OPERATIONS ON FRACTIONS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

Review Mixed Numbers & Improper Fractions To rewrite a Mixed Number as an Improper Fraction, multiply the denominator to the whole number and add it to the numerator. To rewrite an Improper Fraction as a Mixed Number, divide the denominator into the numerator, which is the whole number and write remainder as the numerator.

Addition of Fractions & Mixed Numbers Fractions with the same denominator are added by adding the numerators and placing that sum over the (common) denominator. We are literally counting how many parts in total there are where all the parts are the same size. Ex: Add 2/5 + 1/5 Draw each fraction. 2 + 1 = 3/5 2/5 + 1/5 = 5

When the fractions have the same denominator, the parts of the whole are of the same size so adding fractions is literally counting up the parts. When the fractions have different denominators, the parts of the whole are not the same size so we cannot add fractions by counting the parts  “adding apples and oranges” To add fractions with different denominators, first rewrite the fractions as equivalent fractions with the same denominator  “adding apples and apples” The common denominator we will use for the equivalent fractions is the LCM of the denominators, called the LCD, Least Common Denominator.

Note: different denominators Ex: Add 1/4 + 5/8 Note: different denominators Draw each fraction. Can we further divide each of the pieces so that the pieces of each whole are of the same size? 1/4 5/8 YES! LCM(4, 8) = LCD = 8  divide each whole into 8 pieces Consider the first fraction 1/4 : What times 4 is 8? 2  Divide each part into two pieces  1/4 = 2/8 Consider the second fraction 5/8 : It is already divided into 8 pieces  5/8  1/4 + 5/8 = 2/8 + 5/8 = 7/8

Mathematically: LCD = LCM(4, 8) = 8 Therefore,

Note: different denominators Ex: Add 1/5 + 1/2 Note: different denominators Draw each fraction. 1/5 Can we further divide each of the pieces so that the pieces of each whole are of the same size? 1/2 YES! LCM(5, 2) = LCD = 10  divide each whole into 10 pieces Consider the first fraction 1/5 : What times 5 is 10? 2  Divide each part into two pieces  1/5 = 2/10 Consider the second fraction 1/2 : What times 2 is 10? 5  Divide each part into five pieces  1/2 = 5/10  1/5 + 1/2 = 2/10 + 5/10 = 7/10

Your turn to try a problem.

Ex: Subtract 175/9 - 115/12 Use a vertical format Start right (with the fractions) and work left To subtract fractions, we need a - Common Denominator  LCM(9,12) = 36 Find equivalent fractions 3 * 3 2 * 2 * 3 Subtract 20/36 – 15/36 = 5/36 - Subtract whole numbers 17 - 11 = 6 6

Ex: Subtract 7 – 4 2/5 Use a vertical format 6 Start right (with the fractions) and work left The minuend does not have a fraction part so we have to borrow  take one whole - Cut the borrow whole into parts . . . How many parts? 5 How many parts do we have?  5/5 - Subtract 5/5 – 2/5 = 3/5 2 Subtract whole numbers 6 - 4 = 2

Find equivalent fraction with denominator 21  2/7 = 6/21 Practice: Rewrite the mixed number 5 2/7 as a mixed number with an improper fraction part with denominator 21. Find equivalent fraction with denominator 21  2/7 = 6/21 4 Now borrow a whole Cut the borrowed whole into parts . . . How many parts? 21 How many parts do we have? 21 from the chops plus the original 6  21 + 6 = 27  27/21

Your turn to try a problem.

Multiplication of Fractions & Mixed Numbers Ex: Multiply ½ x ¾ read “ ½ of ¾” and draw it Take ½ of three-fourths by chopping the whole in 2 parts (the other direction) and shading 1 part How many parts are there now? 8 How many parts represent ½ of ¾ ? (i.e. how many parts are doubly-shaded?) 3 Therefore, ½ of ¾ is 3/8  ½ x ¾ = 3/8

Ex: Multiply 9/11 x 2/3 Can we factor the numerator and the denominator? Reduce any factor in the numerator with the same factor in the denominator?

Notice that the multiplication of the numerators and/or denominators can get more complicated. We multiplied first and then simplified. Charge! We can simplify first and then multiply. Tidy Up First!

Ex: Multiply 9/11 x 2/3 Can we simplify the fraction?

Since we know how to multiply fractions, we can now multiply fractions, whole numbers, and mixed numbers together To multiply whole numbers, mixed numbers, and fractions first turn every factor into a fraction.  fraction multiplication: multiply and simplify OR simplify and multiply

Ex: Multiply 2 x 6/7 First rewrite the question as a multiplication of fractions  write the whole number 2 as a fraction 2 = 2/1  improper fraction  mixed number Consider this . . . 2 x 6/7 can be read two times 6/7  “twice” 6/7  draw 6/7 then double it! 12 How many parts are shaded? How many parts make a whole? 7

Ex: Multiply 3 1/5 x 2 3/11 First rewrite the question as a multiplication of fractions  write mixed number as a fraction

Your turn to try a problem.

Division of Fractions & Mixed Numbers Ex: Divide What are we doing with the division? The answer will be how many chunks of size ¼ we can make out of a part of size ½ ? Divide the whole into fourths? How many fourths are in ½?  ½ divided by ¼ = 2

What are we doing with the division? Ex: Divide The answer will be how many chunks of size ¼ we can make out of a part of size 2/3 ? Divide the whole into fourths? How many fourths are in 2/3? 1 2 But now what? Put the two pink pieces together . . . How much of a fourth do we have? 2/3  2/3 divided by 1/4 = 2 and 2/3 = 2 + 2/3 = 2 2/3

What is the process for dividing fractions? First, a definition: the reciprocal of a fraction is the fraction with the numerator and denominator interchanged (“flip it!”) Ex: Find the reciprocals of the following: = 5 5

“divided by” mathematically is the same operation as “times the reciprocal of” To divide fractions, multiply the first fraction (the dividend) by the reciprocal of the second fraction (the division) Ex: Divide = 2 Reciprocal of ¼ is __? 4/1

Ex: Divide Reciprocal of 3/5 is __? 5/3 The division is asking, “how many chunks of size 3/5 can be made from a part of size 8/9?” Answer: 1 whole chunk (of size 3/5) and 13/27 of another chunk (of size 3/5)

Since we know how to divide fractions, we can now divide fractions, whole numbers, and mixed numbers together To divide whole numbers, mixed numbers, and fractions first turn every number into a fraction.  fraction division: multiply the first fraction by the reciprocal of the second fraction.

Ex: Divide = 6 The division is asking, “how many chunks of size 2/3 can be made from a 4 wholes?” 1 2 3 4 Now divide the wholes into 3 How many chunks of size 2/3 are there? 1 2 3 4 5 6

Ex: Divide Reciprocal of 11/6 is __? 6/11