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Lesson 2.1.3

Converting Repeating Decimals to Fractions. Lesson 2.1.3. Lesson 2.1.3. Converting Repeating Decimals to Fractions. California Standard: Number Sense 1.5

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Lesson 2.1.3

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  1. Converting Repeating Decimals to Fractions Lesson 2.1.3

  2. Lesson 2.1.3 Converting Repeating Decimals to Fractions California Standard: Number Sense 1.5 Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into reduced fractions. What it means for you: You’ll see how to change repeating decimals into fractions that have the same value. Key Words: • fraction • decimal • repeating

  3. 0.27 3 11 Lesson 2.1.3 Converting Repeating Decimals to Fractions You’ve seen how to convert a terminating decimal into a fraction. But repeating decimals are also rational numbers, so they can be represented as fractions too. That’s what this Lesson is all about — taking a repeating decimal and finding a fraction with the same value.

  4. Look at the decimal 0.33333..., or 0.3. If you multiply it by 10, you get 3.33333..., or 3.3. 3.33333… – 0.33333… = 3 3.3 – 0.3 = 3 Lesson 2.1.3 Converting Repeating Decimals to Fractions Repeating Decimals Can Be “Subtracted Away” In both these numbers, the digits afterthe decimal point are the same. So if you subtractone from the other, the decimal part of the number “disappears.”

  5. Find 3.3 – 0.3 So 3.3 – 0.3 = 3. The digits after the decimal point in both these numbers are the same, since 0.3 = 0.3333… and 3.3 = 3.3333… 3.3333… – 0.3333… 3.0000… 3.3 – 0.3 3.0 Lesson 2.1.3 Converting Repeating Decimals to Fractions Example 1 Solution So when you subtract the numbers, the result has no digits after the decimal point. or Solution follows…

  6. Lesson 2.1.3 Converting Repeating Decimals to Fractions This idea of getting repeating decimals to “disappear” by subtracting is used when you convert a repeating decimal to a fraction. Solution follows…

  7. Example 2 If x = 0.3, find: (i) 10x, and (ii) 9x.Use your results to write x as a fraction in its simplest form. 3 1 9 3 (i) 10x = 10 × 0.3 = 3.3. (ii) 9x = 10x – x = 3.3 – 0.3 = 3 (from Example 1). x = , which can be simplified to x = . Lesson 2.1.3 Converting Repeating Decimals to Fractions Solution You now know that 9x = 3.So you can divide both sides by 9 to find x as a fraction: Solution follows…

  8. 10x = 10 × 0.4 = 4.4 4 11 9x = 10x – x = 4.4 – 0.4 = 4 9 9 9x = 4, divide both sides by 9 to give x = 9x = 11, divide both sides by 9 to give x = 10y = 10 × 1.2 = 12.2 9y = 10y – y = 12.2 – 1.2 = 11 Lesson 2.1.3 Converting Repeating Decimals to Fractions Guided Practice In Exercises 1–3, use x = 0.4.1. Find 10x. 2. Use your answer to Exercise 1 to find 9x.3. Write x as a fraction in its simplest form. In Exercises 4–6, use y = 1.2.4. Find 10y. 5. Use your answer to Exercise 4 to find 9y.6. Write y as a fraction in its simplest form. Solution follows…

  9. Let x = 2.510x = 10 × 2.5 = 25.59x = 10x – x = 25.5 – 2.5 = 239x = 23, divide both sides by 9 to give x = 23 37 23 9 9 9 Let x = 4.110x = 10 × 4.1 = 41.19x = 10x – x = 41.1 – 4.1 = 379x = 37, divide both sides by 9 to give x = Let x = –2.510x = 10 × –2.5 = –25.59x = 10x – x = –25.5 – –2.5 = –239x = –23, divide both sides by 9 to give x = – Lesson 2.1.3 Converting Repeating Decimals to Fractions Guided Practice Convert the numbers in Exercises 7–9 to fractions.7. 2.5 8. 4.1 9. –2.5 Solution follows…

  10. You May Need to Multiply by 100 or 1000 or 10,000... Lesson 2.1.3 Converting Repeating Decimals to Fractions If two digits are repeated forever, then multiply by 100before subtracting. If three digits are repeated forever, then multiply by 1000, and so on.

  11. Example 3 Convert 0.23 to a fraction. Now subtract: 100x – x = 23.23 – 0.23 = 23. 23 So 99x = 23, which means that x = . 99 100x = 23.23 Lesson 2.1.3 Converting Repeating Decimals to Fractions Solution Call the number x. There are tworepeating digits in x, so you need to multiply by 100before subtracting. Solution follows…

  12. Example 4 Convert 1.728 to a fraction. 1000y = 1728.728 Now subtract: 1000y – y = 1728.728 – 1.728 = 1727. 1727 So 999y = 1727, which means that y = . 999 Lesson 2.1.3 Converting Repeating Decimals to Fractions Solution Call the number y. There are threerepeating digits in y, so you need to multiply by 1000before subtracting. Solution follows…

  13. 99(0.09) = 100(0.09) – 0.09 = 9.09 – 0.09 = 9so 0.09 = 2 1 11 11 101 111 99(0.18) = 100(0.18) – 0.18 = 18.18 – 0.18 = 18so 0.18 = 999(0.909) = 1000(0.909) – 0.909 = 909.909 – 0.909 = 909so 0.909 = Lesson 2.1.3 Converting Repeating Decimals to Fractions Guided Practice For Exercises 10–12, write each repeating decimal as a fraction in its simplest form. 10. 0.09 11. 0.18 12. 0.909 Solution follows…

  14. 999(0.123) = 1000(0.123) – 0.123 = 123.123 – 0.123 = 123so 0.123 = 70 33 41 333 99(2.12) = 100(2.12) – 2.12 = 212.12 – 2.12 = 210so 2.12 = 1234 9999 9999(0.1234) = 10,000(0.1234) – 0.1234= 1234.1234 – 0.1234 = 1234so 0.1234 = Lesson 2.1.3 Converting Repeating Decimals to Fractions Guided Practice For Exercises 13–15, write each repeating decimal as a fraction in its simplest form. 13. 0.123 14. 2.12 15. 0.1234 Solution follows…

  15. Lesson 2.1.3 Converting Repeating Decimals to Fractions The Numerator and Denominator Must Be Integers You won’t always get a whole number as the result of the subtraction. If this happens, you may need to multiplythe numerator and denominator of the fraction to make sure they are both integers.

  16. Using 34.33 rather than 34.3 makes the subtraction easier. 10x = 34.33 Subtract as usual: 10x – x = 34.33 – 3.43 = 30.9. 30.9 So 9x = 30.9, which means that x = . 9 Lesson 2.1.3 Converting Repeating Decimals to Fractions Example 5 Convert 3.43 to a fraction. Solution Call the number x. There is onerepeating digit in x, so multiply by 10. Solution continues… Solution follows…

  17. 30.9 × 10 309 103 30.9 So 9x = 30.9, which means that x = . x = = , or more simply, x = . 9 × 10 90 30 9 Lesson 2.1.3 Converting Repeating Decimals to Fractions Example 5 Convert 3.43 to a fraction. Solution (continued) But the numerator here isn’t an integer, so multiply the numerator and denominator by 10 to get an equivalent fraction of the same value.

  18. 9(1.12) = 10(1.12) – 1.12 = 11.22 – 1.12 = 10.1so 1.12 = 2311 990 99(2.334) = 100(2.334) – 2.334 = 233.434 – 2.334 = 231.1so 2.334 = 18,089 33,300 999(0.54321) = 1000(0.54321) – 0.54321= 543.21321 – 0.54321 = 542.67so 0.54321 = 101 90 Lesson 2.1.3 Converting Repeating Decimals to Fractions Guided Practice For Exercises 16–18, write each repeating decimal as a fraction in its simplest form. 16. 1.12 17. 2.334 18. 0.54321 Solution follows…

  19. 1. 0.8 2. 0.7 3. 1.14. 0.26 5. 4.87 6. 0.2467. 0.142857 8. 3.142857 9. 10.01 26 10 8 22 1 7 99 9 9 7 9 7 901 161 82 333 90 33 Lesson 2.1.3 Converting Repeating Decimals to Fractions Independent Practice Convert the numbers in Exercises 1–9 to fractions. Give your answers in their simplest form. Solution follows…

  20. Lesson 2.1.3 Converting Repeating Decimals to Fractions Round Up This is a really handy 3-step method — (i) multiply by 10, 100, 1000, or whatever, (ii) subtract the original number, and (iii) divide to form your fraction.

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