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Lesson 2.3.6

Adding and Subtracting Fractions. Lesson 2.3.6. Lesson 2.3.6. Adding and Subtracting Fractions. California Standards: Number Sense 1.2

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Lesson 2.3.6

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  1. Adding and Subtracting Fractions Lesson 2.3.6

  2. Lesson 2.3.6 Adding and Subtracting Fractions California Standards: Number Sense 1.2 Add, subtract, multiply, and divide rational numbers(integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers. Number Sense 2.2 Add and subtract fractions by using factoring to find common denominators. What it means for you: You’ll see how to add and subtract fractions. • Key words: • denominator • numerator • common denominator

  3. 1 5 3 2 3 1 2 5 6 2 6 6 3 6 6 6 + = + = + = These fractions have a common denominator,so they can be easily added together. These fractions have different denominators, which means you’ve got to do a bit of work before you can add them together. Lesson 2.3.6 Adding and Subtracting Fractions Adding and subtracting fractions can be quick, or it can be quite a long process — it all depends on whether the fractions already have a common denominator, or whether you have to find it first.

  4. 3 3 2 + 3 5 2 5 2 5 – 3 + 6 6 6 6 6 6 6 6 = = = = – Lesson 2.3.6 Adding and Subtracting Fractions You Can Add Fractions With a Common Denominator If fractions have a common denominator (their denominators are the same), adding them is fairly straightforward. To find the numerator of the sum, you add the numerators of the individual fractions. The denominatorstays the same. You subtractfractions with a common denominator in exactly the same way.

  5. Find + . 2 3 2 3 5 7 7 7 7 7 So + = . Lesson 2.3.6 Adding and Subtracting Fractions Example 1 Solution These two fractions have a common denominator, 7. So 7will also be the denominator of the sum. The numeratorof the sum will be 2 + 3 = 5. Solution follows…

  6. Example 2 Find – . 2 7 7 5 2 9 9 9 9 9 So – = . Lesson 2.3.6 Adding and Subtracting Fractions Solution The denominator of the result will be 9(the fractions’ common denominator). The numeratorof the result will be 7 – 2 = 5. Solution follows…

  7. Find the sums and differences in Exercises 1–8. 1. 2. 3. 4. 5. 6. 7. 8. 1 2 5 3 4 4 7 7 1 1 4 2 11 21 42 2 8 7 6 3 3 + + + 25 15 17 50 21 11 25 5 5 25 11 11 15 15 17 17 21 5 5 5 5 + 9 10 – – – 25 21 23 19 + or 50 50 Lesson 2.3.6 Adding and Subtracting Fractions Guided Practice Solution follows…

  8. 1 3 2 5 1 3 6 2 6 6 1 • 3 1 • 2 2 • 3 3 • 2 + = + = + = Lesson 2.3.6 Adding and Subtracting Fractions You May Need to Find a Common Denominator First Fractions with unlike denominators cannot be directly added or subtracted. You must first find equivalent fractions with a common denominator.

  9. Find + . 6 9 5 5 7 7 9 9 6 6 6 9 42 45 5 × 9 7 × 6 × = = × = = 54 54 6 × 9 9 × 6 Using 54 as a common denominator, the equivalent fractions are: and . Lesson 2.3.6 Adding and Subtracting Fractions Example 3 Solution The denominators are different here. This means you need to find two fractions equivalent to them, but with a common denominator. The common denominator can be any common multiple of 9 and 6. For example you could use 9 × 6 = 54. Solution continues… Solution follows…

  10. Find + . 7 5 9 6 87 29 45 42 54 54 54 18 Now you can add these fractions: + = , which you can simplify to by dividing the numerator and denominator by 3. Lesson 2.3.6 Adding and Subtracting Fractions Example 3 Solution (continues) Or you could find the LCM (least common multiple) using prime factorizations and use this as a common denominator. Since 9 = 32 and 6 = 2 × 3, the LCM is 2 × 3 × 3 = 18. Solution continues…

  11. Find + . 7 3 5 5 2 7 6 6 9 3 9 2 Using a common denominator of 18, the equivalent fractions are: and . 15 14 29 14 15 5 × 3 7 × 2 × = = × = = 18 18 18 18 18 6 × 3 9 × 2 Now you can add these fractions: + = . Lesson 2.3.6 Adding and Subtracting Fractions Example 3 Solution (continues)

  12. Lesson 2.3.6 Adding and Subtracting Fractions You can use the LCM or any other common multiple as your common denominator — you’ll end up with the same answer. But using the LCM means that the numbers in your fractions are smaller and easier to use.

  13. By putting both fractions over a common denominator, find: – . 3 3 11 11 20 20 15 15 5 5 5 3 3 2 2 2 2 15 20 9 44 60 60 LCM Find equivalent fractions with denominator 60: and 11 × 4 3 × 3 = = = = 20 × 3 15 × 4 Lesson 2.3.6 Adding and Subtracting Fractions Example 4 Solution Use a table to find the LCM of 15 and 20 — this is 5 × 3 × 2 × 2 = 60. Solution continues… Solution follows…

  14. By putting both fractions over a common denominator, find: – . 3 3 3 11 11 11 20 20 20 15 15 15 So rewriting the subtraction gives: – = – = and 7 44 35 9 9 44 35 60 60 60 60 60 60 12 This can be simplified to = . = = Lesson 2.3.6 Adding and Subtracting Fractions Example 4 Solution (continued)

  15. 2 2 4 2 12 2 14 8 6 20 41 14 21 + = – = 15 21 15 21 15 21 30 30 30 7 3 3 5 + 14 7 – + 10 15 – + = – Lesson 2.3.6 Adding and Subtracting Fractions Guided Practice Calculate the answers in Exercises 9–11. 9. 10. 11. – Solution follows…

  16. Lesson 2.3.6 Adding and Subtracting Fractions Be Extra Careful if There are Negative Signs As always in math, if there are negative numbers around, you have to be extra careful. Remember — subtracting a negative number is exactly the same as adding a positive number.

  17. 3 –2 Find . – – 5 7 –3 2 This sum can be rewritten as — it means the same. + 5 7 Lesson 2.3.6 Adding and Subtracting Fractions Example 5 Solution This looks tricky because of all the negative signs. So take things slowly and carefully. Now you can find a common denominator and solve the problem in the same way as before. Solution continues… Solution follows…

  18. 3 3 –2 –2 Find . – – – – 5 5 7 7 –3 –21 –3 × 7 2 10 2 × 5 = = = = 5 5 × 7 35 7 7 × 5 35 10 –21 + 10 –11 11 or – + = = 35 35 35 35 –21 = 35 Lesson 2.3.6 Adding and Subtracting Fractions Example 5 Solution (continued) The LCM of 5 and 7 is 5 × 7 = 35. Put both these fractions over a common denominator of 35. Now you can add the two fractions in the same way as before.

  19. 7 4 1 7 4 9 3 7 21 25 2 25 29 21 8 16 48 12 9 48 16 48 18 48 18 24 6 8 3 24 18 24 2 3 – – – – + = – + = – 3 8 7 4 – – + = + = 6 9 25 7 – – – – + = – + = – = – 48 16 Lesson 2.3.6 Adding and Subtracting Fractions Guided Practice Calculate the answers in Exercises 12–14. Simplify your answers. 12. 13. 14. Solution follows…

  20. Calculate the following. Give all your answers in their simplest form. 1. 2. 3. 4. 5. 6. 7. 8. 3 5 2 2 5 5 8 7 2 8 75 10 1 59 6 5 60 56 7 16 5 9 17 8 5 9 7 5 7 9 7 8 + + 7 3 – – – – 16 16 19 7 + 20 30 131 1 – – 19 20 – 272 420 + – 20 21 Lesson 2.3.6 Adding and Subtracting Fractions Independent Practice Solution follows…

  21. Lesson 2.3.6 Adding and Subtracting Fractions Round Up Keep practicing this until it becomes routine... (i) find a common denominator, (ii) put both fractions over this denominator, (iii) do the addition or subtraction, (iv) simplify your answer if possible. You’ll see this again next Lesson.

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