Terminating and Recurring Decimals — Solutions

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1. Convert fractions to decimals

   1. 1/4: 0.25 — T
   2. 1/3: 0.3̇ — R
   3. 3/5: 0.6 — T
   4. 2/9: 0.2̇ — R
   5. 7/8: 0.875 — T
   6. 5/11: 0.4̇5̇ — R
   7. 3/4: 0.75 — T
   8. 4/15: 0.2̇6̇ — R

2. Dot notation

   1. 0.4444…: 0.4̇
   2. 0.727272…: 0.7̇2̇
   3. 0.1666…: 0.16̇
   4. 0.583333…: 0.58̇3̇
   5. 0.142857142857…: 0.1̇4̇2̇8̇5̇7̇
   6. 0.090909…: 0.0̇9̇
   7. 0.3181818…: 0.31̇8̇
   8. 0.6666…: 0.6̇

3. Identify terminators without dividing

   1. 3/16: Terminating — 16 = 2⁴, only factor of 2
   2. 7/15: Recurring — 15 = 3 × 5, has factor of 3
   3. 9/25: Terminating — 25 = 5², only factor of 5
   4. 5/14: Recurring — 14 = 2 × 7, has factor of 7
   5. 11/40: Terminating — 40 = 2³ × 5, only factors of 2 and 5
   6. 3/7: Recurring — 7 is prime and not 2 or 5
   7. 13/50: Terminating — 50 = 2 × 5², only factors of 2 and 5
   8. 4/21: Recurring — 21 = 3 × 7, has factors of 3 and 7

4. Convert recurring decimals to fractions

   1. 0.4̇: Let x = 0.444…, 10x = 4.444…, 9x = 4, x = 4/9
   2. 0.7̇: Let x = 0.777…, 10x = 7.777…, 9x = 7, x = 7/9
   3. 0.1̇2̇: Let x = 0.1212…, 100x = 12.1212…, 99x = 12, x = 12/99 = 4/33
   4. 0.3̇6̇: Let x = 0.3636…, 100x = 36.3636…, 99x = 36, x = 36/99 = 4/11
   5. 0.8̇: Let x = 0.888…, 10x = 8.888…, 9x = 8, x = 8/9
   6. 0.2̇7̇: Let x = 0.2727…, 100x = 27.2727…, 99x = 27, x = 27/99 = 3/11
   7. 0.5̇: Let x = 0.555…, 10x = 5.555…, 9x = 5, x = 5/9
   8. 0.1̇4̇: Let x = 0.1414…, 100x = 14.1414…, 99x = 14, x = 14/99

5. Order mixed decimals

   1. 0.3, 1/3 ≈ 0.333, 0.3̇ = 0.333…, 0.34 → Order: 0.3, 1/3 = 0.3̇, 0.34
   2. 0.666 < 0.6̇6̇ = 0.666… = 2/3 < 0.67 → Order: 0.666, 0.6̇6̇ = 2/3, 0.67
   3. 0.6̇2̇ ≈ 0.6262, 0.625 = 5/8, 0.63 → Order: 0.6̇2̇, 5/8 = 0.625, 0.63
   4. 0.12 < 2/17 ≈ 0.1176 — wait: 2/17 ≈ 0.1176, 0.12, 0.1̇2̇ ≈ 0.1212, 0.125 → Order: 2/17, 0.12, 0.125, 0.1̇2̇
   5. 0.77, 0.7̇ = 0.777…, 0.778, 7/9 = 0.777… → Order: 0.77, 7/9 = 0.7̇, 0.778
   6. 0.16, 1/6 = 0.1666… = 0.16̇, 0.1̇6̇ = 0.1616…, 0.17 → Order: 0.16, 0.1̇6̇, 1/6, 0.17

6. Problem solving

   1. 1 ÷ 3 = 0.3̇ m each. Recurring — in practice you would round to e.g. 0.33 m and accept a tiny difference.
   2. Each sector = 1/9. 1 ÷ 9 = 0.1̇ (dot notation: 0.1̇)
   3. 1. 2/6 = 1/3 cup per person
      2. 1/3 = 0.3̇ — Recurring
      3. Use 0.33 cups. Difference = 0.333… − 0.33 = 0.003̇ ≈ 0.003 cups
   4. Let x = 0.999…, then 10x = 9.999…. Subtract: 9x = 9, so x = 1. Therefore 0.9̇ = 1 exactly. This means the two representations are the same real number.

7. Fraction | Decimal | % table

   1. Row 1: 0.25, 25%, T
   2. Row 2: fraction = 2/3, percentage = 66.6̇%, R
   3. Row 3: decimal = 0.375, T
   4. Row 4: fraction = 11/25, decimal = 0.44, T
   5. Row 5: decimal = 0.5̇, percentage = 55.5̇%, R
   6. Row 6: fraction = 7/20, percentage = 35%, T

8. Classification table

   1. Terminating: 0.75, 7/8, 11/25, 0.625, 9/40 — Recurring: 1/3, 0.1̇, 5/6, 4/15, 0.4̇5̇
   2. Reasoning: a fraction terminates iff its denominator (in lowest terms) has only 2s and/or 5s as prime factors.

9. Number line

   A = 0.25 (halfway between 0 and 0.5), B = 0.3̇ ≈ 0.333 (slightly past 0.25 toward 0.5), C = 0.5 (midpoint), D = 0.6̇6̇ ≈ 0.667 (past midpoint, near 0.75), E = 0.75 (three-quarters)

10. Error analysis

    1. Error: 3/6 = 1/2 in lowest terms; denominator = 2 (only factor of 2), so it TERMINATES. The student failed to simplify the fraction first.
    2. Error: 0.4̇5̇ has a 2-digit repeating block, so multiply by 100, not 10. Let x = 0.454545…, 100x = 45.4545…, 99x = 45, x = 45/99 = 5/11.
    3. Incorrect. 0.7̇ = 0.777… Let x = 0.777…, 10x = 7.777…, 9x = 7, x = 7/9.
    4. Counter-example: 1.3̇ = 1.333… > 1. Any whole number plus a recurring decimal part gives a recurring decimal greater than 1.