Extension 14.4 Repeating Lesson Tutorials

You have written terminating decimals as . Because repeating decimals are rational , you can also write repeating decimals as fractions.

Writing a Repeating as a Let a variable x equal the repeating decimal d. = Rational Numbers Step 1: Write the equation x d. In this extension, you will Step 2: Multiply each side of the equation by 10n to form a new ● write a repeating decimal as a fraction. equation, where n is the of repeating digits. Step 3: Subtract the original equation from the new equation. Step 4: Solve for x.

EXAMPLE 1 Writing a Repeating Decimal as a Fraction (1 Digit Repeats) — Write 0. 4 as a fraction in simplest form. — Let x = 0. 4 . — x = 0. 4 Step 1: Write the equation. — 10 ⋅ x = 10 ⋅ 0. 4 Step 2: There is 1 repeating digit, so multiply each side by 101 = 10. — Check 10x = 4.4 Simplify. — 0.44 . . . − (x = 0.4 ) Step 3: Subtract the original equation. ;‾ 9 4.00 ✓ 3 6 9x = 4 Simplify. 40 4 x = — Step 4: Solve for x. 36 9 40 — 4 So, 0.4 = — . 9

Write the decimal as a fraction or a mixed number. — — — — 1. 0. 1 2. − 0. 5 3. −1. 2 4. 5. 8 5. STRUCTURE In Example 1, why can you subtract the original equation from the new equation after multiplying by 10? Explain why these two steps are performed. 6. REPEATED REASONING Compare the repeating decimals and their equivalent fractions in Exercises 1–4. Describe the pattern. Use the pattern to explain how to write a repeating decimal as a fraction when only the tenths digit repeats.

654 Chapter 14 Real Numbers and the Pythagorean Theorem

mms_accel_pe_1404b.indds_accel_pe_1404b.indd 654654 22/24/15/24/15 9:16:169:16:16 AMAM EXAMPLE 2 Writing a Repeating Decimal as a Fraction (1 Digit Repeats) — Write −0.2 3 as a fraction in simplest form. — Let x = − 0.23 . — x = − 0.23 Step 1: Write the equation. — Check 10 ⋅ x = 10 ⋅ (− 0.23 ) Step 2: There is 1 repeating digit, so multiply each side by 101 = 10. — 10x = − 2.3 Simplify. — − (x = − 0.23 ) Step 3: Subtract the original equation. 9x = − 2.1 Simplify. − 2.1 x = — Step 4: Solve for x. 9

— − 2.1 21 7 So, − 0.23 = — = − — = − — . 9 90 30

EXAMPLE 3 Writing a Repeating Decimal as a Fraction (2 Digits Repeat) — Write 1. 25 as a mixed number. — Let x = 1. 25 . — x = 1.25 Step 1: Write the equation. — Check 100⋅ x = 100 ⋅ 1. 25 Step 2: There are 2 repeating digits, so multiply each side by 102 = 100. — 100x = 125.25 Simplify. — − (x = 1. 25 ) Step 3: Subtract the original equation. 99x = 124 Simplify. 124 x = — Step 4: Solve for x. 99

— 124 25 So, 1.25 = — = 1 — . 99 99

Write the decimal as a fraction or a mixed number. — — — — 7. − 0.43 8. 2.0 6 9. 0. 27 10. − 4.50 11. REPEATED REASONING Find a pattern in the fractional representations of repeating decimals in which only the tenths and hundredths digits repeat. Use the pattern to — explain how to write 9.04 as a mixed number.

Extension 14.4 Repeating Decimals 655

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