Fractions Ms. Crusenberry 9-2013

Vocabulary Fraction – part of a whole number Numerator – the number of parts that are used; the number above the fraction line Denominator – the number of parts to the whole; the number below the fraction line

Write the fraction

Comparing Fractions Cross Product – the answer obtained by multiplying the denominator of one fraction by the numerator of another aka cross multiplication Greater than > or Less than < 2 3 multiply 2 x 5 first = 10 3 > 4 multiply 3 x 3 last = 9

Practice 6 5 8 6 7 5 8 6 16 15 17 16 5 2 19 7

Answers 6 x 6 = 12 < 8 x 5 = 40 7 x 6 = 42 > 8 x 6 = 40

Simplifying Fractions Simplest form – a fraction in which the numerator and denominator have no common factor greater than one Simplify – to express in simplest form

Example 14 ÷ 2 = 7 16 ÷ 2 = 8 What is the greatest common factor? 14 16 1 x 14 1 x 16 2 x 7 2 x 8 4 x 4 2 is the greatest common factor

Practice 4/18 12/42 27/81 13/39 80/120 55/121 12/15 45/50

Answers 2/9 2/7 1/3 2/3 5/11 4/5 9/10

Mixed Number/Improper Fractions Mixed number – number composed of a whole number and a fraction ex. 1 ½ Improper fraction – fractions who numerators are equal to or greater than their denominators ex. 50/10

Renaming Mixed Numbers as Improper Fractions Step 1 – multiple the whole number by the denominator Step 2 – add the numerator Step 3 – Write the answer over the denominator Ex. 3 ½ 3 x 2 + 1 = 7 2

Practice 2 ¾ 8 3/7 5 ½ 6 2/9 23 2/5 10 ¾

Answers 11/4 59/7 11/2 56/9 115/5 43/4

Renaming Improper Fractions as Mixed Numbers Ex. 18/5 Step 1 – divide 18 by 5 3 5 18 -15 so, its 3 3/5 3 **The number left is the new numerator and you always keep the denominator

Practice 32/6 61/4 235/4 79/5 19/2 80/8

Answers 5 1/3 15 ¼ 5/ ¾ 15 4/5 9 ½ 10

Writing Mixed Numbers in Simplest Form 3 2/4 - 2/4 simplifies to ½, so your answer is 3 ½ 5 16/8 – 16/8 simplifies to 2, so you add that whole number to the one in the mixed number to get the answer 7 4 7/6 – 7/6 simplifies to 1 1/6 – so you add that whole number to the one in the mixed number to get the answer 5 1/6

Practice 6 5/15 17 13/10 7 6/3 6 4/3 10 25/6

Answers 6 1/3 18 3/10 9 7 1/3 14 1/6

Multiplying Fractions Ex. 2/7 x 4/5 Step 1 – multiply the numerators 2 x 4 = 8 Step 2 – multiply the denominators 7 x 5 = 35 Answer is 8/35

Continued… Ex. 6 x 4/7 Step 1 – make the whole number a 6 a fraction by putting it over 1 Step 2 – multiply the numerators 6 x 4 = 24 Step 3 – multiply the denominators 1 x 7 = 7 Answer is 24/7 (must make an improper fraction) = 3 3/7

Practice 2/3 x 4/5 4/7 x 3/6 11/13 x 26/33 5/11 x 4/6 6/7 x 5/12

Answers 8/15 2/7 286/429 10/33 5/12

Using Cross Simplification Ex. 9 x 14 10 15 Step 1 - Check the first numerator with the second denominator; simply if possible 3 goes into both so 9 becomes a 3 and the 15 becomes a 5 Step 2 – Check the first denominator with the second numerator; simply if possible 2 goes in to both so 10 becomes 5 and 14 becomes 7 New problem is now: 3 x 7 = 21 **This does not simplify any 5 5 25 further

Practice 4/9 x 7/12 6/12 x 2/12 3/16 of 8/9 4/9 of 7/12 7/10 x 5/28

Answers 7/27 1/12 1/6 1/8

Multiplying Mixed Numbers Ex. 2 ¾ x 1 ½ Step 1: You must change mixed numbers to improper fractions before you can multiply them. 2 ¾ = 11/4 and 1 ½ = 3/2, so 11 x 3 = 33 4 2 8 **you must rename to a mixed number, so the answer is 4 1/8

Practice Cross simplify the improper fractions if you can before multiplying them. 1 ½ x 2 3/5 2 1/6 x 2/3 5 1/3 x 2 ½ 5/6 x 2 3/10 3 5/6 x 3/8

Answers 3 9/10 1 4/9 13 1/3 1 11/12 1 7/16

Dividing Fractions Ex. 5/7 ÷ 4/5 Step 1 – keep the first fraction Step 2 – change the ÷ to x Step 3 – do the reciprocal of 4/5, which is 5/4 5/7 x 5/4 = 25/28

Practice Cross simplify if you can after you do the steps but before you divide 2/7 ÷ 5/6 3/8 ÷ ¾ 4/7 ÷ 5/7 2/3 ÷ 5/6 3/13 ÷ 5/6 9 ÷ 9/10

Answers 5/21 ½ 4/5 18/65 10

Dividing Mixed Numbers Just like multiplying mixed numbers, you must change the mixed numbers to improper fractions before you do anything else Once that is done, change the ÷ to x and then do the reciprocal of the last improper fraction

Example 3 ½ ÷ 2 ½ - change to improper 7/2 ÷ 5/2 – change the sign and use reciprocal 7/2 x 2/5 – cross simplify if possible then multiply 7/1 x 1/5 = 7/5 (this is improper so fix it by changing it to a mixed number) Answer is 1 2/5

Practice 2 ¾ ÷ 5/6 1 1/3 ÷ ¼ 2 2/7 ÷ 2 2/7 7 ½ ÷ 6 2/3 5 2/5 ÷ 1 1/5

Answers 3 3/10 5 1/3 1 1 1/8 4 ½

Adding Fractions with Like Denominators Ex. 2/7 + 4/7 = 6/7 Step 1 – add the numerators Step 2 – carry over the denominator Step 3 – simplify if needed Ex. 2 4/5 + 6 3/5 = 8 7/5 = 9 2/5 Step 1 – add the whole numbers Step 2 – add the numerators Step 3 – carry over the denominator Step 4 – simply if needed

Practice 5/8 + 3/8 5/11 + 4/11 2 5/16 + 5 1/16 3 2/19 + 4 3/19 4 17/20 + 3/20

Answers 1 9/11 7 3/8 7 5/19 5

Subtracting with Like Denominators You do these problems the exact same way as you did the addition, except you are subtracting the numbers Ex. 4/7 – 2/7 = 2/7 13 11/12 – 5 5/12 = 8 6/12 = 8 ½

Practice 7/8 – 3/8 11/12 – 1/12 25 4/5 – 6 3/5 13 3/13 – 2/13 19 19/21 – 5 5/21

Answers ½ 5/6 19 1/5 13 1/13 14 2/3

Adding with Unlike Denominators Common denominators – common multiples of two or more denominators Least common denominator (LCD) – smallest denominator that is a multiple of two denominators

How to Solve these Problems Step 1 – find the least common multiple of the denominators Step 2 – use 12 as a new denominator Step 3 – raise the fractions to higher terms Step 4 – add the fractions, simplify if needed

Example 1 + 3 6 4 find the LCD (the smallest number that both 6 and 4 will go into) 6 x 1 = 6 4 x 1 = 4 6 x 2 = 12 4 x 2 = 8 4 x 3 = 12 2 9 12 + 12 6 goes into 12 twice so 2 x 1 = 2; 4 goes into 12 three times so 3 x 3 = 9 Answer is 11/12

Practice 2/7 + ¾ 3/8 + 2/3 2/9 + 2/3 1/8 + 2/5 5/12 + 2/3

Answers 5/7 1 1/24 8/9 21/40 1 1/12

Subtraction with Unlike Denominators Use solve the problems the exact same way as you did the addition problems except you are subtracting

Practice 8/9 – 1/3 2/3 – 2/5 2/3 – 2/7 4/5 – 1/3 7/10 – 1/2

Answers 5/9 4/15 8/21 7/15 1/5

Addition/Subtraction of Mixed Numbers You will do these the exact same way with the exception of whether you add or subtract. Step 1 – change all mixed numbers to improper fraction Step 2 – find the LCD Step 3 – add or subtract Step 4 – simplify/rename into a mixed number

Example of Addition 6 3/7 + 3 1/3 45/7 + 10/3 – cross simplify when you can 15/7 + 10/1 – find the LCD (it is 7) 15/7 + 70/7 – add the numerators/carry denominator 85/7 – rename into a mixed number = 12 1/7

Example of Subtraction 18 5/6 - 2 3/8 113/6 – 19/8 – cross simplify when you can 113/6 – 19/8 – find the LCD (it is 24) 452/24 – 57/24 – subtract the numerators/carry denominator 395/24 – rename into a mixed number = 16 11/24

Practice 5 1/3 + 4 1/6 5 3/8 + ¼ 2 3/8 + 6 3/10 11 5/8 + 13 2 1/8 + 3 1/6 + 2 1/3

Answers 9 ½ 5 5/8 8 27/40 24 5/8 7 5/8

Practice 5 6/7 – 4 2/3 10 9/10 – 3/20 9 5/6 – 3 8 15/16 – 2 ¼ 5 2/3 – 2 2/5

Answers 1 4/21 10 ¾ 6 5/6 6 11/16 3 4/15

Subtracting with Renaming Step 1 – Rename the whole number as a mixed fraction Ex: 15 = 14 8/8 8/8 is equal to 1, so 1 plus 14 = 15

Example – when missing the top fraction 16 15 8/8 - 2 3/8 - 2 3/8 13 5/8 Step 1 – rename 16 to an improper fraction of 15 8/8. I chose 8/8 because it is the denominator of the mixed number that determines this. Step 2 – perform the subtraction

Example – when you have numerators you cannot subtract 16 2/8 + 8/8 15 10/8 - 2 3/8 - 2 3/8 13 7/8 Step 1 – rename 16 to an improper fraction of 15 8/8. I chose 8/8 because it is the denominator of the fractions. Step 2 – add the 8/8 to the 2/8 to get 10/8 Step 3 – perform the subtraction

Example – when you have different denominators 15 2/7 = 15 6/21 + 21/21 = 14 27/21 4 8/21 4 8/21 4 8/21 10 19/21 Step 1 – carry the whole numbers and then find LCD Step 2 - rename 15 to an improper fraction of 14 21/21. I chose 21/21 because it is the denominator of the fractions. Step 2 – add the 21/21 to the 6/21 to get 27/21 Step 3 – perform the subtraction