Fractions

Fraction = Part of a Whole What is a Fraction? Fraction = Part of a Whole Numerator = tells how many parts you have Top Number? 3 4 Denominator = tells how many parts are in the whole Bottom Number? Note: the fraction bar means to divide the numerator by the denominator

One Way To Remember 3 4 Numerator = North # you have Divided by # of parts in whole Divided by Denominator = Down

What Fraction is Shaded? ¾ ⅝ 7/16

Identifying Forms of Fractions There are three forms of fractions: Proper fraction: The numerator (top number) is always less than the denominator. The value of a proper fraction is less than 1. Improper fraction: The numerator is equal to or greater than the denominator. When the numerator is equal to the denominator, an improper fraction is equal to 1. Mixed Number: A whole number & a proper fraction are written next to each other. A mixed number always has a value of more than 1.

Example Tell whether each of the following is a proper fraction (P), an improper fraction (I) or a mixed number (M). 1 2 3 2 7 110 75 10 9

Group Work Tell whether each of the following is a proper fraction (P), an improper fraction (I) or a mixed number (M). 1 2 3 1 2 17 9 6 4

Reducing Reducing a fraction means dividing both the numerator & the denominator (top & bottom) by a number that goes into each evenly. Reducing changes the numbers in a fraction, but it does not change the value of a fraction. Always check to see if you can continue to reduce. Sometimes a fraction can be reduced more than once to reach the lowest terms

Shortcuts For Reducing Are the numerator & denominator both even? Divide by 2 both end in a 0 or 5? Divide by 5 both end in 0? Divide by 10 Add the digits of the numerator separate from the digits of the denominator. Do they add up to a number that is divisible by 3? Divide by 3 If no to all previous questions: You just have to try 7, 11, 13 & so on

Example Reduce each fraction to lowest terms 1 6 12 = 3 33 77 = 2 25 30 =

Group Work Reduce each fraction to lowest terms 1 75 80 = 3 35 49 = 2 25 50 =

Raising Fractions to Higher Terms Raising to higher terms is the opposite of reducing. To reduce a fraction, you must divide both the numerator & denominator by the same number. To raise a fraction to higher terms, multiply both the numerator & the denominator by the same number. This is an important skill in the addition & subtraction of fractions

Example Raise each fraction to higher terms by filling in the missing numerator. 1 4 5 30 = 2 4 7 35 =

Group Work Raise each fraction to higher terms by filling in the missing numerator. 1 1 3 45 = 2 5 6 42 =

Changing Improper Fractions to Whole or Mixed Numbers The answers to many fraction problems are improper fractions. These answers are easier to read if you change them to whole numbers or mixed numbers. To change an improper fraction, divide the denominator into the numerator. Remember that an improper fraction is any fraction where the numerator is larger than or equal to the denominator.

Example Change each fraction to a whole number or a mixed number. Reducing any remaining fractions. 1 14 8 = 2 30 9 =

Group Work Change each fraction to a whole number or a mixed number. Reducing any remaining fractions. 1 13 12 = 2 36 12 =

Changing Mixed Numbers to Improper Fractions When you multiply & divide fractions, you will have to change mixed numbers to improper fractions. To change a mixed number to an improper fraction, follow these steps: Multiply the denominator (bottom number) by the whole number. Add that product to the numerator (top number) Write the sum over the denominator.

Example Change each mixed number to an improper fraction 1 3 4 = 2 2 1 2 9 =

Group Work Change each mixed number to an improper fraction 1 1 3 = 10 2 4 5 3 =

Comparing Fractions This can be done in two ways Change the fractions so they have common denominators. Then compare Compare each fraction with ½. The size of the numerator compared to the size of the denominator tells you: It is equal to ½ when the numerator is exactly half of the denominator It is less then ½ when the numerator is less than half of the denominator It is greater than ½ when the numerator is more than half of the denominator The symbol = means “is equal to” The symbol < means “is less than” The symbol > means “is greater than”

Example In the box between each pair of fractions, write a symbol that makes the statement true. 3 5 1 2 8 16 1 2

Group Work In the box between each pair of fractions, write a symbol that makes the statement true. 7 20 1 2 9 15 1 2

Addition of Fractions with the Same Denominators To add fractions with the same denominators (bottom numbers), first add the numerators. Then write the total (or sum) over the denominator. Don’t forget to check to see if you can reduce your answer.

Example Add 3 8 12 2 1 7 13 1 8 8 9 4 13 2 8 6 10 + +

Addition of Fractions with Different Denominators If the fractions in an addition problem do not have the same denominators, you must find a common denominator. common denominator = a number that can be divided evenly by every denominator in the problem. lowest common denominator or LCD = The lowest denominator that can be divided evenly by every denominator in the problem.

Finding a Common Denominator Method 1: Multiply the denominators together. Brute force method: List the multiples of the larger number until you find a multiple of the smaller number Prime factorization method: find prime factors of both numbers. Circle the numbers they have in common. Write those once then write in the rest of the numbers and multiply to find the LCM

Example Add 11 16 2 1 7 10 1 3 3 4 7 8 + +

Group Work 5 7 2 1 5 12 4 9 5 9 2 3 + +

Subtraction of Fractions To subtract fractions with the same denominator, subtract the numerators & put the difference (the answer) over the denominator. When fractions do not have the same denominators, first find a common denominator. Change each fraction to a new fraction with the common denominator. Then subtract.

Example 2 1 5 6 5 9 23 1 6 2 9 – 7 –

Example Subtract & Reduce 2 1 5 8 11 18 25 13 2 5 1 2 – 22 – 8

Group Work Subtract & Reduce 2 1 3 5 5 6 18 16 3 10 7 10 – 9 – 9

Subtraction with Regrouping Sometimes there is no top fraction to subtract the bottom fraction from. Other times the top fraction is not big enough to subtract the bottom fraction from. To get something in the position of the top fraction, you must borrow. To borrow means to write the whole number on top as a whole number and an improper fraction. For example, 12 = 11 . The numerator and denominator of the improper fraction should be the same as the denominator of the other fraction in the problem. This is also called Borrowing 8

Example Subtract. 2 1 2 9 3 16 13 24 5 6 2 3 – 7 – 9

Group Work Subtract. 2 1 1 6 1 3 12 30 7 12 8 11 – 10 – 16

Multiplication of Fractions When you multiply whole numbers (except 1 & 0), the answer is bigger than the two numbers you multiply. When you multiply two proper fractions, the answer is smaller than either of the two fractions. When you multiply two fractions, you find a fraction of a fraction or a part of a part. To multiply fractions, multiply the numerators together & the denominators. Then reduce.

Example Multiply & Reduce 1 2 3 4 5 x = 2 1 3 4 7 2 3 x x =

Group Work Multiply & Reduce 1 5 7 2 9 x = 2 2 5 7 9 1 3 x x =

Canceling Canceling is a way of making multiplication of fractions problems easier. Canceling is similar to reducing. To cancel, divide a numerator and a denominator by a number that goes evenly into both of them. You don’t have to cancel to get the right answer, but it makes the multiplication easier.

Example Multiply & Reduce 1 15 28 7 16 12 45 x x = 2 17 21 14 51 7 11 x x =

Group Work Multiply & Reduce 1 11 39 10 11 13 18 x x = 2 19 36 7 10 3 7 x x =

Multiplication with Fractions and Whole Numbers To multiply a whole number & a fraction, first write the whole number as a fraction. Write the whole number as the numerator & 1 a the denominator.

Example Multiply & Reduce 1 9 10 x 2 = 2 7 12 x = 36

Group Work Multiply & Reduce 1 5 24 x 16 = 2 7 30 x = 35

Multiplying Mixed Numbers To multiply mixed numbers, first change the mixed numbers to improper fractions. Then multiply the improper fractions. Reduce the answer if necessary.

Example Multiply. 1 2 15 1 4 1 2 2 x 5 x 7 = 2 2 5 3 8 7 9 2 x 3 x 2 =

Group Work Multiply. 1 3 4 8 9 1 5 3 x x 1 = 2 1 3 5 14 16 x 2 =

Division by Fractions To divide fractions, invert* the divisor (the number at the right of the ÷ sign) Then follow the rules for multiplying fractions. To invert means to write the numerator on the bottom and the denominator on the top. Invert = take the reciprocal

Example Divide & Reduce 1 16 21 3 4 ÷ = 2 12 19 18 38 ÷ =

Group Work Divide & Reduce 1 3 10 6 7 ÷ = 2 5 11 25 33 ÷ =

Division of Fractions and Mixed Numbers by Whole Numbers & Division by Mixed Numbers In fraction division problems, change whole numbers & mixed numbers to improper fractions. Then take the reciprocal of the fraction you are dividing by & follow the rules for multiplying fractions.

Example Divide & Reduce 1 24 25 ÷ 56 = 3 2 5 8 1 2 10 ÷ 4 = 7 24 ÷ 35 =

Group Work Divide & Reduce 1 21 25 ÷ 7 = 3 2 5 6 5 12 5 ÷ 3 = 5 18 ÷ 15 =