Practice Maths

Topic Review — Real Numbers

Mixed Practice — L01 – L04

This review covers all lessons in the Real Numbers topic. Try each question before checking your answers.

Review Questions

  1. Classifying and converting real numbers. Fluency

    1. Classify each as rational or irrational:   √36,   √15,   0.&overline;4,   π/2,   −3/8
    2. Convert 0.&overline;27 to a fraction in simplest form.
    3. Convert 0.5&overline{4} to a fraction in simplest form.
    4. List all sets (ℕ, ℤ, ℚ, Irrational, ℝ) that contain each number:   −11,   √100,   5/3

  2. Operations and rounding. Fluency

    1. Round 0.048 762 to 3 significant figures.
    2. Round 5 499 to 2 significant figures.
    3. Calculate (3/5 + 1/4) × (2 − 7/8), giving your answer as a fraction.
    4. Calculate 4.8² − 2.3 × 1.6 to 3 significant figures.
    5. Arrange in ascending order: √7,   2.6,   13/5,   √8

  3. Surds — simplifying and operating. Understanding

    1. Simplify: √80
    2. Simplify: 3√12 − √27 + 2√3
    3. Simplify: 2√6 × 3√6
    4. Simplify: √5 × √45 − (√3)²
    5. Simplify: (2√7)² − √196

  4. Estimating and ordering surds. Understanding

    1. State the consecutive integers between which √40 lies.
    2. Estimate √40 to 1 decimal place. Show your working.
    3. Compare without a calculator:   5√2   versus   3√5
    4. Place in ascending order:   2√6,   5,   √26,   3√3
    5. A student estimates √90 ≈ 9.5. Check by squaring whether this is an overestimate or underestimate. What is a better 1 d.p. estimate?

  5. Mixed problem solving. Problem Solving

    1. A square courtyard has an area of 75 m². Find the exact perimeter. If fencing costs $48 per metre, estimate the total cost (to the nearest dollar) by first approximating the perimeter to 1 d.p.
    2. Two right triangles share a common hypotenuse. Triangle A has legs 3 cm and √7 cm. Triangle B has legs √5 cm and √11 cm. Which triangle has the longer hypotenuse? Show exact working.
    3. A student claims that √(a + b) = √a + √b for all positive a and b. Test this claim with a = 9, b = 16. Is the student correct? Provide a general counterexample or proof.
    4. The diagonal of a rectangle is √(l² + w²). A rectangle has length 2√5 cm and width √5 cm. Find the exact diagonal length, and verify your answer is reasonable by estimating each measurement first.
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