Lesson 8: Working with Fractions One area of math that is important to understand is fractions. This lesson covers the basics of what fractions are and how to work with them. Lesson Objectives After completing this lesson, you will be able to:   

Identify the parts of a fraction. Describe what a fraction means. Perform simple calculations with fractions.

Definitions Here are some standard definitions to start the discussion of fractions. A unit is a single quantity with which another quantity of the same kind can be measured or compared. A fractional unit is one of the equal parts into which a unit has been divided. A fraction is one or more fractional units. Fractions allow you to count parts of a whole. A fraction is made up of two parts: the numerator, or top number, and the denominator, or bottom number. The numerator and denominator are separated by what’s called a fraction bar. Here is an example:

1 4 This fraction can be read as “one-fourth.” The numerator is 1 and the denominator is 4. The fraction can also be read as “one part of four parts” or “one divided by four.” Basic Concepts In the previous example, the number 4 represents the denominator of the fraction. The denominator is the bottom portion of the fraction. The denominator represents the size of the fractional unit; it shows how many parts the unit has been divided into. In this example, the unit is divided into 4 parts (as represented by the denominator). In this example, the number 1 represents the numerator of the fraction 1/4. The numerator is the top portion of the fraction. The numerator represents the number of fractional units. It shows how many parts are taken. In this example, 1 part is taken out of the 4 parts that make up the whole unit.

A fraction can also be looked at as a division problem. To solve the division problem, the numerator (dividend) is divided by the denominator (divisor) to solve for the quotient. In our example, 1 is divided by 4. 1 ÷ 4 = .25 This example results in a quotient with a value less than 1. This result is read as “twenty-five hundredths.” As you will learn in the next lesson, this value can be converted into a percentage by multiplying the decimal result by 100. For this example, the percentage is 25%. 1 ÷ 4 = .25 × 100 = 25% Therefore, 1/4 represents 25% of a whole unit. Adding of Fractions When adding or subtracting (“combining”) fractions, all of the fractions must have a common denominator. When fractions have a common denominator, simply add or subtract the numerators and put the result on top of the common denominator.

1 3 1 3 4    5 5 5 5 These fractions have the common denominator of 5. To solve, we added the numerators of the two fractions (1 + 3) and put the sum (4) over the common denominator. If the fractions do not have a common denominator, they are not compatible. To combine then, you must find the common denominator of the fractions and rename them all with that denominator. The first step is to find the lowest common denominator (LCD) so that you can rename the fractions and make them compatible. The lowest common denominator is the lowest, or least, number that both denominators will divide into evenly. It is also called the least common multiple. Often times, the denominators can all divide evenly into several numbers; however, you always want the lowest of these numbers. If you do not choose the lowest common denominator, you will need to continue reducing your result after combining the fractions. Here is an example: 2/5 + 1/3 The denominators of this example are not common. To find the LCD, list the multiples of each denominator. 2/5 1/3

List the multiples of 5: 5, 10, 15, 20, 25, 30… List the multiples 3: 3, 6, 9, 12, 15, 18…

What is the smallest number that is the same in both lists? That is right—15. The number 15 is the least common denominator of these two fractions. It is the smallest number that both 3 and 5 can divide into evenly. Now, to add the fractions together, both original fractions must be renamed with this new denominator. Both fractions must be “over” the lowest common denominator. The first fraction is 2/5. The denominator of 5 must be multiplied by 3 to get the LCD value of 15 (5 × 3 = 15). The rule of working with fractions is that the numerator must also be multiplied by this same value. In this case, the numerator is 2; 2 × 3 = 6. 2/5 = Numerator: 2 × 3 = 6 Denominator: 5 × 3 = 15 = 6/15 The second fraction is 1/3. The denominator of 3 must be multiplied by 5 to get the LCD value of 15 (3 × 5 = 15). What happens next? That is right! You must also multiply the numerator by 5, also (1 × 5 = 5). 1/3 = Numerator: 1 × 5 = 5 Denominator: 3 × 5 = 15 = 5/15 To perform the addition, add the numerators together over the common denominator. Then carry the denominator over. Here is the result:

2 1 2(3)  1(5) 6  5 11     5 3 15 15 15 Subtracting of Fractions Remember: when adding or subtracting (“combining”) fractions, all of the fractions must have a common denominator. Once the fractions have a common denominator, you can simply subtract the numerators and place the result on top of the common denominator. Subtracting fractions is very similar to the addition of fractions. Follow these steps: Step 1. Make sure the denominators are the same. Step 2. Place the numerators over the common denominator. Then subtract the numerators. Step 3. Simplify the fraction (if needed).

Let’s look at an example. 2/5 – 1/3 Step 1. Find the least common denominator. To find the LCD, list the multiples of each denominator. 2/5 1/3

List the multiples of 5: 5, 10, 15, 20, 25, 30… List the multiples 3: 3, 6, 9, 12, 15, 18…

What is the smallest number that is the same in both lists? That is right—15. The number 15 is the least common denominator of these two fractions. It is the smallest number that both 3 and 5 can divide into evenly. To subtract the fractions, both fractions must be renamed with this new denominator. In other words, both fractions must be “over” the lowest common denominator. The first fraction is 2/5. The denominator of 5 must be multiplied by 3 to get the LCD value of 15 (5 × 3 = 15). The rule of working with fractions is that the numerator must also be multiplied by this same value. In this case, the numerator is 2; 2 × 3 = 6. 2/5 = 6/15 The second fraction is 1/3. The denominator of 3 must be multiplied by 5 to get the LCD value of 15 (3 × 5 = 15). What happens next? That is right! You must also multiply the numerator by 5, also (1 × 5 = 5). 1/3 = 5/15 Step 2. Subtract the numerators. Place the result over the common denominator. 2/5 – 1/3 = 6/15 – 5/15 = 1/15 Step 3. Simplify. Since 1/15 cannot be simplified any further, the answer is 1/15. Multiplication and Division of Fractions Multiplying fractions is a little different. Follow these three steps: Step 1. Multiply the numerators. Step 2. Multiply the denominators. Step 3. Simplify the fraction if needed. (“Simplifying” means to reduce the expression to a simpler form.) Here is an example: 1/2 × 2/5

Step 1. Multiply the numerators. 1/2 × 2/5 1 × 2 = 2 Step 2. Multiply the denominators. 1/2 × 2/5 2 × 5 = 10 The answer is 2/10. However, this fraction can be simplified to a lower form. Step 3. Simplify the fraction. 2/10 = 1/5 Simplifying Fractions Fractions can be simplified in two ways. The first way is to divide both the numerator and the denominator by a value until you cannot divide any further. (Try dividing by the low numbers 2, 3, 5, 7, etc.) Here is an example: 24/48 ÷ 2/2 = 12/24 ÷ 2/2 = 6/12 ÷ 2/2 = 3/6 ÷ 2/2 = 1/2 (Alternately, you could have divided both the 6 and 12 of 6/12 by 6 to get the same results.) Here is a more complex example: 33/99 ÷ 3/3 = 11/33 ÷ 11/11 = 1/3 The second method involves identifying the greatest common factor and dividing the numerator and denominator by this value. The greatest common factor is simply the highest or largest number that exactly divides both the numerator and the denominator. Remember, factors are the numbers you multiply together to get another number. For example: 6 × 2 = 12 the factors are 6 and 2 3 × 7 = 21 the factors are 3 and 7 To determine the greatest common factor, follow these steps: Step 1. Find all the factors of each number. That means all of the values that can be multiplied together to get the number. Step 2. Identify the common factors. Which values appear in both lists? These are the common factors. Step 3. Choose the greatest of the common factors—the number with the highest value.

For example, you are looking for the greatest common factors for the following fraction: 12/16 Step 1. Find the factors of the numerator and the denominator. 12: 1, 2, 3, 4, 6, 12 (Why? Because 1 × 12 = 12, 2 × 6 = 12, 3 × 4 = 12) 16: 1, 2, 4, 8, 16 (Why? Because 1 × 16 = 16, 2 × 8 = 16, 4 × 4 = 16) Step 2. Identify the common factors. 12: 1, 2, 3, 4, 6, 12 16: 1, 2, 4, 8, 16 Step 3. Choose the greatest common factor. 12: 1, 2, 3, 4, 6, 12 16: 1, 2, 4, 8, 16 Finally, to simplify 12/16 using the greatest common factor, divide both the numerator (12) and the denominator (16) by the greatest common factor (4). 12/16 ÷ 4 = 3/4 No matter which simplification method you use, the idea is to simplify the fraction down to its lowest value possible.

Dividing Fractions Division of fractions is actually a form of multiplication. To divide two fractions, follow three steps. Step 1. Invert the second fraction (the divisor). This is a fancy way of saying reverse the order of the numerator and the denominator of the second fraction, or turn the fraction upside down. Step 2. Multiply the two fractions together, following the process you already learned. Step 3. Simplify the result. Let’s look at an example. 5/14 ÷ 10/21

Step 1. Invert the order of the second fraction. (Remember, this is the “divisor,” or the value you’re dividing by.) 5/14 ÷ 21/10 Step 2. Multiply the two fractions together. Remember, to multiply fractions, you first multiply the numerators together. Then you multiply the denominators together. 5/14 × 21/10

5 × 21 = 105 14 × 10 = 140

Step 3. Simplify the result. 105/140 = 105 ÷ 5 = 21 21 ÷ 7 = 3 140 ÷ 5 = 28 28 ÷ 7 = 4 The answer? 3/4 Let’s move on to the next lesson.