Topic Review — Decimals and Real Numbers

Review Questions

1. Terminating and recurring decimals: Fluency

   1. Convert 5/8 to a decimal. Is it terminating or recurring?
   2. Convert 4/9 to a decimal using dot notation.
   3. Without dividing, explain why 7/40 terminates but 3/14 recurs.
   4. Convert 0.6̇ to a fraction in simplest form. Show all algebraic steps.
   5. Convert 0.4̇5̇ to a fraction in simplest form.
   6. Order from smallest to largest: 0.4̇, 4/9, 0.45, 0.4

2. Simplifying surds: Fluency

   1. Simplify √20
   2. Simplify √63
   3. Simplify √150
   4. Simplify √288
   5. Estimate √55 between two consecutive integers without a calculator.
   6. Classify each: √36, √11, √(4/25), √13 as rational or irrational.

3. Classifying real numbers: Understanding

   1. True or False: All whole numbers are rational numbers.
   2. True or False: √49 is irrational.
   3. Name one irrational number between 4 and 5.
   4. Explain why 0.010010001… (where the number of zeros increases by 1 each time) is irrational.
   5. What type of real number is √5 × √5? Explain.

4. Ordering mixed types: Understanding

   1. Order from smallest to largest: √2, 1.5, π/2, 4/3
   2. Insert <, >, or = : √8  □  2√2
   3. Insert <, >, or = : π  □  √10
   4. Find a rational number and an irrational number between 2 and √5.
   5. Is √3 + 1 greater than or less than 2? Show working.

5. Real-world problems: Problem Solving

   1. A student scores 5 out of 12 on a test. Express this as a recurring decimal. What is the nearest whole-number percentage?
   2. A square swimming pool has area 72 m². Write the exact side length as a simplified surd, and estimate the length to one decimal place.
   3. Three lengths are measured: A = 2√5 m, B = 4.5 m, C = 9/2 m.
      1. Order them from shortest to longest.
      2. Are B and C the same length? Explain.
   4. A recipe divides 1 litre among 6 servings. A second recipe divides 1 litre among 8 servings. Write each serving as a decimal. Which serving is larger, and by exactly how much (as a fraction)?