Rational and Irrational Numbers: math test solutions, Grade 8 – download the PDF

1 Classify the numbers / 4 pts

\(-\dfrac{9}{4}\): rational (0.5 pt). \(\sqrt{64} = 8\): rational (0.5 pt). \(\sqrt{45}\): irrational (0.5 pt). \(0.\overline{27} = \dfrac{3}{11}\): rational (0.5 pt). \(5\pi\): irrational (0.5 pt). \(0.3030030003\ldots\): irrational, the zeros keep growing so nothing repeats (0.5 pt).

Explanation: \(6^2 = 36 \lt 45 \lt 49 = 7^2\), so \(\sqrt{45}\) lies strictly between \(6\) and \(7\) and is not an integer (1 pt).

2 Repeating decimals / 4 pts

a) \(x = 0.1515\ldots\), \(100x = 15.1515\ldots\) (1 pt). \(99x = 15\) (0.5 pt), so \(x = \dfrac{15}{99} = \dfrac{5}{33}\) (0.5 pt).

b) \(x = 0.7222\ldots\), \(100x = 72.222\ldots\) and \(10x = 7.222\ldots\) (1 pt). \(90x = 65\) (0.5 pt), so \(x = \dfrac{65}{90} = \dfrac{13}{18}\) (0.5 pt).

3 Roots and equations / 4 pts

a) \(13\) (0.5 pt). b) \(-6\) because \((-6)^3 = -216\) (0.5 pt). c) \(\dfrac{7}{8}\) (1 pt). d) \(x = 11\) or \(x = -11\) (1 pt, 0.5 pt per solution). e) \(x = 0.3\) because \(0.3^3 = 0.027\) (1 pt).

4 Estimating a root / 3 pts

a) \(49 \lt 57 \lt 64\), so \(7 \lt \sqrt{57} \lt 8\) (1 pt).

b) \(7.5^2 = 56.25\) and \(7.6^2 = 57.76\), so \(7.5 \lt \sqrt{57} \lt 7.6\) (0.5 pt). The halfway point is \(7.55\): \(7.55^2 = 57.0025 \gt 57\), so \(\sqrt{57} \lt 7.55\) (1 pt). Therefore \(\sqrt{57} \approx 7.5\) (0.5 pt).

5 Order the numbers / 3 pts

Decimals: \(-\sqrt{10} \approx -3.162\) (1 pt), \(-3.2\), \(-\pi \approx -3.1416\), \(-\dfrac{10}{3} = -3.333\ldots\), \(-3.1\) (1 pt).

Order: \[-\dfrac{10}{3} \lt -3.2 \lt -\sqrt{10} \lt -\pi \lt -3.1\] (1 pt).

6 The rug / 2 pts

a) Side \(= \sqrt{30} \approx 5.477\), so about \(5.5\) ft (1 pt).

b) \(30\) is not a perfect square (\(5^2 = 25\), \(6^2 = 36\)), so \(\sqrt{30}\) is irrational (1 pt).