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

1 Sort the numbers ★★★

Rational: \(5 = \dfrac{5}{1}\); \(-\dfrac{2}{3}\) is already a fraction; \(\sqrt{49} = 7\); \(0.\overline{3} = \dfrac{1}{3}\); \(2.75 = \dfrac{11}{4}\) (terminating decimal).

Irrational: \(\sqrt{11}\) because \(11\) is not a perfect square, and \(\pi\), whose digits never end and never repeat.

2 Perfect squares and cubes ★★★

a) \(9\) because \(9^2 = 81\). b) \(12\) because \(12^2 = 144\). c) \(\dfrac{1}{2}\) because \(\left(\dfrac{1}{2}\right)^2 = \dfrac{1}{4}\). d) \(0.6\) because \(0.6^2 = 0.36\).

e) \(3\) because \(3^3 = 27\). f) \(-4\) because \((-4)^3 = -64\). g) \(\dfrac{2}{3}\) because \(\dfrac{2^3}{3^3} = \dfrac{8}{27}\).

3 Fractions to decimals ★★★

a) \(\dfrac{3}{8} = 0.375\): terminates. b) \(\dfrac{5}{9} = 0.\overline{5}\): repeats. c) \(\dfrac{7}{11} = 0.\overline{63}\): repeats (the block \(63\)). d) \(\dfrac{1}{6} = 0.1\overline{6}\): repeats (only the \(6\) repeats).

All four are rational numbers, as they come from fractions.

4 Between two integers ★★★

a) \(16 \lt 20 \lt 25\), so \(4\) and \(5\). b) \(49 \lt 50 \lt 64\), so \(7\) and \(8\). c) \(81 \lt 90 \lt 100\), so \(9\) and \(10\). d) \(4 \lt 5 \lt 9\), so \(2\) and \(3\).

e) \(27 \lt 30 \lt 64\), so \(3\) and \(4\). f) \(64 \lt 100 \lt 125\), so \(4\) and \(5\).

5 True or false? ★★★

a) True: \(n = \dfrac{n}{1}\). b) False: \(\sqrt{25} = 5\), which is rational. c) False: \(\dfrac{22}{7} = 3.142857\ldots\) repeats, while \(\pi = 3.14159\ldots\) does not; \(\dfrac{22}{7}\) is only an approximation.

d) True: for example \(0.62 = \dfrac{62}{100} = \dfrac{31}{50}\). e) False: irrational means "not rational", so no number is both.

6 Solve the equations ★★★

a) \(x = \pm 7\). b) \(x = \pm 1.5\) because \(1.5^2 = 2.25\). c) \(x = -2\). d) \(x = \dfrac{1}{5}\).

e) \(17\) is not a perfect square, so \(x = \pm\sqrt{17}\). f) \(20\) is not a perfect cube, so \(x = \sqrt[3]{20}\) (only one real solution).

7 Garden, box and tile ★★★

a) Side \(= \sqrt{169} = 13\) ft, which is \(13 \times 0.3048 \approx 3.96\) m.

b) Edge \(= \sqrt[3]{216} = 6\) in, since \(6^3 = 216\).

c) Side \(= \sqrt{50}\). Since \(7.07^2 = 49.9849\) and \(7.08^2 = 50.1264\), the side is about \(7.07\) cm.

8 Repeating decimals to fractions ★★★

a) \(x = 0.2727\ldots\), \(100x = 27.2727\ldots\), so \(99x = 27\) and \(x = \dfrac{27}{99} = \dfrac{3}{11}\).

b) Three digits repeat: \(1000x - x = 504\), so \(x = \dfrac{504}{999} = \dfrac{56}{111}\).

c) \(10x - x = 14 - 1 = 13\), so \(x = \dfrac{13}{9}\).

d) \(99x = 12\), so \(x = \dfrac{12}{99} = \dfrac{4}{33}\).

9 A digit in front of the block ★★★

a) \(x = 0.2777\ldots\); \(100x = 27.777\ldots\) and \(10x = 2.777\ldots\) Subtract: \(90x = 25\), so \(x = \dfrac{25}{90} = \dfrac{5}{18}\).

b) \(100x = 83.333\ldots\), \(10x = 8.333\ldots\), so \(90x = 75\) and \(x = \dfrac{75}{90} = \dfrac{5}{6}\).

c) \(1000x = 145.4545\ldots\), \(10x = 1.4545\ldots\), so \(990x = 144\) and \(x = \dfrac{144}{990} = \dfrac{8}{55}\).

10 Squeeze the root ★★★

a) \(3.6^2 = 12.96\) and \(3.7^2 = 13.69\), so \(3.6 \lt \sqrt{13} \lt 3.7\). Because \(13\) is much closer to \(12.96\), \(\sqrt{13} \approx 3.6\).

b) \(6.4^2 = 40.96\) and \(6.5^2 = 42.25\), so \(6.4 \lt \sqrt{41} \lt 6.5\), and \(\sqrt{41} \approx 6.4\).

c) \(8.66^2 = 74.9956\) and \(8.67^2 = 75.1689\), so \(8.66 \lt \sqrt{75} \lt 8.67\). \(75\) is much closer to \(74.9956\), so \(\sqrt{75} \approx 8.66\).

11 Which is bigger? ★★★

a) \(5.5^2 = 30.25 \gt 30\), so \(\sqrt{30} \lt 5.5\). b) \(\left(\dfrac{7}{2}\right)^2 = 12.25 \gt 12\), so \(\sqrt{12} \lt \dfrac{7}{2}\). c) \(2.1^3 = 9.261 \gt 9\), so \(\sqrt[3]{9} \lt 2.1\).

d) \(\dfrac{22}{7} = 3.142857\ldots\) and \(\pi = 3.14159\ldots\), so \(\pi \lt \dfrac{22}{7}\).

12 Rational or irrational results ★★★

a) \(3 + 4 = 7\): rational. b) \(\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4\): rational, even though both factors are irrational. c) \(2\pi\): irrational (nonzero rational times irrational).

d) \(\sqrt{0.09} = 0.3\): rational. e) \(4 + \sqrt{5}\): irrational (rational plus irrational).

13 Name that point ★★★

Estimate each number. \(-\sqrt{3} \approx -1.73\) is negative, so it is \(A\). \(\sqrt{2} \approx 1.41\) is \(B\). \(\sqrt{5} \approx 2.24\) is \(C\). \(\sqrt[3]{20} \approx 2.71\) (since \(2.7^3 = 19.683\) and \(2.8^3 = 21.952\)) is \(D\). \(\sqrt{12} \approx 3.46\) is \(E\).

14 The aquarium ★★★

The volume is \(64 \times 1000 = 64{,}000 \text{ cm}^3\). The edge is \(\sqrt[3]{64000} = 40\) cm since \(40^3 = 64{,}000\).

In inches: \(40 \div 2.54 \approx 15.7\) in. The edge of the aquarium is \(40\) cm, about \(15.7\) inches.

15 Pattern or no pattern? ★★★

a) The number of zeros keeps growing, so no block repeats and the decimal never ends: irrational.

b) A block repeats, so it is rational; in fact it equals \(\dfrac{142857}{999999} = \dfrac{1}{7}\).

c) The groups of \(2\)s keep growing, so there is no repeating block: irrational.

d) \(3.14 = \dfrac{314}{100} = \dfrac{157}{50}\): rational, a terminating decimal.

16 A tricky repeating decimal ★★★

Let \(x = 2.0363636\ldots\) Then \(10x = 20.363636\ldots\) and \(1000x = 2036.3636\ldots\) Subtract: \(990x = 2016\), so \(x = \dfrac{2016}{990} = \dfrac{112}{55}\).

Check: \(112 \div 55 = 2\) with remainder \(2\), and \(\dfrac{2}{55} = 0.0363636\ldots\), so \(\dfrac{112}{55} = 2.0\overline{36}\).

17 Is 0.999… equal to 1? ★★★

Let \(x = 0.999\ldots\) Then \(10x = 9.999\ldots\) Subtract: \(9x = 9\), so \(x = 1\). Therefore \(0.\overline{9} = 1\): there is no gap between them, because the difference would have to be smaller than every positive number.

So \(0.\overline{9}\) is just another way to write \(1 = \dfrac{1}{1}\), a rational number.

18 Order the negatives ★★★

Decimals: \(-\sqrt{7} \approx -2.646\) (since \(2.646^2 \approx 7.0013\)), \(-2.6\), \(-\dfrac{5}{2} = -2.5\), \(-\sqrt[3]{15} \approx -2.466\) (since \(2.466^3 \approx 14.996\)), \(-2.4\).

The farther from zero, the smaller the negative number: \[-\sqrt{7} \lt -2.6 \lt -\dfrac{5}{2} \lt -\sqrt[3]{15} \lt -2.4.\]

19 How many integers? ★★★

a) Square everything: \(49 \lt n \lt 64\), so \(n = 50, 51, \ldots, 63\). That is \(63 - 50 + 1 = 14\) integers; the smallest is \(50\) and the largest is \(63\).

b) \(6.2^2 = 38.44\) and \(6.3^2 = 39.69\), so \(38.44 \lt n \lt 39.69\). Only \(n = 39\) works: exactly one integer.

20 The hypotenuse ★★★

a) By the Pythagorean theorem, \(BC^2 = 9^2 + 5^2 = 81 + 25 = 106\), so \(BC = \sqrt{106}\) ft.

b) \(10.2^2 = 104.04\) and \(10.3^2 = 106.09\), so \(10.2 \lt \sqrt{106} \lt 10.3\), and the ramp is about \(10.3\) ft long.

c) \(106\) is not a perfect square (\(10^2 = 100\) and \(11^2 = 121\)), so \(\sqrt{106}\) is irrational.

21 Find the mistakes ★★★

a) \(25^2 = 625\), not \(50\). In fact \(7^2 = 49\) and \(8^2 = 64\), so \(\sqrt{50} \approx 7.07\).

b) The fraction \(\dfrac{45}{100}\) is the terminating decimal \(0.45\). The repeating decimal is \(\dfrac{45}{99} = \dfrac{5}{11}\).

c) \(16 \lt 20 \lt 25\), so \(\sqrt{20}\) is between \(4\) and \(5\).

d) Cube roots of negative numbers exist: \(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\).