Topic Review Solutions — Decimals and Real Numbers

Answers

1. Terminating and recurring decimals Fluency

   1. 5/8 = 0.625 — Terminating (8 = 2³)
   2. 4/9 = 0.4̇
   3. 7/40: 40 = 2³ × 5 — only factors 2 and 5, so terminates. 3/14: 14 = 2 × 7 — factor of 7, so recurs.
   4. Let x = 0.666…; 10x = 6.666…; 9x = 6; x = 6/9 = 2/3
   5. Let x = 0.454545…; 100x = 45.454545…; 99x = 45; x = 45/99 = 5/11
   6. 0.4 < 0.4̇ = 4/9 ≈ 0.444 < 0.45

2. Simplifying surds Fluency

   1. √20 = √(4×5) = 2√5
   2. √63 = √(9×7) = 3√7
   3. √150 = √(25×6) = 5√6
   4. √288 = √(144×2) = 12√2
   5. 7²=49, 8²=64 → 7 < √55 < 8 (closer to 7)
   6. √36 = 6 (rational); √11 (irrational); √(4/25) = 2/5 (rational); √13 (irrational)

3. Classifying real numbers Understanding

   1. True
   2. False — √49 = 7, which is rational
   3. e.g. √17 (since 4²=16 and 5²=25, so 4 < √17 < 5)
   4. The decimal is non-terminating and the pattern never repeats with a fixed period, so it cannot be written as p/q and is irrational.
   5. √5 × √5 = 5, which is a rational number (in fact an integer). The product of two irrationals can be rational.

4. Ordering mixed types Understanding

   1. √2≈1.414, π/2≈1.571, 4/3≈1.333 → 4/3, √2, 1.5, π/2
   2. √8 = √(4×2) = 2√2 → =
   3. π≈3.142, √10≈3.162 → π < √10
   4. Rational example: 2.1. Irrational example: √(4.5) = √(9/2) ≈ 2.12. (Many answers possible.)
   5. √3≈1.732, so √3+1≈2.732 > 2

5. Real-world problems Problem Solving

   1. 5/12 = 0.416̇. As a percentage: 5/12 × 100 ≈ 41.7% → nearest whole number = 42%.
   2. Side = √72 = √(36×2) = 6√2 m ≈ 8.5 m
   3. 1. 2√5≈4.472 m, 4.5 m, 9/2=4.5 m → Order: 2√5, 4.5 = 9/2
      2. Yes, B and C are the same length: 4.5 = 9/2 exactly.
   4. 1/6 = 0.16̇ L per serving (first recipe); 1/8 = 0.125 L per serving (second recipe). First recipe serving is larger. Difference = 1/6 − 1/8 = 4/24 − 3/24 = 1/24 L.