Operations with Rational Numbers — Solutions

Click any answer to watch the solution video.

1. Add and Subtract Fractions

   1. ½ + ⅓ = ⁵⁄₆
   2. ¾ − ⅙ = ⁷⁄₁₂
   3. ²⁄₅ + ³⁄₁₀ = ⁷⁄₁₀
   4. ⁵⁄₆ − ¹⁄₄ = ⁷⁄₁₂
   5. ³⁄₈ + ⁵⁄₁₂ = ¹⁹⁄₂₄
   6. ⁷⁄₉ − ¹⁄₃ = ⁴⁄₉
   7. ²⁄₃ + ³⁄₇ = ²³⁄₂₁ = 1²⁄₂₁
   8. ⁹⁄₁₀ − ³⁄₄ = ³⁄₂₀

2. Multiply Fractions

   1. ½ × ¾ = ³⁄₈
   2. ²⁄₃ × ³⁄₅ = ²⁄₅
   3. ⁴⁄₇ × ⁷⁄₈ = ¹⁄₂
   4. 2½ × ⅘ = ⁵⁄₂ × ⁴⁄₅ = 2
   5. 1⅓ × 1½ = ⁴⁄₃ × ³⁄₂ = 2
   6. ⁵⁄₆ × ³⁄₁₀ = ¹⁄₄
   7. 2¾ × ⅔ = ¹¹⁄₄ × ²⁄₃ = ¹¹⁄₆ = 1⁵⁄₆
   8. ⁷⁄₉ × ³⁄₁₄ = ¹⁄₆

3. Divide Fractions

   1. ¾ ÷ ½ = 1½
   2. ²⁄₅ ÷ ⁴⁄₅ = ¹⁄₂
   3. ⁵⁄₆ ÷ ⁵⁄₃ = ¹⁄₂
   4. 1½ ÷ ¾ = 2
   5. ⁷⁄₈ ÷ ⁷⁄₄ = ¹⁄₂
   6. 2⅓ ÷ 1¾ = ⁷⁄₃ ÷ ⁷⁄₄ = ⁴⁄₃ = 1¹⁄₃
   7. ³⁄₁₀ ÷ ⁹⁄₅ = ¹⁄₆
   8. 4 ÷ ²⁄₃ = 6

4. Mixed Operations

   1. ½ + ⅓ × ¾ = ½ + ¼ = ³⁄₄
   2. 1½ − ⅔ = ³⁄₂ − ²⁄₃ = ⁵⁄₆
   3. ¾ × 2 + ⅛ = ³⁄₂ + ⅛ = ¹³⁄₈ = 1⁵⁄₈
   4. 2⅓ ÷ ⁷⁄₉ = ⁷⁄₃ × ⁹⁄₇ = 3
   5. 3 − 1⅖ = 1⅗
   6. ⅚ × ⅗ ÷ ½ = ¹⁄₂ ÷ ½ = 1
   7. 2½ + 1¾ − ⅚ = ⁵⁄₂ + ⁷⁄₄ − ⁵⁄₆ = ³⁰⁄₁₂ + ²¹⁄₁₂ − ¹⁰⁄₁₂ = ⁴¹⁄₁₂ = 3⁵⁄₁₂
   8. ⁴⁄₉ ÷ ²⁄₃ + ⅙ = ²⁄₃ + ⅙ = ⁵⁄₆

5. Order of Operations with Fractions

   1. (½ + ¼) × ²⁄₃ = ³⁄₄ × ²⁄₃ = ¹⁄₂
   2. ¾ ÷ (³⁄₈ + ³⁄₈) = ³⁄₄ ÷ ³⁄₄ = 1
   3. ⁵⁄₆ − (¹⁄₃ × ½) = ⁵⁄₆ − ⅙ = ⁴⁄₆ = ²⁄₃
   4. 2 × (³⁄₅ − ¹⁄₁₀) = 2 × ¹⁄₂ = 1
   5. (⁴⁄₅ + ²⁄₅) ÷ ³⁄₅ = ⁶⁄₅ ÷ ³⁄₅ = 2
   6. ¼ + ½ × (²⁄₃ ÷ ⅓) = ¼ + ½ × 2 = ¼ + 1 = 1¼
   7. (⁵⁄₈ − ⅜) × (1 + ¾) = ²⁄₈ × ⁷⁄₄ = ¼ × ⁷⁄₄ = ⁷⁄₁₆
   8. ⁷⁄₁₀ ÷ (⁷⁄₅ × ½) = ⁷⁄₁₀ ÷ ⁷⁄₁₀ = 1

6. Real-World Fraction Problems

   1. 2¼ × 1½ = ⁹⁄₄ × ³⁄₂ = 3⅜ cups of flour
   2. 3½ ÷ 7 = ¹⁄₂ kg each
   3. ¾ + 1⅓ + ⅚ = ⁹⁄₁₂ + ¹⁶⁄₁₂ + ¹⁰⁄₁₂ = ³⁵⁄₁₂ = 2¹¹⁄₁₂ km
   4. 4½ ÷ ³⁄₄ = ⁹⁄₂ × ⁴⁄₃ = 6; remainder = 4.5 − 6 × 0.75 = 0 — 6 pieces, no remainder
   5. 72½ × 2⅔ = ¹⁴⁵⁄₂ × ⁸⁄₃ = ⁵⁸⁰⁄₆ = 96²⁄₃ km ≈ 96.67 km
   6. 120 × ⁵⁄₈ = 75 L; 75 × (1 − ¼) = 75 × ¾ = 56.25 L (or 56¼ L)

7. Add and Subtract Mixed Numbers

   1. 1½ + 2¼ = ⁶⁄₄ + ⁹⁄₄ = ¹⁵⁄₄ = 3¾
   2. 3⅔ − 1⅙ = ¹¹⁄₃ − ⁷⁄₆ = ²²⁄₆ − ⁷⁄₆ = ¹⁵⁄₆ = 2½
   3. 4⅕ + 2¾ = ²¹⁄₅ + ¹¹⁄₄ = ⁸⁴⁄₂₀ + ⁵⁵⁄₂₀ = ¹³⁹⁄₂₀ = 6¹⁹⁄₂₀
   4. 5½ − 2⅔ = ¹¹⁄₂ − ⁸⁄₃ = ³³⁄₆ − ¹⁶⁄₆ = ¹⁷⁄₆ = 2⁵⁄₆
   5. 3⅜ + 1½ = ²⁷⁄₈ + ³⁄₂ = ²⁷⁄₈ + ¹²⁄₈ = ³⁹⁄₈ = 4⁷⁄₈
   6. 6¼ − 3⅚ = ²⁵⁄₄ − ²³⁄₆ = ⁷⁵⁄₁₂ − ⁴⁶⁄₁₂ = ²⁹⁄₁₂ = 2⁵⁄₁₂
   7. 2&frac79; + 1⅓ = ²⁵⁄₉ + ⁴⁄₃ = ²⁵⁄₉ + ¹²⁄₉ = ³⁷⁄₉ = 4¹⁄₉
   8. 4⅕ − 1½ = ²¹⁄₅ − ³⁄₂ = ⁴²⁄₁₀ − ¹⁵⁄₁₀ = ²⁷⁄₁₀ = 2⁷⁄₁₀

8. Rational Number Reasoning

   1. ¹⁷⁄₅ between which whole numbers: ¹⁷⁄₅ = 3.4, which lies between 3 and 4. (5 × 3 = 15 < 17 < 20 = 5 × 4)
   2. Ascending order: 1⅔ ≈ 1.667; ⁷⁄₄ = 1.75; 1.6; ¹¹⁄₆ ≈ 1.833. Order: 1.6, 1⅔, ⁷⁄₄, ¹¹⁄₆.
   3. ⁵⁄₃ ÷ ⁵⁄₆: ⁵⁄₃ × ⁶⁄₅ = ³⁰⁄₁₅ = 2. The student’s answer of ½ is incorrect; the correct answer is 2.
   4. Dividing by a fraction < 1: When you divide by a number less than 1, you are asking “how many of these small pieces fit into the original?” — which is more than the original number. e.g. 4 ÷ ½ = 8, because 8 halves fit into 4.

9. Multi-Step Fraction Problems

   1. Rope 4½ m ÷ ¾ m: ⁹⁄₂ ÷ ³⁄₄ = ⁹⁄₂ × ⁴⁄₃ = 6. 6 full pieces; remainder = 4.5 − 6 × 0.75 = 4.5 − 4.5 = 0 m (no remainder).
   2. n × ⅔ = ¹⁰⁄₉: n = ¹⁰⁄₉ ÷ ²⁄₃ = ¹⁰⁄₉ × ³⁄₂ = ³⁰⁄₁₈ = ⁵⁄₃ = 1⅔
   3. (1½ + ⅔) × (2 − ¾): = (³⁄₂ + ²⁄₃) × ⁵⁄₄ = (⁹⁄₆ + ⁴⁄₆) × ⁵⁄₄ = ¹³⁄₆ × ⁵⁄₄ = ⁶⁵⁄₂₄ = 2¹⁷⁄₂₄
   4. Container 5¼ L, cup ⅜ L: ²¹⁄₄ ÷ ³⁄₈ = ²¹⁄₄ × ⁸⁄₃ = ¹⁶⁸⁄₁₂ = 14. Exactly 14 full cups.

10. Extended Real-World Problems

    1. Baker: 6½ ÷ 1¾ = ¹³⁄₂ ÷ ⁷⁄₄ = ¹³⁄₂ × ⁴⁄₇ = ⁵²⁄₁₄ = ²⁶⁄₇ = 3⁵⁄₇. So 3 full batches. Sugar used: 3 × 1¾ = 5¼ cups. Remaining: 6½ − 5¼ = 1¼ cups.
    2. Pizza fractions: Zara + Marcus + Priya = ³⁄₈ + ²⁄₈ + ⁵⁄₁₆ = ⁶⁄₁₆ + ⁴⁄₁₆ + ⁵⁄₁₆ = ¹⁵⁄₁₆. Remaining = 1 − ¹⁵⁄₁₆ = ¹⁄₁₆. As a percentage: ½ ÷ 16 × 100 = 6.25%.
    3. Factory: 1⅔ × 7½ = ⁵⁄₃ × ¹⁵⁄₂ = ⁷⁵⁄₆ = 12.5 tonnes.