FRACTION REVIEW. 3 and. Any fraction can be changed into an equivalent fraction by multiplying both the numerator and denominator by the same number

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1 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. Business Math Study Guide Page

2 . 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. 0 Business Math Study Guide Fractions FB/0 Page

3 . EXERCISE : Introduction to Fractions a) Find the missing part of these equivalent fractions x ) ) ) 0 Example: 0 x ) ).) Since x 0, multiply the numerator by, also. ) ) 0 00 So, 0 b) Find the missing part of these equivalent fractions. ) ) Example: 0 ) ) 0 Since 0 divide the numerator by, also. ) 0 ) So, 0 c) Simplify the following fractions (reduce to lowest terms). ) ) ) 0 ) ) 0) 0 Business Math Study Guide Fractions FB/0 Page ) ) ) ) 0

4 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 or So, the fraction 0 is really the whole number. 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) (always reduce fractions) Business Math Study Guide Fractions FB/0 Page

5 . 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. 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 0. Simplifying fractions All types of fractions must always be simplified (reduced to lowest terms). e.g.,, Note that many fractions can not be reduced since they have no common factors. e.g.,, Business Math Study Guide Fractions FB/0 Page

6 . EXERCISE : Types of Fractions a) Which of the following are common fractions (C), whole numbers (W), mixed numbers (M) or improper fractions (I)? ) ) ) ) ) ) ) ) ) 0) b) Change the following to mixed numbers: ) ) ) 0 ) ) 00 ) c) Change the following to improper fractions: ) ) ) ) ) ) d) Simplify the following fractions: ) 0 ) 0 0 ) ) ) ) 0 Business Math Study Guide Fractions FB/0 Page

7 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. 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? 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 () divides into it. Notice that, when we multiply x, we get, which 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. x ( doesn t divide into ) x ( doesn t divide into ) x LCD x x x x 0 Business Math Study Guide Fractions FB/0 Page

8 Which is larger: or? Examine multiples of the larger denominator () until the smaller denominator divides into it. This tells us that the LCD is. Now, we change each fraction to equivalent fractions with the LCD of. x x x (LCD) LCD is. x x x 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 0 is the number that all denominators divide into. x x 0 x x 0 (LCD) x x x x x x So, is the largest fraction. Business Math Study Guide Fractions FB/0 Page

9 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 ) or 0 0 ) or 0 ) or ) or ) or or ) or ) or or ) or Business Math Study Guide Fractions FB/0 Page

10 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. + Notice that we must reduce the answer, if possible.. Adding When One Denominator is a Multiple of the Other Add and x 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 ). + + Business Math Study Guide Fractions FB/0 Page 0

11 . Adding Any Fraction Add and We must find a common denominator by examining multiples of the largest denominator. We find that the LCD Add and When adding mixed numbers, add the whole numbers and the fractions separately. Find common denominators and add. + 0 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 Business Math Study Guide Fractions FB/0 Page

12 . EXERCISE : Adding Fractions a) Add the following: ) + ) + ) + ) + + b) Find the sum of: ) + ) + ) + ) + c) Add the following: ) + ) + ) ) ) ) ) ) d) Evaluate the following: ) and ) total of and ) plus ) combined with ) greater than ) sum of and Business Math Study Guide Fractions FB/0 Page

13 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 -. 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, first change the common fractions to equivalent fractions with the LCD. Since is smaller than, borrow from Business Math Study Guide Fractions FB/0 Page

14 . The Language of Subtraction minus from. EXERCISE : Subtracting Fractions a) Subtract the following: (NOT minus) subtracted from decreased by or lowered by - CAN BE WORDED less than the difference of and NOTE: Unlike addition, we can not reword the above with the fractions in reverse order: i.e. - is NOT the same as - ) - ) - ) - ) - ) - ) - ) - ) - b) Subtract the following: ) - ) - ) - ) - ) - ) - c) Find the following: ) What is minus? ) decreased by is what? ) What is less than? ) What is from? Business Math Study Guide Fractions FB/0 Page

15 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 () has the number understood in its denominator. x x 0 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 for instructions on changing a mixed number to an improper fraction. x x x or x Business Math Study Guide Fractions FB/0 Page

16 . The Language of Multiplication multiplied by of x CAN BE WORDED. EXERCISE : 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 ) x ) ) 0) x x x 0 x 0 x x x x x ) ) ) x x x ) ) ) x x x x b) Find the following: ) What is of? ) by is what number? ) What is times? 0 ) of is what number? ) of is what number? Business Math Study Guide Fractions FB/0 Page

17 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. Business Math Study Guide Fractions FB/0 Page

18 . 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: ) ) ) ) ) ) 0 0 ) 0) ) 0 ) b) Find the following: ) divided by ) divided by ) into Business Math Study Guide Fractions FB/0 Page

19 ) Divide by ) into ) Divide by Business Math Study Guide Fractions FB/0 Page

20 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?? ) and equals.? ) 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? ) of equals.? ) What is greater than? Business Math Study Guide Fractions FB/0 Page 0

21 ANSWER KEY FRACTION REVIEW EXERCISE : Introduction to Fractions (Page ) a) ) ) ) 0 ) ) ) 0 ) 00 ) b) ) ) ) ) ) ) 0 c) ) ) ) ) ) ) ) ) ) 0) EXERCISE : Types of Fractions (Page ) a) ) C ) M ) I ) W ) W ) M ) W ) I ) C 0) W b) ) ) ) ) ) ) c) ) ) ) ) ) ) d) ) 0 ) ) ) ) or ) or EXERCISE : Comparing Fractions (Page ) ) ) ) 0 0 ) ) ) ) ) ) EXERCISE : Adding Fractions (Page ) a) ) ) or ) ) or b) ) ) ) ) c) ) ) ) ) 0 ) ) ) ) d) ) ) ) 0 ) ) ) Business Math Study Guide Fractions FB/0 Page

22 EXERCISE : Subtracting Fractions (Page ) a) ) ) 0 ) ) ) ) ) ) b) ) ) ) ) ) 0 ) c) ) ) ) ) EXERCISE : Multiplying Fractions (Page ) a) ) ) ) ) ) ) ) ) 0 ) or 0) ) ) or b) ) 0 ) ) or ) or ) EXERCISE : Dividing Fractions (Page ) a) ) ) ) ) ) ) ) 0 ) ) 0) b) ) ) ) ) ) ) 0 Fraction Review (Page ) a) ) ) ) ) ) ) ) 0 ) 0 ) 0) ) 0 ) ) ) Business Math Study Guide Fractions FB/0 Page

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