BASIC MATH The purpose of this booklet is to refresh the reader s skills in basic mathematics. There are basic mathematical processes, which must be followed throughout all areas of math applications. While this is not a fully comprehensive mathematics course, it will review areas in which most students need assistance. CONTENTS Section...................... Whole Number Review Section...................... Decimal Review Section...................... Fraction Review Section...................... Percents Review Section...................... Basic Algebra Section 6...................... Geometry Review
WHOLE NUMBER REVIEW This is one of the most basic areas, but if you don t follow correct processes, may prove to be the most frustrating. The first thing we learned how to do in mathematics was to add columns of numbers. The quickest way is to align the numbers on the right side and simply add down each column from right to left, taking care with the carrying if there is any. For example, if we add the following numbers:,, 98, 68,. A second method is add only numbers at one time. + 98 8 68 + 98 + 06 +68 + 06 When subtracting two numbers, simply line the numbers up on the right side and subtract the smaller number from the larger. Some times you don t have to borrow, but if the bottom number is larger than the top, you must. 9 9 9 86 6 000-6 - -88 99 These examples show borrowing done correctly. You must be careful.
Multiplication When you multiply whole numbers, simply set one under the other and multiply every number in the first by every number in the second. If you are multiplying two or three digit numbers by another two or three digit number, you must multiply every number on the top by every number on the bottom. The process is more complex, so care must be given to make sure that the product is placed in the correct position. 0 6 9 x 8 x x6 0 99 0 68 6 (positions 089 lined to 6 hold the place value) For some adding zeros is the easier way to be sure you put the numbers in the correct positions, while for others remember if you multiply by the second digit start under the second digit, third digit starts under the third digit and so on. The choice is yours. 0 6 9 x 8 x x6 0 99 0 680 60 (zeros added 089 00 to hold the 6 place value) see page # 6 for multiplication problems.
DIVISION Division is probably the most misunderstood mathematical operation. It is so simple because it is just multiplication in reverse. For example the problem: 6 means 6 divided by It is worked as shown below. quotient divisor dividen d 6 9 6 9 6 - - So the answer is 9 remainder, written 9r. The division process is easily remembered by the simple loop: BN This means: divide, multiply, subtract, and bring down the next number. When dividing by a two or more digit number, the process does not change. 0 How many times can be divided evenly by 0? Think ( 8) so 0 8 0 This means that can be divided evenly by 0, 8 times and have a remainder of, written 8r.
6 Think 60 instead of 6. Think!! 60 multiplied by is 0 so try! 6 The answer is R. Think 00 6 - The remainder of 88 is large, but it 6 can be as large as one less than the -0 divisor or 6. Answer is R88. 88 Work the sample problems below: 9 6 8 86 Answers R 66R R6 9 6 8 86
Whole Number Review These problems should enable you to become familiar with the processes shown previously.. x 8. 8 x. x 9. + 8 + 96. -89 6. 89 9. 68 8. 89 689 9. 96 68 0. 8. 9. 6 86. 00 986. 98 + 89. 6 866 6. x 6. x 6 8. 8 886 6
Answers to Whole Number Review. x 8. 8 x. x 9 9,6 0,9,. + 8 + 96. -89 6. 89 9, 9. 68 8. 89 689 9. 96 68 66R8 0R R9 0. 8. 9. 6 86 9 9R 0R. 00 986. 98 + 89. 6 866 8 9,9 0 6. x 6. x 6 8. 8 886 0,60 0,00 88R
Decimals Decimals are worked mathematically just like whole numbers with the exception of doing addition and subtraction, where the numbers are aligned to keep place value in position. Place value table thousands hundreds tens ones and tenths hundredths thousandths Tenthousandths Hundredthousandths. All whole numbers have an understood decimal to the right of the last digit. Whole number Decimal number.0 or.000 6 6.0 or 6.000 You may add as many zeros as necessary for the problem you are doing. Example:.6++.69 Line up on the decimal.6.60 add zeros..000 to keep +.69 +.69 yourself.6.6 straight. When subtracting, you must add zero so borrowing (renaming) can be accomplished. Ex. 8.68 9 9 0 8.000. -.68 -.00 0. 0.9 8
Decimal Addition:. +.+.00..+.+.00..00+.+.0+. +. +. +.0. +.8+.0 6. 6 + 6.+.0..00 +.0 +.0 + 8..8 + 8. 9.. +. + 0. 0.00+.. 89.+ +.00. +.+.+. Decimal Subtraction:. 0 0... -.. 6. - 6. 6. 8.9 96.689.. 9.8 8...69 9. 89 -. 0. - 9.. 0 8.. 0-9.9. 99-89.. - 0. Answers are given on the next page.
Answers: Decimal Addition:. +.+.00..+.+.00..00+.+.0+ 8. 6. 80.. +. +. +.0. +.8+.0 6. 6 + 6.+.0. 0.68...00 +.0 +.0 + 8..8 + 8. 9.. +. +.0 0.0. 0. 0.00+.. 89.+ +.00. +.+.+..0 9.0 6.0 Decimal Subtraction:. 0 0...-.. 6. - 6. 9. 0.8 88.8 6. 8.9 96.689.. 9.8 8...69 8.6..0 9. 89 -. 0. - 9.. 0 8. 6.6... 0-9.9. 99-89.. - 0. 0.0 9. 6.
Multiplying Decimals To multiply decimals, you multiply as with whole numbers. When you get the product, go back to the original numbers being multiplied and count the number of digits to the right of the decimal point in each number. Count this number of digits to the left in the product and insert the decimal point. (Note: Whole numbers have a decimal following them so there are no numbers to the right.) Example Example. count.00 count x. count x.0 count.0 move left digits.000 move left digits. x 6.8 use a short cut 6.8 count Commutative property of multiplication x. count 06 same 060 product 0600 each time 8. move left digits When multiplying a whole number by a decimal number, count the digits to the right of the decimal in the decimal number. x.0 count 0 x.0 count.0 move left digits When multiplying by a power of ten (numbers with trailing zeros), multiply the non- zero digits and count the trailing zeros and place that many zeros to the right of the product. x 0 So 00 x.0 000 with decimal places or 0.00
0000 x.00 0000 with decimal places or 0.000 00 x 00 60000 with 0 decimal places or 60,000. If you encounter a larger problem, the process is easily extended.. x. x x. THINK!!! Multiplication is an associative operation so x and x 00 and x 00 00 with decimal places or.00 Thus you did this problem in you head.. x. x x..00 Try these:.. x 00..00 x 600.. x 00..0 x. Answers:..00..000. 60.00. 0.090 Multiplication Problems. 0.60. 0.0. 0.6. 0.9 x0.0 x0.0 x0.9 x0... 6. 0.08. 0. 8.. x 0.6 x0.8 x0.0 x0.00 9. 0.0 0. 9.. 0.06. 0.0 x 0.0 x 0.0 x. x 0.6. 00. 00. 00 6. 0000 x 0.0 x. x.000 x 0.00
Answers to Multiplication Problems:. 0.60. 0.0. 0.6. 0.9 x0.0 x0.0 x0.9 x0..800.000.6 0... 6. 0.08. 0. 8.. x 0.6 x0.8 x0.0 x0.00.86.06.0090.006 0. 0.0 0. 9.. 0.06. 0.0 x 0.0 x 0.0 x. x 0.6.000.8.08.06. 00. 00. 00 6. 0000 x 0.0 x. x.000 x 0.00.00 60.0.00 60.000
Division of Decimals When you perform division with decimals, you must move the decimal before you start the problem. You can t divide by a decimal number so you have to move the decimal all the way to the right in the divisor (front number). In the example below,.0 must be made into a whole number. Move the decimal all the way to the right and count the digits from the original position (places). Move the decimal in the dividend (inside the bracket) places (digits) to the right and straight up into the quotient. Now divide as with whole numbers and your answer will have the decimal in the correct position. 0..0.0.0.00. -0 0-0 Place the decimal in the correct position in these examples.. 6. 600.00.00. 0 6. Answers:.. 6..00 600. 0..00. 6. 0 The numbers don t always come out even when performing division. In decimal division, you cannot leave a remainder so the remainder must be rounded off at or to a specific digit. Use digits to the right of the decimal unless otherwise noted. In this problem, where the numbers don t repeat or come out even, you may add another zero and get an extra digit to round off ( digits to round to ).
.9.0. 900 which would round to 0.0-9 -0 0-9 0 (Second method: think x, > 9 so the.0-9 would round up to 0.0) A second method is to go to digits, then look at the remainder. If you double the remainder and get a number larger than the number you are dividing by, you simply add one to the third digit and you are done. 9..00 If a problem repeats, simply put a bar over the second -9 0-9 Try these: repeated digit..00.0 6..0 9.... 6..0.....6.... 6. 8.8..9. 9...0...
Answers 00.. 0. 6..00.0 6..0 9.... 6. 0... 6..0.... 90. 0. 0. 0..6.... 6. 8.8.. 0. 0. 0..9. 9...0...
Fractions Fractions are a different way of expressing a relationship or a ratio. Parts 6 Whole 9 8 For instance: Trout are put into a stream at the rate of brown trout to brook trout for every one hundred fish put into the stream. What fractional part of the fish were brown trout? Reduce your answer to lowest terms. # of brown reducing : 9 total # of fish 00 00 0 Similarly, #of brook total # of fish 00 0 The sum of any group of objects is the total or whole. Write each of the following as a fraction. pounds is what part of a ton? ( ton 000 pounds) inches is what part of a foot? ( foot inches) 6 inches is what part of a yard? (6 inches yard) 6 Answers:,, 000 00 6 9 A fraction may also be referred to as a ratio. If a fraction is not in lowest terms, it must be reduced. If both the numerator (top) and denominator (bottom) are even, the number may be reduced or divided by. If the last digit is a five or zero, the number may be reduced by a. If the number is a multiple of or 9 or if the digits add to a multiple of or 9 then you may reduce by or 9.
9 since + 9 and + 6, both numbers are multiples of so reduce by. A proper fraction is one, which has a numerator less than the denominator. An improper fraction is one with the numerator equal to or greater than the denominator. 6 is a proper fraction. is an improper fraction. 9 If a proper fraction contains common factors in the numerator and denominator, the fraction must be reduced to lowest terms or simplest form. This is accomplished by dividing the both the numerator and denominator by the common factor. This is illustrated below. These common factors may be any number, not just,,,, etc. may be reduced by. 8 Any fraction with even last digits, may be reduced by. Many times, the fraction may be reduced by a larger number. 6 may be reduced to which may be reduced to. 8 Or it may be reduced by. 8 6 giving the same result. The important thing is to look for is similar numbers in both the numerator and denominator. Reduce the following: 6 6 6 6 8 Answers:, 6,,,,,, 8
To reduce an improper fraction, simply divide the numerator by the denominator. so you write the answer as -6 (the divisor goes under the remainder to make it a fraction) If the improper fraction comes out evenly, then the answer is just a whole number. DON T WRITE 0 as a numerator of a fraction! Try These: 9 6 6 8 Answers:,..,,..,,..,,..,,..,,.., 6 6 Fractions with the same denominator are called common fractions. For you to be able to add or subtract the denominators must be the same. If they are not, you must convert them to a common denominator. + and + are Common fractions and may be added. 6 6 + + 6 6 6 6 Those denominators, which are different, must be converted to a common denominator. + since and have nothing in common, the fractions must be converted a common denominator. x so this is the common denominator.
+ 0 9 9 8 + 8 8 9 8 8 When adding the denominator does not change, just add the numerator, reduce if necessary. When adding mixed numbers, remember to treat the problem as if it were small problems. The whole number portion you have done since first grade, so you already know how to do that. The fractions you treat as we did the previous examples. 8 9 + 8 Since add to 8 When subtracting fractions, the process is precisely the same. Convert fractions to common denominators and instead of adding the numerator, you simply subtract. 6 9 9 9 9 6 6
The only time you need to worry or do anything different is when the numerator of the first number is less than the bottom number (subtrahend). (ie: larger number is on the bottom) When you borrow you must convert the borrowed to the common denominator over itself or CD ie: 8 0,,, CD 8 0 The basic example is 9 0 9 0 Since you can t subtract 0 from 9 you must borrow from the and add to 9. is the answer. 8 6 6 6
Here are some addition and subtraction problems for you.. +. +. 8 + 6 9. 6 +. 6 + 6 6.. 8 8. 8 9. + 0. 6.. Answers:. 6. 0. 6 8. 9. 9. 0 6 9. 0. 8 0.. 6.. 6 0
Multiplying and dividing To multiply a simple fraction, multiply the numerators (top) together and the denominators (bottom) together, reduce if possible. 6 6 8 8 6 96 6 0 To multiply a fraction by a whole number, a one () is placed under the whole number to make it a fraction. Then proceed as before. To make any number a fraction, put a under it.!!!! 6 Multiplying by mixed numbers requires an additional process. The mixed number must be converted to an improper fraction before the problem is worked. Multiply the whole number by the denominator of the fraction and add the numerator of the fraction. Place this number over the original denominator and you are ready to proceed. Work the problem as described previously. 8 9 8 6 Short Cut (Cross Cancellation) When certain conditions exist, you may cancel out a common factor from a denominator and the opposite numerator or vice versa. Mixed numbers must be converted to improper fractions before this process is done. The problem from the above problem is done using this process. 9 since and are common factors, they can be cancelled out and 9 and have a factor of which can be cancelled out. Then multiply the numbers on top and bottom and get the same answer. This reduces the problem to lowest terms before you multiply it out and then reduce.
Dividing fractions How do you divide this problem? YOU CAN T!!! This problem must be converted to a multiplication problem before it can be completed. The second number you must invert or make it into a reciprocal. 8 After the problem is made into a multiplication problem then proceed as in multiplication. 8 9 8 8 0 8 0 Then the problem must be converted to a multiplication problem before it can be done. Division is changed to multiplication and the second fraction is inverted. (reciprocated, flipped, etc). The problem is now a multiplication problem and you have already accomplished that task. Note: If the problem has a mixed number, it must be converted to an improper fraction before you start the process. 8 9 6 6 6 9 You try this one. 9 Answer: 8 8 9 9 Try these multiplication and division problems.
. 6 9.. 0.. 6. 9. 8 8. 9 9. 9 0..... 8. 8 6. 8. 8. 9. 0.
Answers:. 6 9.. 0. 8. 6. 9. 8 8. 9 0 9. 9 0... 98 0.. 8. 8 6. 8 6 6. 8. 9. 0 0. 6
Applications. Ann bought a lot of yarn. She bought pink, green, blue, red, and black skeins of yarn. How many skeins did she buy in all? If each skein was $, how much did she spend?. Berry had $8 on payday. If he spent $ on shoes, $8 on a haircut, and $9 to repair his car, how much money does he have left for the week? To the nearest dollar, how much per day could he spend for the next week?. Charles has to fill in a -inch space with washers that are 0. inch thick. How many washers will he need? If each washer costs $0., how much will he spend to finish the job?. Darcie has rock CD s, 8 country CD s, jazz CD s, and classical CD s. What fractional part of her total CD collection is country CD s?. Edward has to add the fractional areas of a complex figure. The areas are.-sq. in.,.- sq. in.,.- sq. in., 0.- sq. in., and 9- sq. in. What is the total area of the figure? 6. Francis measures a window with the following sides: 6 in., in., 6 in., in. How much trim 8 8 would be needed to go around this window?. Gabriel needs to use 8 inch washers to fill a space inches long. How many washers does he have to use?
8. Harriett has a piece of clay, which weighs. kg. After she removes pieces which weigh. kg,. kg, and. kg, how much will she have left? 9. Ira has a board which is 8 long. If he cuts off pieces of,, and, how much of the board will he 8 have left? 0. Juan earns $. per hour working as a cashier. If he works 8 hours each week, how much will he make in one month(-weeks)?. Lora spends $ out of every $00 she makes on rent. If she makes $80 every two weeks, how much would go toward her rent?. Mickey sells items for $.. If he and others are working their stand and split the money equally, how much will each one get?
Answers to Applications. Ann bought a lot of yarn. She bought pink, green, blue, red, and black skeins of yarn. How many skeins did she buy in all? If each skein was $, how much did she spend? ++++ 0 skeins of yarn 0 X $ $0. Berry had $8 on payday. If he spent $ on shoes, $8 on a haircut, and $9 to repair his car, how much money does he have left for the week? To the nearest dollar, how much per day could he spend for the next week? $8-(+8+9) 8-0 $98 $98 $ per day. Charles has to fill in a -inch space with washers that are 0. inch thick. How many washers will he need? If each washer costs $0., how much will he spend to finish the job? 0. 6 washers 6 washers X $0. $.00. Darcie has rock CD s, 8 country CD s, jazz CD s, and classical CD s. What fractional part of her total CD collection is country CD s? +8++ 6 total CD s 8 9 are country CD s 6
. Edward has to add the fractional areas of a complex figure. The areas are.-sq. in.,.- sq. in.,.- sq. in., 0.- sq. in., and 9- sq. in. What is the total area of the figure?. +. +. + 0. + 9.98 6. Francis measures a window with the following sides: 6 in., in., 6 in., in. How much trim 8 8 would be needed to go around this window? 6 8 8 6 8 6 8 + 8 8 6 8 8 inches 8 8. Gabriel needs to use inch washers to fill a space 8 inches long. How many washers does he have to use? 8 washers 8
8. Harriett has a piece of clay, which weighs. kg. After she removes pieces which weigh. kg,. kg, and. kg, how much will she have left?. - (.+.+.).8kg 9. Ira has a board which is 8 long. If he cuts off pieces of,, and, how much of the board will he 8 have left? 8 ( + + ) 8-8 inches 8 8 8 0. Juan earns $. per hour working as a cashier. If he works 8 hours each week, how much will he make in one month(-weeks)? $. X 8 $9.0 $9.0 X $8.00 0. Lora spends $ out of every $00 she makes on rent. If she makes $80 every two weeks, how much would go toward her rent? 00 0 80 $0
. Mickey sells items for $.. If he and others are working their stand and split the money equally, how much will each one get? X $. $. $. $6.