Module 2: Fractions: Addition & Subtraction
Fractions with Different Denominators
Adding & Subtracting Fractions with Different Denominators To add or subtract fractions with different denominators, you need to find the least common denominator. This is the smallest number that can be divided by both denominators. Different Denominators 1. Find the least common denominator (LCD). 2. Use the LCD to find the equivalent fractions and rewrite the expression. 3. Add or subtract the numerators of all the fractions in the expression. 4. Keep the denominator the same—do not add them. If necessary, reduce the
identity property
Addition also has the identity property*: adding 0 to any number does not change the original number. For example, 0.924+0=0.924. The identity property says a+0=a for any real number a.
Identities
Addition is commutative*, meaning the order in which the numbers appear in the sum can be reversed. 3+2=2+3. If you are adding two or more numbers, it does not matter which order you add the numbers
Addition of Positive Numbers
Addition* is denoted by the plus sign, +. When multiple numbers are added to one another, we refer to the result as a sum*. Using the number line, a sum can be found by moving that many places to the right along the number line. For example, we can use the number line to find the sum, 3+2. Starting at 3, we can move 2 places to the right along the number line: Here, we visualize the sum (3+2=5) by moving along the number line.
Why do you need the same denominator?
Consider the following addition problem. You and your friends pool your left-over pizzas the next night and consider whether you have enough to invite a third friend over. You have 13 of a pizza and your friend has 12. How much pizza do you have altogether? The problem with unlike denominators is there's no way to express an answer. You and your friend have a fair amount of pizza—almost a whole pizza—but how do you tell someone who is not looking at the pizza how much it is? The answer is you find like denominators so that you have equal parts to count up. In the graphic below, you can see that a circle divided into 3 and a circle divided into 2 can be further divided into 6 equal parts. In the circle divided into 3, each section will be further divided into 2. So what used to be 1 part out of 3 is now 2 parts out of 6. In the circle divided into 2, each section will be further divided into 3. You now have 2 circles with the same number of equal parts. You can count how many pizza slices you have out of 6. Since each pizza was divided into 6 equal pieces, you can count the like objects. You can tell others that you have 56 of a pizza. Before you can add or subtract fractions, they must have the same denominator. That is, the numerators you are adding have to be divided by the same number. The easiest common denominator to work with is the least common denominator. Click on the tabs below to learn how to add and subtract different kinds of fractions.
Basic Arithmetic Refresher
Familiar with addition* and subtraction. Be using number. These will also be important for our work with negative.
Ex. 2 Subtract the following fractions:
In this example, let's take a look at an example where the final answer needs to be reduced. 5/6−1/6 5/6 - 1/6=4/6 Step 1: Subtract the numerators of each fraction. Remember the denominator stays In this example, let's take a look at an example where the final answer needs to be reduced. 56−16 5/6 - 1/6 Remember the denominator stays the same. (5−1)6=4/6 Step 2: Reduce the fraction (if necessary). 4/6 Notice that the both the numerator, 4 , and the denominator, 6, are both divisible by 2. Therefore we can reduce this fraction to: (4÷2)/(6÷2)=2/3 Therefore: 5/6−1/6=2/3
A calculator
Is a great resource here. Addition and subtraction expressions can be evaluated using the operations on any standard calculator. Click the addition, + , and subtraction, −, commands on the right side of the calculator to solve multi-step expressions like the ones in this section. Click the CE or AC command to clear any expression and enter a new one.
Example 1
Step 1. Add the following fractions: 2/7+3/7=5/7 Remember the denominator stays the same. (2+3)7=57 Step 2. Reduce the fraction (if necessary). 5/7 This fraction is already in reduced form. Therefore: 2/7+3/7=5/7
Ex 2 Let's add the following two mixed numbers: 6 2/3+7 3/4.
Step 1: Change the mixed number to an improper fraction 6 2/3=20/3 and 7 3/4=31/4 Step 2: Find equivalent fractions with the least common denominator. Convert the fractions to equivalent fractions with the LCD. The lowest common denominator is 3×4=12, therefore: Multiply the numerator and denominator of 203 by 4 Multiply the numerator and denominator of 314 by 3 20/3+31/4=80/12+93/12 Step 3: Add like fractions. Next, add the fractions. If this were a subtraction problem, you would simply subtract instead of adding. 8012+9312=17312 Step 4: To complete the problem, convert any improper fractions to lowest terms. 173/12=14 5/12 In this example, the fraction is already reduced to its lowest form.
Ex 3 Subtract the following mixed numbers: 8 5/6−5 1/2
Step 1: Change the mixed number to an improper fraction. 8 5/6=53/6 and 5 1/2=11/2 Step 2: Find equivalent fractions with the least common multiple. Convert the fractions to equivalent fractions with the LCM. The least common multiple is 6, therefore: 53/6 does not need to change Multiply the numerator and denominator of 11/2 by 3 53/6−11/2=53/6−33/6 Step 3: Subtract like fractions. Next, subtract the fractions. 53/6−33/6=20/6 Step 4: To complete the problem, convert any improper fractions to lowest terms. 20/6=3 2/6 Finally, reduce the fraction to its lowest form. 3 2/6=3 1/3
Example 1. Let's add the following two mixed numbers: 1 2/5−3/2
Step 1: Change the mixed number to an improper fraction. Here is a reminder of how to change a mixed number to an improper fraction. 1 2/5=7/5 Step 2: Find equivalent fractions with the least common denominator. Convert the fractions to equivalent fractions with the LCD. The lowest common denominator is 5×2=10, therefore: Multiply the numerator and denominator of 75 by 2 Multiply the numerator and denominator of 3/2 by 5 7/5−3/2=14/10−15/10 Step 3: Subtract Like Fractions Next, subtract the numerators of the fractions. Remember to keep the same denominator—do not subtract them! Step 4: To complete the problem, convert any improper fractions to lowest terms. In this example, the fraction is already reduced to its lowest form.
Example 1. Let's say we have an expression that is as follows: 1/2+2/3
Step 1: Find the least common denominator. Looking at the example above, the smallest possible multiple of 3 and 2 (our denominators) is 6; 6 is the smallest, common number that is the product* of an operation for both 2 and 3. Step 2: Write equivalent fractions. With that knowledge, we can manipulate both fractions to create equivalent fractions that have common denominators. This graphic shows both the numerator and denominator of one over two being multiplied by three and the numerator and denominator of two over three being multiplied by two to see that both fractions have the same denominator — in other words, the Least Common Multiple. The fractions now equal three over six and four over six, so the equation reads three over six plus four over six. 1/2+2/3 1X2/2X3+2X2/3X2 3/6+4/6 1/2+2/3=3/6+4/6 LCM 6 Step 3: Add the numerators of the fractions. Now that our fractions have common denominators, we can add the numerators and keep the same denominator. one half plus two thirds equals three sixths plus four sixths equals seven sixths. 1/2+2/3=3/6+4/6=7/6
Example 3. Subtract the following fractions: 2/3−5/9=
Step 1: Find the least common multiple (LCM) of both 3 and 9. The LCM of 3 and 9 is 9. Step 2: Convert fractions to have like denominators. 2/3−5/9= (2⋅3)/(3⋅3)−5/9= 6/9−5/9 Step 3: Subtract the like fractions. 6/9−5/9=1/9 Therefore: 2/3−5/9=1/9
Example 2. Add the following fractions: 1/4+1/3=
Step 1: Find the least common multiple (LCM) of both 4 and 3 The LCM of 4 and 3 is 12. Step 2: Convert fractions to have like denominators: Multiplying both the numerator and denominator by 3 is equivalent to dividing each of the old parts into 3 parts. Multiplying the numerator and denominator by 4 is equivalent to dividing each of the old parts into 4 parts. The amount of each fraction(shaded region of the pies) remains the same, but we have the thinner slices in each pie. The point is we have the same thinner slices for each of the pies. 1/4+1/3=(1⋅3)/(4⋅3)+(1⋅4)/(3⋅4)=3/12+4/12 Step 3: Now, we can add the like fractions: 3/12+4/12=7/12 Therefore: 1/4+1/3=7/12 adding one fourth and one third by finding a common denominator, splitting each of the parts into three and four parts respectively.
Example 4. Subtract the following fractions: 4/5−2/4=
Step 1: The LCD of 5 and 4 is 20. (If you happen to notice that you can reduce 24 to 12, you can do that step first and then find the LCD of 5 and 2. (It will make no difference to the final answer which version of the fraction you work with.) Step 2: Convert both fractions to the common denominators of 20. 4/5−2/4=(4⋅4)/(5⋅4)−(2⋅5)/(4⋅5)=16/20−10/20 Step 3: Now, we can subtract the like fractions. 16/20−10/20=6/20 Step 4: Reduce the answer to simplest form. Both 6 and 20 are divisible by 2. The final answer is 3/10 Therefore: 4/5−2/4=3/10
Subtraction of Positive Numbers
Subtraction* is denoted by the minus sign, −. When multiple numbers are subtracted from one another, we refer to the result as a difference*. Using the number line, a difference can be found by moving that many places to the left along the number line. For example, we can use the number line to find the difference, 7−4. Starting at 7, we can move 4 places to the left along the number line: Here, we visualize the difference 7−4=3 by moving along the number line. Note: Subtraction is not commutative. 7−3≠3−7 .
Additive inverses
There is a special relationship between values that are equally far from 0 on opposite sides of the number line, such as −3 and 3. These numbers are called additive inverses*. The sum of any number and its inverse is 0.
In order to add or subtract fractions or mixed numbers, the denominators must be the same.
When you encounter fractions or mixed numbers with different denominators, you must first transform them into fractions with a common denominator.
Fractions with Different Denominators
Adding & Subtracting Fractions with Different Denominators To add or subtract fractions with different denominators, you need to find the least common denominator. This is the smallest number that can be divided by both denominators. Different Denominators 1. Find the least common denominator (LCD). 2. Use the LCD to find the equivalent fractions and rewrite the expression. 3. Add or subtract the numerators of all the fractions in the expression. 4. Keep the denominator the same—do not add them. 5. If necessary, reduce the answer. 1/4+1/3=or less than 2/7 Notice that you cannot just add 1+1=2 in the numerator and 4+3=7 in the denominator to get an answer of 2/7. 1/3 alone is bigger than 2/7
Fractions with the Same Denominator
Adding & Subtracting Fractions with the Same Denominator To add fractions with the same denominator, follow these two steps: Same Denominators 1. Add or subtract the numerators of all the fractions in the expression 2. Keep the same denominator! (The temptation to add the two denominators is very strong—resist.) 3. If necessary, reduce the answer.
-,+ Mixed Numbers
Whether you are subtracting or adding mixed numbers, the steps are the same. Mixed Numbers 1. Change the mixed numbers* to improper fractions*. 2. Find the least common denominator (LCD) if the fractions have different denominators and convert to equivalent fractions with the LCD. 3. Add or subtract the numerators of all the fractions in the expression. 4. Keep the denominator the same. 5. Change improper fractions to a mixed number (if needed). 6. If necessary, reduce the fraction to lowest form.
Mathematical Expressions
Will often see mathematical expressions*. Expressions are a group of symbols, such as numbers* and operators*, that has mathematical validity. The most basic expressions are arithmetic expressions, such as: 3+3 6÷2 In math, you may be asked to "evaluate an expression." This statement is often synonymous with the statement "solve the problem."
