Converting Repeating Decimals to Fractions Lesson 2.1.3

2.1.3 Converting Repeating Decimals to Fractions California Standard: 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

2.1.3 Converting Repeating Decimals to Fractions 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. 0.27 3 11

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

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

2.1.3 Converting Repeating Decimals to Fractions 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…

2.1.3 Converting Repeating Decimals to Fractions Lesson 2.1.3 Converting Repeating Decimals to Fractions 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. Solution (i) 10x = 10 × 0.3 = 3.3. (ii) 9x = 10x – x = 3.3 – 0.3 = 3 (from Example 1). You now know that 9x = 3. So you can divide both sides by 9 to find x as a fraction: 3 9 1 x = , which can be simplified to x = . Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Guided Practice 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. 10x = 10 × 0.4 = 4.4 9x = 10x – x = 4.4 – 0.4 = 4 4 9 9x = 4, divide both sides by 9 to give x = 10y = 10 × 1.2 = 12.2 9y = 10y – y = 12.2 – 1.2 = 11 11 9 9x = 11, divide both sides by 9 to give x = Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Guided Practice 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 23 9 Let x = 2.5 10x = 10 × 2.5 = 25.5 9x = 10x – x = 25.5 – 2.5 = 23 9x = 23, divide both sides by 9 to give x = 37 9 Let x = 4.1 10x = 10 × 4.1 = 41.1 9x = 10x – x = 41.1 – 4.1 = 37 9x = 37, divide both sides by 9 to give x = 23 9 Let x = –2.5 10x = 10 × –2.5 = –25.5 9x = 10x – x = –25.5 – –2.5 = –23 9x = –23, divide both sides by 9 to give x = – Solution follows…

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

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

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

2.1.3 Converting Repeating Decimals to Fractions Guided Practice 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 99(0.09) = 100(0.09) – 0.09 = 9.09 – 0.09 = 9 so 0.09 = 1 11 99(0.18) = 100(0.18) – 0.18 = 18.18 – 0.18 = 18 so 0.18 = 2 11 999(0.909) = 1000(0.909) – 0.909 = 909.909 – 0.909 = 909 so 0.909 = 101 111 Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Guided Practice 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 999(0.123) = 1000(0.123) – 0.123 = 123.123 – 0.123 = 123 so 0.123 = 41 333 99(2.12) = 100(2.12) – 2.12 = 212.12 – 2.12 = 210 so 2.12 = 70 33 9999(0.1234) = 10,000(0.1234) – 0.1234 = 1234.1234 – 0.1234 = 1234 so 0.1234 = 1234 9999 Solution follows…

2.1.3 Converting Repeating Decimals to Fractions 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 multiply the numerator and denominator of the fraction to make sure they are both integers.

2.1.3 Converting Repeating Decimals to Fractions Lesson 2.1.3 Converting Repeating Decimals to Fractions Example 5 Convert 3.43 to a fraction. Solution Call the number x. There is one repeating digit in x, so multiply by 10. 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. So 9x = 30.9, which means that x = . 30.9 9 Solution continues… Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Lesson 2.1.3 Converting Repeating Decimals to Fractions Example 5 Convert 3.43 to a fraction. Solution (continued) So 9x = 30.9, which means that x = . 30.9 9 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. x = = , or more simply, x = . 30.9 × 10 9 × 10 309 90 103 30

2.1.3 Converting Repeating Decimals to Fractions Guided Practice 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 101 90 9(1.12) = 10(1.12) – 1.12 = 11.22 – 1.12 = 10.1 so 1.12 = 2311 990 99(2.334) = 100(2.334) – 2.334 = 233.434 – 2.334 = 231.1 so 2.334 = 18,089 33,300 999(0.54321) = 1000(0.54321) – 0.54321 = 543.21321 – 0.54321 = 542.67 so 0.54321 = Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Independent Practice 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. 1. 0.8 2. 0.7 3. 1.1 4. 0.26 5. 4.87 6. 0.246 7. 0.142857 8. 3.142857 9. 10.01 8 9 7 9 10 9 26 99 161 33 82 333 1 7 22 7 901 90 Solution follows…

2.1.3 Converting Repeating Decimals to Fractions Round Up 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.