Decimals - Exercise 3(E)

Express the following as a recurring decimal:

$\dfrac{20}{3}$

Answer

By actual division, we get:

$\begin{array}{r} 6.666... \\ 3 \overline{\smash{)} 20.000 } \\ \underline{18} \phantom{.000} \\ 20 \phantom{.00} \\ \underline{18} \phantom{.00} \\ 20 \phantom{.0} \\ \underline{18} \phantom{.0} \\ 2 \phantom{.} \end{array}$

∴ $\dfrac{20}{3} = 6.666... = 6.\overline{6}$

Express the following as a recurring decimal :

$\dfrac{3}{11}$

Answer

By actual division, we get:

$\begin{array}{r} 0.2727... \\ 11 \overline{\smash{)} 3.0000 } \\ \underline{22} \phantom{.000} \\ 80 \phantom{.00} \\ \underline{77} \phantom{.00} \\ 30 \phantom{.0} \\ \underline{22} \phantom{.0} \\ 80 \phantom{} \\ \underline{77} \phantom{} \\ 3 \phantom{} \end{array}$

∴ $\dfrac{3}{11} = 0.2727... = 0.\overline{27}$

Express the following as a recurring decimal :

$\dfrac{5}{6}$

Answer

By actual division, we get:

$\begin{array}{r} 0.833... \\ 6 \overline{\smash{)} 5.000 } \\ \underline{48} \phantom{.00} \\ 20 \phantom{.0} \\ \underline{18} \phantom{.0} \\ 20 \phantom{} \\ \underline{18} \phantom{} \\ 2 \phantom{} \end{array}$

∴ $\dfrac{5}{6} = 0.833... = 0.8\overline{3}$

Express the following as a recurring decimal :

$\dfrac{17}{90}$

Answer

By actual division, we get:

$\begin{array}{r} 0.188... \\ 90 \overline{\smash{)} 17.000 } \\ \underline{90} \phantom{.00} \\ 800 \phantom{.0} \\ \underline{720} \phantom{.0} \\ 800 \phantom{} \\ \underline{720} \phantom{} \\ 80 \phantom{} \end{array}$

∴ $\dfrac{17}{90} = 0.188... = 0.1\overline{8}$

Express the following as a recurring decimal :

$\dfrac{1}{37}$

Answer

By actual division, we get:

$\begin{array}{r} 0.027027... \\ 37 \overline{\smash{)} 1.000000 } \\ \underline{0} \phantom{.00000} \\ 100 \phantom{.0000} \\ \underline{74} \phantom{.0000} \\ 260 \phantom{.000} \\ \underline{259} \phantom{.000} \\ 10 \phantom{.00} \\ \underline{0} \phantom{.00} \\ 100 \phantom{.0} \\ \underline{74} \phantom{.0} \\ 26 \phantom{} \end{array}$

∴ $\dfrac{1}{37} = 0.027027... = 0.\overline{027}$

Express the following as a recurring decimal :

$\dfrac{22}{7}$

Answer

By actual division, we get:

$\begin{array}{r} 3.142857... \\ 7 \overline{\smash{)} 22.000000 } \\ \underline{21} \phantom{.00000} \\ 10 \phantom{.0000} \\ \underline{7} \phantom{.0000} \\ 30 \phantom{.000} \\ \underline{28} \phantom{.000} \\ 20 \phantom{.00} \\ \underline{14} \phantom{.00} \\ 60 \phantom{.0} \\ \underline{56} \phantom{.0} \\ 40 \phantom{} \\ \underline{35} \phantom{} \\ 50 \phantom{} \\ \underline{49} \phantom{} \\ 1 \phantom{} \end{array}$

∴ $\dfrac{22}{7} = 3.14285714... = 3.\overline{142857}$

Express the following as a recurring decimal :

$\dfrac{2}{13}$

Answer

By actual division, we get:

$\begin{array}{r} 0.153846... \\ 13 \overline{\smash{)} 2.000000 } \\ \underline{13} \phantom{.00000} \\ 70 \phantom{.0000} \\ \underline{65} \phantom{.0000} \\ 50 \phantom{.000} \\ \underline{39} \phantom{.000} \\ 110 \phantom{.00} \\ \underline{104} \phantom{.00} \\ 60 \phantom{.0} \\ \underline{52} \phantom{.0} \\ 80 \phantom{} \\ \underline{78} \phantom{} \\ 2 \phantom{} \end{array}$

∴ $\dfrac{2}{13} = 0.153846153846... = 0.\overline{153846}$

Convert the following into a vulgar fraction :

${0.\overline{6}}$

Answer

Let $x = 0.\overline{6}$. Then,

x = 0.6666... (i)

⇒ 10x = 6.6666... (ii)

On subtracting (i) from (ii), we get:

$9x = 6 \\[1em] \Rightarrow x = \dfrac{6}{9} = \dfrac{2}{3}$

Hence, $0.\overline{6} = \dfrac{2}{3}$

Convert the following into a vulgar fraction :

${0.\overline{8}}$

Answer

Let $x = 0.\overline{8}$. Then,

x = 0.8888... (i)

⇒ 10x = 8.8888... (ii)

On subtracting (i) from (ii), we get:

$9x = 8 \\[1em] \Rightarrow x = \dfrac{8}{9}$

Hence, $0.\overline{8} = \dfrac{8}{9}$

Convert the following into a vulgar fraction :

${0.\overline{34}}$

Answer

Let $x = 0.\overline{34}$. Then,

x = 0.343434... (i)

⇒ 100x = 34.343434... (ii)

On subtracting (i) from (ii), we get:

$99x = 34 \\[1em] \Rightarrow x = \dfrac{34}{99}$

Hence, $0.\overline{34} = \dfrac{34}{99}$

Convert the following into a vulgar fraction :

${2.\overline{13}}$

Answer

Let x = $2.\overline{13}$.

Then,

x = 2.131313... (i)

⇒ 100x = 213.131313... (ii)

On subtracting (i) from (ii), we get:

$99x = 211 \\[1em] \Rightarrow x = \dfrac{211}{99} \\[1em] \Rightarrow x = 2\dfrac{13}{99}$

Hence, $2.\overline{13} = 2\dfrac{13}{99}$

Convert the following into a vulgar fraction :

${1.\overline{243}}$

Answer

Let $x = 1.\overline{243}$. Then,

x = 1.243243243... (i)

⇒ 1000x = 1243.243243... (ii)

On subtracting (i) from (ii), we get:

$999x = 1242 \\[1em] \Rightarrow x = \dfrac{1242}{999} \\[1em] \Rightarrow x = 1\dfrac{243}{999} \\[1em]$

Hence, $1.\overline{243} = 1\dfrac{243}{999}$

Convert the following into a vulgar fraction :

${0.1\overline{6}}$

Answer

Let $x = 0.1\overline{6} = 0.1666...$

Multiplying by 10 to move the non-repeating digit:

10x = 1.6666... $\qquad$..... (i)

Multiplying by 100 to move the decimal past the first repeating digit:

100x = 16.6666... $\qquad$..... (ii)

On subtracting (i) from (ii), we get:

$90x = 15 \\[1em] \Rightarrow x = \dfrac{15}{90} = \dfrac{1}{6}$

Hence, $0.1\overline{6} = \dfrac{1}{6}$.

Convert the following into a vulgar fraction :

${0.1\overline{43}}$

Answer

Let $x = 0.1\overline{43} = 0.1434343...$

Multiplying by 10 to move the decimal past the non-repeating digit:

10x = 1.434343... $\qquad$ ..... (i)

Multiplying by 1000 to move the decimal past the first repeating block:

1000x = 143.434343... $\qquad$ ..... (ii)

Subtracting (i) from (ii), we get:

$990x = 142 \\[1em] \Rightarrow x = \dfrac{142}{990} = \dfrac{71}{495}$

Hence, $0.1\overline{43} = \dfrac{71}{495}$

Convert the following into a vulgar fraction :

${0.57\overline{4}}$

Answer

Let $x = 0.57\overline{4} = 0.57444...$

Multiplying by 100 to move the decimal past the non-repeating digits:

100x = 57.444..... (i)

Multiplying by 10 to move the decimal past the first repeating block:

1000x = 574.444..... (ii)

Subtracting (i) from (ii):

$900x = 517 \\[1em] \Rightarrow x = \dfrac{517}{900}$

Hence, $0.57\overline{4} = \dfrac{517}{900}$

Convert the following into a vulgar fraction :

${0.12\overline{34}}$

Answer

Let $x = 0.12\overline{34} = 0.12343434...$

Multiplying by 100 to move the decimal past the non-repeating digits:

100x = 12.343434..... (i)

Multiplying by 100 to move the decimal past the first repeating block:

10000x = 1234.343434..... (ii)

Subtracting (i) from (ii):

$9900x = 1222 \\[1em] \Rightarrow x = \dfrac{1222}{9900} = \dfrac{611}{4950}$

Hence, $0.12\overline{34} = \dfrac{611}{4950}$