ESCC Mathematics Tutorials - Eastern Shore Community College

Subtraction with borrowing If during the course of perform a subtraction on a large number, you are attempting to subtract a large number from a smaller number you must use borrowing. Subtract 692 – 378 692 -378 you can not subtract 8 from 2 so we need to borrow 1 ten from the 9 tens in the tens column leaving 8 tens and 12 ones. 81 692 -378 12 – 8 4 4 81 692 -378 8 – 7 1 14 81 692 -378 6 – 3 3 314 When do we use Subtraction? There are many key words that lead us to perform subtraction. Subtraction Key Words Minus Less Less than The difference between Decreased by 8 minus 5 9 less 3 2 less than 7 the difference between 8 and 2 5 decreased by 1 8–5 9–3 7–2 8–2 5–1

Multiplying Whole Numbers Multiplication is basically repeated additions. 3 2 2 2 2 6 6 8 8 8 8 8 8 8 8 8 48 The numbers that are multiplied are called factors (6 and 8) and the result is the product (48). There are three basic ways to represent multiplication a b a b all mean the same thing (a multiplied by b) a(b) As is addition the best way to learn multiplication is to memorize the basic facts. Multiplication Table 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81 To use the table, place one finger on the top row on the first factor and place another finger on the second factor on the first column. Bring the finger together and you have the product. 0 1 2 3 4 5 6 7 8 9 Properties of Multiplication There are several useful properties of multiplication that will help us in our computations Multiplication Property of Zero The product of any number and 0 is 0 0 4 0 7 0 0 Multiplication Property of One The product of any number and one is the number 1 5 5 6 1 6

Commutative Property of Multiplication Two numbers can be multiplied in either order and the product is unchanged. 4 3 3 4 12 Associative Property of Multiplication Grouping the numbers in a multiplication problem in any order gives the same result (4 2) 3 8 3 24 4 (2 3) 4 6 24 Multiplying Larger Numbers Multiplying large numbers involves the repeated usage of basic one-digit multiplication facts. Multiply 37 4 37 4 above the ten’s column 2 37 4 8 4 7 28 write the 8 in the one’s column and carry the 8 3 4 12, add the carry digit – 12 2 14 37 4 148 Multiply 47 23 47 23 47 3 141 47 23 141 20 47 940 47 23 141 940 1081 141 940 1081

Multiply 439 206 439 203 3 439 2634 439 203 2634 0 439 000 439 203 2634 0000 439 203 2634 00000 87800 90434 200 x 439 87800 2634 0000 87800 90434 When do we use multiplication? There are key words that indicate the use of multiplication Multiplication Key Word Times 7 times 3 7 3 The product of The product of 6 and 9 6 9 Multiplied by 8 multiplied by 2 8 2

Division of Whole Numbers Division is used to separate objects into groups of equal size. Division is the inverse of multiplication (3 4 12 and 12 3 4 and 12 4 3) We right division in two different ways 12 4 is the same as 4 12 6 Look at the division 4 24 , we refer to 4 as the divisor, 24 as the dividend and 6 as the quotient. In general, we have quotient divisor dividend 6 Also, we can see the relationship between division and multiplication 4 24 because 4 6 24 Important Division Rules Any number, except zero, divided by itself equals 1 1 1 88 2 2 Any number divided by 1 is the number itself 8 27 18 1 27 Zero divided by any number is zero 0 0 7 0 101 0 Division by zero is not allowed ? 0 8 , there is no number whose product with 8 is 0

Dividing single digits into larger numbers Divide 3192 4 4 divides into 31 - 7 times since 4 * 7 28 4 3192 7 4 3192 28 39 79 4 3192 28 39 36 3 subtract 28 from 31, bring down the 9 7 4 3192 28 39 36 3 7 4 3192 28 39 36 32 32 0 4 divides into 39 - 9 times since 4*9 36, subtract 39-36 bring down the 2, 4 divides into 32-8 times since 4*8 32, subtract 32 – 32 0 Dividing by single digit with a remainder Divide 3 14 4 3 14 3*4 12 4 3 14 12 2 4 r2 3 14 12 2 subtract 14 – 12 2 so the result is 4 with a remainder of 2 (3*4 2 14)

Divide larger numbers Divide 34 1598 34 1598 4 34 1598 4 34 1598 136 238 47 34 1598 136 238 238 Think about 3 * 5 15 but 5 * 34 170, which is larger than 159, so use 4 4*34 136 subtract 159 – 136 23 bring down the 8 7*34 238 subtract 238-238 0 So the solution is 37 since 47 * 34 1598 When do we use Division? There are a couple of key words that indicate the use of division. The quotient of Divided by Division Key Words The quotient of 9 and 3 6 divided by 2 9 3 6 2

Order of Operations Many times, in math classes, the problems involve more than one operation in the same problem. We need a system to determine the order in which we perform our operations. There is a hierarchy of operations that keep us from being confused by the messy problems. The Order of Operations can be remembered by learning the phrase Please Excuse My Dear Aunt Sally. Parenthesis – Exponents – Multiplication & Division – Addition & Subtraction (4 levels) Ex. 4 5*6 from left to right we see addition and multiplication (multiplication first priority) 4 30 now we can add 34 Ex. 55 – 2*10 43 (multiplication first) 55 – 20 43 priority) 35 43 78 subtraction – multiplication – addition now left to right (subtraction and division equal Ex. (3 4)2 72 49 parenthesis and exponent (p first) now exponent

Exponential Notation Repeated multiplication of the same number can be written in two different ways 3*3*3*3 or 34 exponent The exponent shows how many time 3 is multiplied by itself. 34 is in a format called exponential notation. Examples of exponential notation 6 61 6*6 62 6*6*6 63 etc. six to the first power (usually don’t write the 1) six squared or six to the second power six cubed or six to the third power 3*3*3*3*5*5*5 34*53 Place values are actually powers of 10 Ten Hundred Thousand Ten-Thousand Hundred-Thousand Million 101 102 103 104 105 106

Factoring Numbers We can divide whole numbers into two categories (prime and composite). Prime numbers are numbers that are only divisible by 1 and itself such as 3, 5, 11, 13. Composite numbers are numbers that are products of prime numbers such as 6, 15, 20. One of the major things that we need to do with whole numbers is to factor the composite numbers into their prime parts, called factoring. Ex. 10 2*5 20 2*2*5 Ex. Factor 105 Start with the small primes and check for divisibility 2 does not work since 2 does not divide 105 evenly but 3 works 105 3*35 now factor 35 as 5*7 so we get 105 3*5*7 Ex. Factor 129 2 won’t work but 3 does 129 3*43, 43 is prime so 129 3 *43 Ex. Factor 400 400 2*200 FACTOR 200 400 2*2*100 FACTOR 100 400 2*2*2*50 FACTOR 50 400 2*2*2*5*5 DONE

Roman Numerals Prior to the development of our number system, there have been many other civilizations who have had their own unique way of handling mathematics and arithmetic. The one system that has survived to this day and is still in wide use is the Roman Numeral System. The Roman Numeral System A few major things to realize about the Roman Numeral System There is no zero It uses what we think of as letters (I, V, X, L, C, M) Placing a lower value to the left of a higher value subtracts Placing a lower value to the right of a higher value adds I 1 V 5 X 10 L 50 C 100 D 500 M 1000 Notice the significance of 5 in this system (like 10 in our system) See a pattern 1 I 2 II 3 III 4 IV 5 – 1 you add through three higher then for 4 you subtract 6 VI 7 VII 8 VIII 9 IX 10 – 1 LX 60 XL 40 50 10 50 – 10 Starting on the left you build the value MCMLXVI M 1000 CM 900 1000-100 LX 60 50 10 VI 6 5 1 1000 900 60 6 1966 you see these type of things on movie dates.

Let’s go from our system to Roman 2121 2000 MM 100 C 20 XX 1 I MMCXXI

Least Common Multiple and Greatest Common Factor When working with a group of two or more numbers, we sometimes have to find two specials numbers, the least common multiple and the greatest common factor. Least Common Multiple The Least Common Multiple is the smallest number that is a multiple of each number in the group. The least common multiple of 2 and 3 is 6 since it is the smallest number both 2 and 3 divide evenly Finding the least common multiple. Factor each number Write down all the factors of the first number Add in the factors from the other numbers that are not in the LCM already Multiply all the numbers together Ex. Find LCM for 30 and 45 30 2*3*5 45 3*3*5 LCM – start with 30 and write down 2*3*5 Look at 45, it has 2-3’s and a 5, the LCM has a 5 but only one 3 so we put in the other 3 to get LCM 2*3*5*3 90 Ex. Find LCM for 6. 8.15 6 2*3 8 2*2*2 15 3*5 LCM – start with 6 and get 2*3 Go to 8 and put in 2 3’s and get 2*3*2*2 Go to 15 and put in the 5 and get 2*3*2*2*5 LCM 2*3*2*2*5 120 Greatest Common Factor The greatest common factor is the largest number that is a factor of each number in the set of numbers. It is used in the reduction of fractions. Finding the Greatest Common Factor Factor each number

Look at each factor in the first number and if it occurs in the other numbers (if it occurs in all numbers then include it in the GCF) Ex. Find the GCF of 8 and 12 8 2*2*2 12 2*2*3 GCF – look at first 2 in 8 it is also in 12 so get 2 so far 8 2*2*2 12 2*2*3 now go to the next 2 of 8 it is also in 12 so we get 2*2 so far 8 2*2*2 12 2*2*3 now go to the next 2 of 8 it is not in 12 so we have 2*2 4 as the GCF Ex. Find the GCF of 60 and 200 60 2*2*3*5 200 2*2*2*5*5 look at things in common 60 2*2*3*5 200 2*2*2*5*5 so the GCF is 2*2*5 20

Reducing Fractions Whenever we are dealing with numbers in the terms of fractions, we like to have them reduced to lowest terms. The lowest terms of a fraction is the terms when the numerator and the denominator have no factors in common (relatively prime). Ex. 3 is in lowest terms since 3 and 4 are relatively prime 4 8 is not in lowest terms since they have 4 as a common factor 20 To reduce fractions to lowest terms Factor numerator and denominator Cancel out factors in common Ex. Reduce 8 to lowest terms 20 2 2 2 becomes 2 2 5 or 2 5 Ex. Reduce 44 to lowest terms 100 2 2 11 becomes 2 2 5 5 or 11 25

Adding and Subtracting Fractions There will be occasions where will be necessary to add or subtract numbers that are fractions. (I know we don’t like fractions, but they are necessary).To add fractions they must have the same denominator (bottom). If they do not have the same denominator then we must convert each of them to a fraction with a common denominator. Fractions with same denominators Ex. 1 2 1 2 3 , just add the numerators (tops) 4 4 4 4 Ex. 4 2 4 2 2 , just subtract numerators 15 15 15 15 Fractions with different denominators. Ex. 1 1 , different denominators. We must find a common multiple for the denominators to 2 3 use as a common denominator. The least common multiple of 2 and 3 is 6, so we use 6 as the common denominator. We convert each fraction to a new one with the denominator of 6. 1 1 3 3 3 multiply by 1 in the form of 2 2 3 6 3 1 1 2 2 2 multiply by in the form of 3 3 2 6 2 1 1 3 2 5 2 3 6 6 6

Ex. 5 1 different denominators. The Least Common Multiple of 8 and 3 is 24 8 3 5 5 3 15 8 8 3 24 1 1 8 8 3 3 8 24 therefore 5 1 15 8 7 8 3 24 24 24

Multiplying and Dividing Fractions Multiplication of fractions is very simple, just multiply numerators and denominators Ex. 1 3 1 3 3 2 4 2 4 8 Ex. 2 3 6 1 3 4 12 2 Division of fractions is not much harder but has one thing important to remember. You must invert the divisor and then multiply (Flip the last guy and multiply) Ex. 2 3 2 4 2 4 8 3 4 3 3 3 3 9 Ex. 15 5 15 4 60 3 8 4 8 5 40 2

Mixed Numbers and Improper Fractions There are two ways of expressing fractions representing numbers greater than one, mixed number and improper fractions. Mixed numbers are expressed as a whole number part and a fractional part in the form B 1 A like 3 C 2 Improper Fractions are fractions whose numerator is larger than the denominator A where A B B Mixed Numbers to Improper Fractions To convert a mixed number to an improper fraction Multiply denominator by whole number part Add numerator Place over the denominator Ex. Convert 3 1 to an improper fraction 2 Multiply denominator by whole number part – 3*2 6 Add numerator – 6 1 7 Place over the denominator 7 2 Ex. Convert 5 3 to an improper fraction 4 5*4 20 20 3 23

so we get 23 4 Improper Fractions to Mixed Numbers To convert improper fractions to mixed numbers, we have to remember the long division that we learned in elementary school – division with remainders. To convert from Improper Fractions to Mixed Numbers Perform implied division with remainder Write quotient as whole number part Place remainder over divisor as fractional part Ex. Write 19 as a mixed number 9 2 1 9 19R1 so we get 2 9 Ex. Write 23 as a mixed number 4 5 3 4 23R3 so we get 5 4

Introduction to Decimals Numbers that cannot be represented as whole numbers are written as either fractions or in decimal notation. We are familiar with the concept of decimal notation from numerous examples in our lives, namely the use of money ( 3.12 is decimal notation for 3 dollars and 12 cents) We can think of decimal notation as another way of writing certain special types of fractions (those with multiples of ten in the denominator) 3 10 3 100 239 1000 Three tenths Three hundredths Two hundred thirty-nine thousandths 0.3 Note: 1 zero in the denominator and 1 decimal place 0.03 Note: 2 zeroes in the denominator and 2 decimal places 0.239 Note: 3 zeroes in the denominator and 3 decimal places We should be able to note that there are exactly three parts to a decimal number. 3.12 whole number part decimal point decimal part Writing decimals numbers in words 0.03 is read as 3 hundredths since the 3 is in the second decimal place (1/100) 0.6481 is read as six thousand four hundred eighty-one ten-thousandths since the 1 is in the fourth decimal place (1/10000) Writing decimal numbers in standard form Five and thirty-eight hundredths – hundredths implies a total of 2 decimal places to be filled by the 38 so we get 5.38 Nineteen and four thousandths – thousandths implies a total of 3 decimal places to be filled by 4 so we add two leading zeroes to make 004 and get 19.004

Rounding decimals Sometimes we are called upon to limit the number of decimal places that can be used in a specific application (it makes no sense to take money out to 3 places). This process is known as Rounding. Rounding rules If the number to the right of the given place value is less than 5, drop that number and all numbers to the right of it. If the number to the right of the given place value is 5 or greater, increase the number in the given place value by one and drop all numbers to the right of it Round 26.3799 to the nearest hundredth Look at 26.3799, 7 is in the hundredths (second) place and 9 5 so increase 7 to 8 and drop the 99 and get 26.38 Round 42.0237412 to the nearest hundred thousandth Look at 42.0237412, 4 is in the hundred thousandth (fifth) place and 1 5 so drop the 12 and get 42.02374

Addition of decimals The addition of decimal numbers is almost identical to the addition of whole numbers. The only difference is that we need to remember to line up the vertical columns with respect to the decimal point. Add 0.237 4.9 27.32 0.237 4.9 27.32 Remember to line up on the decimal points 0 0.237 04.900 27.320 Rewrite with zeroes added in appropriate place values to make things line up properly 0 0.237 04.900 27.320 32.457 7 0 0 7 3 0 2 5 2 9 3 14 carry 1 0 4 7 (1) 12 carry 1 0 0 2 (1) 3

Subtraction of decimals Subtraction of decimals is almost identical to subtraction of whole numbers. The only difference is to remember to line up the columns on the decimal point. (all rules of borrowing in subtraction apply) 6.93 3.7 6.93 3.70 Subtract 6.93 – 3.7 Add necessary zero to line up columns 3–0 3 9–7 2 6–3 3 6.93 3.70 3.23 Subtract 39.047 – 7.96 39.047 7.960 39.047 7.960 7 39/ 8.0/ 9 1 47 7. 9 60 31. 0 87 7–0 7 We need to borrow from the 9 (changed to 8) to change the 0 to 10 so that we can borrow from the 10(changed to 9) to change the 4 to 14 So the result is 31.087

Multiplication of Decimal Numbers Decimal numbers are multiplied in the same way as whole numbers, with special consideration given to the number of decimal places in each of the factors. (number of decimal places in first factor number of decimal places in second factor number of decimal places in product) Multiply 21.14 0.36 21.4 0.36 1284 642 7.704 1 decimal place 2 decimal places 1 2 3 3 decimal places Multiply 0.037 0.08 0.037 has 3 decimal places 0.08 has 2 decimal places 37 8 296 to make this number have 5 decimal places, we need to add 2 addition zeroes 0.00297 (the idea of keeping track of decimal places was necessary in the days when we used slide rules to perform our multiplications. The slide rule would do the whole number multiplication for us (3 8 297) but you had to put the decimal places in for yourself) Multiplication by multiples of ten To multiply by multiples of ten, move the decimal point to the left the same number of places as there are zeroes in the multiple of ten factor. 3.82 10 38.2 3.82 100 382. 3.82 1000 3820. 1 zero and 1 move to the left 2 zeroes and 2 moves to the left 3 zeroes and 3 moves to the left (needed to add a 0 at the right end to make the move)

Division of Decimal Numbers To divide decimal numbers, move the decimal point in the divisor to the right enough place to make it a whole number. Also move the decimal point in the dividend an equal number of places. (remember to keep the decimal point in the quotient

Mathematics Tutorials These pages are intended to aide in the preparation for the Mathematics Placement test. They are not intended to be a substitute for any mathematics course. Arithmetic Tutorials Algebra I Tutorials Algebra II Tutorials Word Problems