEW: Decide which operation ( +, -, x, ) by the wording in the question. Then find the answer. ) What is of 0? ) How much is from ) How much is from? ) How much is of? 6? ) and equals.? 6) divided by is what number? ) What is 0 of 00? ) What is into? ) from equals.? 0) What is by? ) How much is of? 0 ) Find the total of and and? 6 ) of equals.? ) What is greater than?

36 P a g e 6 ANSWER KEY FRACTION REVIEW EXERCISE : Introduction to Fractions (Page ) a) ) 6 ) ) 0 ) 6 ) 6) 0 ) 00 ) b) ) ) ) ) ) 6 6) 0 c) ) ) ) ) ) 6) ) ) ) 0) EXERCISE : Types of Fractions (Page 6) a) ) C ) M ) I ) W ) W 6) M ) W ) I ) C 0) W b) ) ) ) ) 6 ) 6) c) ) ) ) 6 6 ) ) 6) d) ) 0 ) ) ) ) or 6) or EXERCISE : Comparing Fractions (Page ) ) ) ) 0 0 ) ) 6) ) ) ) EXERCISE : Adding Fractions (Page ) a) ) ) or ) ) or b) ) ) ) 6 ) c) ) ) ) 6 ) 0 ) 6) 6 ) ) 6 d) ) ) ) 0 ) ) 6)

37 P a g e EXERCISE : Subtracting Fractions (Page ) a) ) ) 0 ) ) ) 6) ) ) b) ) ) 6 ) ) ) 6 0 6) c) ) 6 ) ) ) EXERCISE 6: Multiplying Fractions (Page 6) a) ) ) ) ) 6 ) 6) ) ) 0 ) or 6 0) ) ) or b) ) 0 ) ) or ) or ) 6 EXERCISE : Dividing Fractions (Page ) a) ) ) ) ) ) 6) ) 0 ) ) 0) b) ) ) 6 ) ) ) 6) 0 Fraction Review (Page ) a) ) ) ) 6 ) ) 6) ) 0 ) 0 ) 6 0) ) 0 ) ) )

38 P a g e PERCENT REVIEW The use of percentage is another way of expressing numbers (usually fractions) in such a way as to make comparisons between them more obvious. For instance, if you get out of 0 in test A and out of 0 in Test B, it may not be clear whether you have improved or not. The use of percentage will allow this comparison, because a percent is part of 00. (i.e. a percent is a fraction with a denominator of 00). A. CHANGING % TO FRACTIONS / DECIMALS A percent means a part of 00. For example, if you get % on a test, your mark was out of 00. A percent can be changed to a fraction or decimal by simply dividing the percentage number by 00.. Changing % to Fractions Divide by 00. (i.e. put the % number over 00 and reduce if necessary) % 00 0% 0 00 % 00 6 If a decimal appears in the fraction, multiply the fraction by 0, 00, 000 etc. to produce an equivalent fraction without decimals. 6.% x 0 0.% x Changing % to Decimals Simply divide by 00. (i.e. move the decimal 0%.0 or. point places to the left).%.0 %.0 %.

39 P a g e B. CHANGING FRACTIONS / DECIMALS TO PERCENTS. Changing Fractions to Percent If you get out of 0 on a test, it is convenient to change this mark to a percentage.? This means changing to an equivalent fraction with 00 as denominator (i.e. ) To change fractions to %, simply multiply the fraction by 00%. 00 x 00% % x 00% 0% x 00% % 0 x 00% % or.% 0 Note: The mathematical wording for changing a fraction ( ) to a percent would normally be: 0 is what % (out) of 0? or What % is (out) of 0? The word out is usually not included. e.g. is what % of? What % is of? x 00% % x 00% 0%. Changing Decimals to Percents To change decimals to percents, simply multiply by 00% (i.e. move the decimal point places to the right.).. x 00% %.6.6 x 00%.6%.. x 00% 0%

40 P a g e 0 C. USING PERCENTS When percents are used in calculations, they are first converted to either fractions or decimals. Usually it is more convenient to change % to decimals.. Multiplying With Percents If a test mark was 0% and it was out of 0 total marks, what was the test score? 0% (out) of 0 is what number? 0% test score? 0 (total marks ) 0% x 0. x 0 0 So 0% 0 marks 0 What number is 0% (out) of 0? To find the test score, or the part, we multiply the % by the total. e.g. % of is what number? % x. x. What number is 0% of.? 0% x.. x..6. Dividing with Percents If a test mark was 0% and you received a score of 0 marks, what was the test out of? 0% 0 marks total? 0 0% 0. 0 or 0 0 x 0 0% (out) of what number is 0? 0 is 0% (out) of what number? So, 0% 0 0 total marks To find the total marks, we divide by the %. e.g. 0% of what number is? 0%.0 6. is % of what number? %.

41 P a g e D. SUMMARY AND EXERCISE. Three types of Percent Problems In summary, there are three things that we can do with percent. We will use the example on the right side of the page to summarize. 0% 0 (part) 0 total. Finding % or what % of 0 is 0? x 00% 0%. Finding the Part or 0% of 0 is what number? 0% x 0. x 0 0. Finding the total or 0% of what 0 0% 0. 0 number is 0?. EXERCISE: PERCENT PROBLEMS. Change to Fractions a) % b) % c) 0% d).% e).% f) 0% g) %. Change to Decimals a) % b).% c) % d).% e) 0.%. Change to % a) 0 b) c) d) e) f) 0 g) 0.6 h). i).00 j)

42 P a g e. Finding % a) What % of is? b) 6 is what % of 0? c) What % of 0 is.?. Finding the Part a) 0% of is what number? b) What number is 6.% of 0.? c) 6% of is what? 6. Finding the Total a) 0% of what number is? b) is % of what number? c) 0 is 0% of what number?. Percent Problems Combined a) What % of is? b) 0% of is what number? c) is 0% of what number? d) 0 is what % of? e) 0% of what number is 0? f) What number is % of 0?

43 P a g e ANSWERS. a) 00 b) 0 c) d) 000 e) 00 f) g) 00. a). b).0 c).0 d). e).00. a) % b) % or 66 % c) % d). % e). % f) % g) 6.% h) % i).% j) %. a) % b) 0% c) 6. 6 %. a). b). c). 6. a) 0 b). c) 0. a) 0% b) 0. c) d) 6. 6 % e) f)

44 P a g e Section ALGEBRA REVIEW PART - SIGNED NUMBERS In the REAL NUMBER SYSTEM, numbers can be either positive or negative. Positive can be written as: +, or (+), or just. Negative can be written as:, or ( ). Part - ADDING and SUBTRACTING SIGNED NUMBERS. When adding numbers of the same sign, simply put the numbers together and carry the sign. ( + ) + ( + ) +6 ( ) + ( ) 6 6 ( ) Note: This reads combined with. When adding numbers of different signs, take the difference between the two numbers; carry the sign of the number with the largest absolute value. For example: +. The difference is. Take the sign of the. Answer ( ) ) NOTE: These two examples are the ) + same, just switched around! ) + ) + ( ) EXERCISE ON PART : ADDING AND SUBTRACTING SIGNED NUMBERS ) + ) + (-) ) / + / ) 6 + (-) ) 0 0 6) ) 6 ) + (-) ) / - / 0) ( ) + () ).. + (.0) ) + ( ) + + ( )

45 P a g e Part : MULTIPLYING AND DIVIDING WITH SIGNED NUMBERS. The rules for multiplying and dividing two numbers at a time are the same. They are as follows: I. When the signs are the same the answer is positive. II. When the signs are different the answer is negative. ) 6 ( ) 0 ) 6 ( ) 0 ) ( )() ) ) ) ( ). - ) ) - 0. REMOVING BRACKETS When doing operations with more than two signed numbers, remove brackets first before doing any operations. Then, simply take two numbers at a time. To remove brackets: if a positive sign is outside the brackets simply remove the brackets. e.g. +() becomes and +( ) becomes if a negative sign is outside the brackets, change the sign of the number in the brackets. e.g. (+) becomes and ( ) becomes ) + ( ) ( ) + ) ( ) () + (-)

46 EXERCISE ON PART : OPERATIONS WITH SIGNED NUMBERS ) ) ) ) 6. x. ) 6) - ) ( ) ( ) ) + ( ) (6) ) ( ) + ( ) ( ) P a g e 6 0) ( ) ( ) (6) ) ( ) + ( ) + ( ) ) + ( ) + ( ) PART : EXPONENTS Before looking at ORDER OF OPERATIONS, it is necessary to understand how exponents function. Consider the following: is the exponent is the base This is read to the third power, and it means: x x As an introduction to algebra, the same is done with letters: Y Y Y Y Special case : Anything to the zero power is equal to. a) () 0 b) ( ) 0 c) x 0 d) ( x) 0 Examples: ) ( ) - ) Note ) ( x ) but ( ) x + ) (x) x EXERCISE ON PART : EXPONENTS Evaluate or write without brackets. ) ) ) ( ) 6 ) ( ) ) (x) 6) (6x) ) ) ) (y) 0) ( xy) ) ( z)

47 P a g e PART : ORDER OF OPERATIONS When different operations are combined in one problem, they must be done in a certain order. The acronym for remembering the order of operations is BEMA or BEDMAS. B - Brackets. All operations inside brackets must be done first. For example: ( + ) () 6. E - Exponents. An exponent indicates how many times to multiply a number by itself. Any exponents must be carried out next. For example: ( + ) ( + ) () M - Multiplication and Division must be carried out before adding and subtracting. For example: A - Adding and Subtracting are done last, as in the previous example. For example: Fill in the missing steps to this example: (0 6 ) (0 ) ( 6 ) 0 EXERCISE ON PART : ORDER OF OPERATIONS ) ( )( ) ) ( ) + -0 ) ( ) ) ( ) + ) 6 [ ( )] 6) 6 + ) ( ) + 6 (-) ) ) ( ) ( ) ( ) () 0) [ ( ) ] ( )( 0) ) (6) [ ( ) ] ) (.) + ( )

48 P a g e PART 6: THE THREE RULES OF EXPONENTS Rule : WHEN MULTIPLYING WITH THE SAME BASE, ADD THE POWERS. ) x x x + x ) + ) a a a a ++ a 6 ) y y y + y NOTICE, EVEN THOUGH THE EXPONENT OF IS NOT WRITTEN, IT IS STILL THERE AND MUST BE ADDED. When multiplying two terms such as x and x, multiply the numbers and add the powers. ) (x ) (x ) 0X + 0x 6 6) ( ab ) ( a b) a b Rule : WHEN DIVIDING WITH THE SAME BASE, SUBTRACT THE EXPONENTS ) - or ) a a - a a ) b 0 b or ) 0x a a x b x a Rule : WHEN A POWER IS OUTSIDE BRACKETS, THE POWERS ARE MULTIPLIED.) (c ) c c 6.) (x ) x 6.) (x y ) x y EXERCISE ON PART 6: USING EXPONENTS ) ) x x x ) a a ) z z ) z z 6) 6a z a z ) ( ab ) ( b) ) a a a ) (m ) 0) (x ) ) ( xy) ( 6x ) ) ) y (x y) x (x y ) ) x xy ) ( a b) + ( a b ) xy 0xy

49 P a g e PART : SOLVING EQUATIONS - ADDITION AND SUBTRACTION An equation is solved when the unknown letter is isolated on one side of the equal sign. When isolating x, the equation must be kept balanced. To maintain balance, you must always do the same thing to both sides of the equation. For example: Solve for x: x + 6 to isolate x, we must subtract 6 from the left, and thus from the right. x x 6 Another example: x + x x [Note: we want to solve for +x, not x] x EXERCISE ON SECTION : EQUATIONS WITH ADDITION AND SUBTRACTION Solve for the missing letter. ) z ) a ) 6 y ) 0 y 6 ) ( x) + 6).6 0. x ) 6 a ).6 x +. ) (6 + x ) + ( ) 0) x + 0.

50 P a g e 0 PART : SOLVING EQUATIONS - MULTIPLICATION AND DIVISION These types of equations always have a number greater than in front of the letter, or unknown. In the previous type of equation all you had to do was get your letter on one side of the equals sign and all the numbers on the other, and you were done! This is always your first step! Now, if your coefficient is greater than, you must divide both sides of the equal sign by the number in front of the letter. Here are three examples of how to tell what the coefficient is: e.g. x e.g. x e.g. x coefficient ( ) coefficient coefficient In this section, we will just concentrate on the second step for solving equations. In the next section, we will combine both steps. ) 6z ) x ) x 6 z 6 6 x x z x x EXERCISE ON PART : EQUATIONS WITH MULTIPLICATION AND DIVISION ) x 6 ) y ) 6 z ) x ).6b. 6) x ) t ) 0 x ) r 0) x (.06)

51 P a g e PART : SOLVING EQUATIONS BOTH METHODS TOGETHER Before we look at the combination of both methods, we must first review adding and subtracting like terms. When adding terms that have the same letter and same exponent, add the coefficients and carry the letter. e.g. x + x x e.g. 6x x x e.g. y y y TWO STEPS TO SOLVE EQUATIONS:.) COLLECT ALL NUMBERS ON ONE SIDE OF THE EQUAL SIGN, AND COLLECT ALL LETTERS ON THE OTHER SIDE..) SIMPLIFY AND DIVIDE BY THE NUMBER IN FRONT OF THE LETTER. EXERCISE ON PART : EQUATIONS WITH BOTH PRINCIPLES. ) x + 6 ) x + 6 ) y 0 ) Solve: x + x x x x ) x x + x + x + + x x + x + x x + x + x x x ) t + ) x + x 6) y y ) x + x ) x + x + ) x + x 0) + x 6 x + x ) y + y y + y

52 P a g e PART 0: INTRODUCTION TO FACTORING Key Concept: In algebra, when we refer to a term it means to add and subtract to find a sum. In algebra, when we refer to a factor in algebra it means to multiply to find a product. Factoring is a process for changing a term or group of terms into a multiplication. A number has factors, and the factors for the number 0 are: and 0; and 0, and and because * 0 0 and * 0 0 and * 0. Algebraic expressions or groups of terms that contain numbers and letters can also be factored. ) Common Factoring: Always look for a common factor when factoring algebraic expressions a) Common factoring can involve a common number (the highest number which divides evenly into every term in the expression. Example: x (x - ) note the number is common to both terms b) Common factoring can involve a common letter (a letter that appears in every term in the expression Example: x - xy x ( y) note that the letter x is common to both terms c) Common factoring can involve common letter, common letters and exponents (see section ) Example: x x x (x ) note that number and letters are common, and the lowest power (exponent) to appear is its common factor Try these: ) 6x 6 ) a 60 ) ab + ac ) 6z y ) x 6x 6) 0x y 0xy + 0y

53 P a g e PART : FACTORING DIFERENCE OF SQUARES A square is a number, or letter (variable) multiplied by itself. Here are some examples of perfect squares: * * x x * x a b ab * ab Here are two examples of difference of squares, and you will notice that the squares are subtracted from one another. This is why this type of factoring is called the difference of squares. x ( ) ( ) (always set up your brackets first) (x ) (x + ) or (x + ) (x - ) (Note: the order of your brackets does not matter but one bracket always contains + and the other bracket contains -) y ( ) ( ) (y - ) (y + ) or (y + ) (y - ) When factoring a difference of squares, always look for a common factor first: x 00 (x ) (x - ) (x + ) Try factoring these: ) x ) a b ) x + ) a y ) x y 6) a 6

54 P a g e PART : FACTORING A BASIC TRINOMIAL The multiplications below produce trinomials (three terms) with an x term, an x term, and a number term. For example: ) (x + ) (x + ) ) (x ) (x - ) ) (x ) (x + ) (FOIL) x + x + x + 0 x x x + 6 x + x x (collect like terms) x + x + 0 x x + 6 x x Steps to factor a basic trinomial: a) Set up two brackets with the appropriate letter ie. (x ) (x ) or (b ) (b ) b) Decide which signs go in the brackets. If the number term is positive the numbers to multiply will be either both positive if x term is positive (see example ), and both negative if the x term is negative (see example ). If the number term is negative, the numbers which multiply must have opposite signs (see example ) c) The number which goes in the brackets must multiply to be the number term and add up to be the coefficient of the x term. ) x + x + (x + ) ( x + ) ( x + ) (x + ) ) x x + (x - ) (x - ) (x ) ( x ) ) x + x 0 (x + ) (x - ) (x + ) (x - ) or (x ) (x + ) ) x x 0 (x - ) (x + ) (x ) (x + ) or (x + ) (x ) When you factor trinomials, always look for a common factor first. Try factoring these trinomials: ) x + x + 0 ) x x + 0 ) Y + Y ) Z Z ) X - 0 X + 00

55 P a g e ANSWERS PART ) ) ) ) ) 0 6) ) ) ) 0) 0 ). ) ANSWERS PART ) ) ) ). ). 6) ) ) ) 0 ) ) 0) ANSWERS PART ) 6 ) ) ) 6 ) x 6) 6x ) ) ) 6y 0) x y ) z ANSWERS PART ) 6 ) ) ) ) 60 6) ) 6 ) ) or 0) 6 ) 0 ). ANSWERS PART 6 ) ) x 6 ) 0a ) z ) z 6) a z ) ab ) a 6 ) m 0) X ) x y ).y ) x y ) 0 x y ) a b ANSWERS PART ) ) -6. ) ) ) 0 6) 0. ) ) -. ) 0).0

56 P a g e 6 ANSWERS PART ) ). ). ) ) 0. 6) ) ) 0 ) 0 0) 60. ANSWERS PART ) ) ) ) ) 6 6) ) 6 ) ) 0) ) ANSWERS PART 0 ) 6(x - 6) ) (a 0) ) a(b + c) ) no common factor ) x ( x 6) 6) 0y ( x x + ) ANSWERS PART ) (x + ) (x ) ) (a b) (a + b) ) does not factor sum of squares ) (a y) (a + y) ) (xy ) (xy + ) 6) (a + 6) (a - 6) ANSWERS PART ) ( X +) ( X + 0) ) ( X 0) ( X ) ) ( Y + 6) ( Y ) ) ( Z ) ( Z + ) ) ( X ) ( X 0) ( X - 0) (remember to look for a common factor first!) ANSWERS PART please see instructor for answer to graphing question

57 P a g e APPENDIX Equation of a Straight Line The equation of a straight line is usually written this way: y mx + b What does it stand for? y vertical or how far up x horizontal or how far along m Slope or Gradient (how steep the line is) b the Y Intercept (where the line crosses the Y axis) With this example equation you can now choose any value for x and find the matching value for y Y x + For example, when x is : y And so, when x, y And, when x is : y And so, when x, y And, when x is 0: y And so, when x 0, y And, when x - y And so, when x -, y Figure The ordered pairs (x, y) for this line are (-, ) (0, ) (, ) & (, ). Try plotting these ordered pairs on Figure For more practice and help with graphing lines try doing a google search for equation of a line or visit the Khan Academy at: