Rational Numbers - Exercise 4(H)

Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal :

(i) $\dfrac{11}{16}$

(ii) $\dfrac{17}{20}$

(iii) $\dfrac{44}{125}$

(iv) $\dfrac{9}{80}$

(v) $\dfrac{123}{200}$

(vi) $\dfrac{129}{320}$

(vii) $\dfrac{431}{500}$

(viii) $\dfrac{807}{1250}$

Answer

(i) $\dfrac{11}{16}$

The given number is $\dfrac{11}{16}$.

Its denominator is $16 = 2^4$.

Thus, the denominator of $\dfrac{11}{16}$ has no prime factor other than 2.

∴ $\dfrac{11}{16}$ is expressible as a terminating decimal.

(ii) $\dfrac{17}{20}$

The given number is $\dfrac{17}{20}$.

Its denominator is $20 = 2^2 \times 5^1$.

Thus, the denominator of $\dfrac{17}{20}$ has no prime factor other than 2 and 5.

∴ $\dfrac{17}{20}$ is expressible as a terminating decimal.

(iii) $\dfrac{44}{125}$

The given number is $\dfrac{44}{125}$.

Its denominator is $125 = 5^3$.

Thus, the denominator of $\dfrac{44}{125}$ has no prime factor other than 5.

∴ $\dfrac{44}{125}$ is expressible as a terminating decimal.

(iv) $\dfrac{9}{80}$

The given number is $\dfrac{9}{80}$.

Its denominator is $80 = 2^4 \times 5^1$.

Thus, the denominator of $\dfrac{9}{80}$ has no prime factor other than 2 and 5.

∴ $\dfrac{9}{80}$ is expressible as a terminating decimal.

(v) $\dfrac{123}{200}$

The given number is $\dfrac{123}{200}$.

Its denominator is $200 = 2^3 \times 5^2$.

Thus, the denominator of $\dfrac{123}{200}$ has no prime factor other than 2 and 5.

∴ $\dfrac{123}{200}$ is expressible as a terminating decimal.

(vi) $\dfrac{129}{320}$

The given number is $\dfrac{129}{320}$.

Its denominator is $320 = 2^6 \times 5^1$.

Thus, the denominator of $\dfrac{129}{320}$ has no prime factor other than 2 and 5.

∴ $\dfrac{129}{320}$ is expressible as a terminating decimal.

(vii) $\dfrac{431}{500}$

The given number is $\dfrac{431}{500}$.

Its denominator is $500 = 2^2 \times 5^3$.

Thus, the denominator of $\dfrac{431}{500}$ has no prime factor other than 2 and 5.

∴ $\dfrac{431}{500}$ is expressible as a terminating decimal.

(viii) $\dfrac{807}{1250}$

The given number is $\dfrac{807}{1250}$.

Its denominator is $1250 = 2^1 \times 5^4$.

Thus, the denominator of $\dfrac{807}{1250}$ has no prime factor other than 2 and 5.

∴ $\dfrac{807}{1250}$ is expressible as a terminating decimal.

By actual division, express each of the following rational numbers as a terminating decimal :

(i) $\dfrac{11}{8}$

(ii) $\dfrac{23}{16}$

(iii) $\dfrac{76}{125}$

(iv) $\dfrac{103}{40}$

(v) $\dfrac{17}{80}$

(vi) $\dfrac{2}{25}$

(vii) $\dfrac{1}{125}$

(viii) $\dfrac{309}{1250}$

Answer

(i) $\dfrac{11}{8}$

By actual division, we have:

$\begin{array}{r} 1.375 \\ 8 \overline{) 11.000} \\ \underline{-8\phantom{.000}} \\ 30\phantom{00} \\ \underline{-24\phantom{00}} \\ 60\phantom{0} \\ \underline{-56\phantom{0}} \\ 40 \\ \underline{-40} \\ 0 \end{array}$

∴ $\dfrac{11}{8}$ = 1.375

(ii) $\dfrac{23}{16}$

By actual division, we have:

$\begin{array}{r} 1.4375 \\ 16 \overline{) 23.0000} \\ \underline{-16\phantom{.0000}} \\ 70\phantom{000} \\ \underline{-64\phantom{000}} \\ 60\phantom{00} \\ \underline{-48\phantom{00}} \\ 120\phantom{0} \\ \underline{-112\phantom{0}} \\ 80 \\ \underline{-80} \\ 0 \end{array}$

∴ $\dfrac{23}{16}$ = 1.4375

(iii) $\dfrac{76}{125}$

By actual division, we have:

$\begin{array}{r} 0.608 \\ 125 \overline{) 76.000} \\ \underline{-0\phantom{.000}} \\ 760\phantom{00} \\ \underline{-750\phantom{00}} \\ 100\phantom{0} \\ \underline{-0\phantom{0}} \\ 1000 \\ \underline{-1000} \\ 0 \end{array}$

∴ $\dfrac{76}{125}$ = 0.608

(iv) $\dfrac{103}{40}$

By actual division, we have:

$\begin{array}{r} 2.575 \\ 40 \overline{) 103.000} \\ \underline{-80\phantom{.000}} \\ 230\phantom{00} \\ \underline{-200\phantom{00}} \\ 300\phantom{0} \\ \underline{-280\phantom{0}} \\ 200 \\ \underline{-200} \\ 0 \end{array}$

∴ $\dfrac{103}{40}$ = 2.575

(v) $\dfrac{17}{80}$

By actual division, we have:

$\begin{array}{r} 0.2125 \\ 80 \overline{) 17.0000} \\ \underline{-0\phantom{.0000}} \\ 170\phantom{000} \\ \underline{-160\phantom{000}} \\ 100\phantom{00} \\ \underline{-80\phantom{00}} \\ 200\phantom{0} \\ \underline{-160\phantom{0}} \\ 400 \\ \underline{-400} \\ 0 \end{array}$

∴ $\dfrac{17}{80}$ = 0.2125

(vi) $\dfrac{2}{25}$

By actual division, we have:

$\begin{array}{r} 0.08 \\ 25 \overline{) 2.00} \\ \underline{-0\phantom{.00}} \\ 20\phantom{0} \\ \underline{-0\phantom{.0}} \\ 200 \\ \underline{-200} \\ 0 \end{array}$

∴ $\dfrac{2}{25}$ = 0.08

(vii) $\dfrac{1}{125}$

By actual division, we have:

$\begin{array}{r} 0.008 \\ 125 \overline{) 1.000} \\ \underline{-0\phantom{.000}} \\ 10\phantom{00} \\ \underline{-0\phantom{.00}} \\ 100\phantom{0} \\ \underline{-0\phantom{0}} \\ 1000 \\ \underline{-1000} \\ 0 \end{array}$

∴ $\dfrac{1}{125}$ = 0.008

(viii) $\dfrac{309}{1250}$

By actual division, we have:

$\begin{array}{r} 0.2472 \\ 1250 \overline{) 309.0000} \\ \underline{-0\phantom{.0000}} \\ 3090\phantom{000} \\ \underline{-2500\phantom{000}} \\ 5900\phantom{00} \\ \underline{-5000\phantom{00}} \\ 9000\phantom{0} \\ \underline{-8750\phantom{0}} \\ 2500 \\ \underline{-2500} \\ 0 \end{array}$

∴ $\dfrac{309}{1250}$ = 0.2472

Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal :

(i) $\dfrac{23}{24}$

(ii) $\dfrac{79}{30}$

(iii) $\dfrac{100}{9}$

(iv) $\dfrac{205}{27}$

(v) $\dfrac{461}{60}$

(vi) $\dfrac{1003}{112}$

(vii) $\dfrac{127}{225}$

(viii) $\dfrac{219}{440}$

Answer

(i) $\dfrac{23}{24}$

The given rational number is $\dfrac{23}{24}$.

Its denominator is $24 = 2^3 \times 3$.

Thus, the denominator of $\dfrac{23}{24}$ has at least one prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{23}{24}$ is expressible as a repeating decimal.

(ii) $\dfrac{79}{30}$

The given rational number is $\dfrac{79}{30}$.

Its denominator is $30 = 2 \times 3 \times 5$.

Thus, the denominator of $\dfrac{79}{30}$ has a prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{79}{30}$ is expressible as a repeating decimal.

(iii) $\dfrac{100}{9}$

The given rational number is $\dfrac{100}{9}$.

Its denominator is $9 = 3^2$.

Thus, the denominator of $\dfrac{100}{9}$ has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{100}{9}$ is expressible as a repeating decimal.

(iv) $\dfrac{205}{27}$

The given rational number is $\dfrac{205}{27}$.

Its denominator is $27 = 3^3$.

Thus, the denominator of $\dfrac{205}{27}$ has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{205}{27}$ is expressible as a repeating decimal.

(v) $\dfrac{461}{60}$

The given rational number is $\dfrac{461}{60}$.

Its denominator is $60 = 2^2 \times 3 \times 5$.

Thus, the denominator has at least one prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{461}{60}$ is expressible as a repeating decimal.

(vi) $\dfrac{1003}{112}$

The given rational number is $\dfrac{1003}{112}$.

Its denominator is $112 = 2^4 \times 7$.

Thus, the denominator has at least one prime factor, namely 7, other than 2 and 5.

∴ $\dfrac{1003}{112}$ is expressible as a repeating decimal.

(vii) $\dfrac{127}{225}$

The given rational number is $\dfrac{127}{225}$.

Its denominator is $225 = 3^2 \times 5^2$.

Thus, the denominator has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{127}{225}$ is expressible as a repeating decimal.

(viii) $\dfrac{219}{440}$

The given rational number is $\dfrac{219}{440}$.

Its denominator is $440 = 2^3 \times 5 \times 11$.

Thus, the denominator has a prime factor, namely 11, other than 2 and 5.

∴ $\dfrac{219}{440}$ is expressible as a repeating decimal.

By actual division, express each of the following as a repeating decimal :

(i) $\dfrac{103}{9}$

(ii) $\dfrac{7}{12}$

(iii) $\dfrac{101}{15}$

(iv) $\dfrac{303}{11}$

(v) $\dfrac{212}{143}$

(vi) $\dfrac{16}{7}$

(vii) $\dfrac{227}{30}$

(viii) $\dfrac{2000}{33}$

Answer

(i) $\dfrac{103}{9}$

By actual division, we have:

$\begin{array}{r} 11.444... \\ 9 \overline{) 103.000} \\ \underline{-9\phantom{.000}} \\ 13\phantom{.000} \\ \underline{-9\phantom{.000}} \\ 4.0\phantom{00} \\ \underline{-3.6\phantom{00}} \\ 40\phantom{0} \\ \underline{-36\phantom{0}} \\ 40 \\ \underline{-36} \\ 4... \end{array}$

∴ $\dfrac{103}{9} = 11.444... = 11.\overline{4}$

(ii) $\dfrac{7}{12}$

By actual division, we have:

$\begin{array}{r} 0.5833... \\ 12 \overline{) 7.0000} \\ \underline{-0\phantom{.0000}} \\ 70\phantom{000} \\ \underline{-60\phantom{000}} \\ 100\phantom{00} \\ \underline{-96\phantom{00}} \\ 40\phantom{0} \\ \underline{-36\phantom{0}} \\ 40 \\ \underline{-36} \\ 4... \end{array}$

∴ $\dfrac{7}{12} = 0.5833... = 0.58\overline{3}$

(iii) $\dfrac{101}{15}$

By actual division, we have:

$\begin{array}{r} 6.733... \\ 15 \overline{) 101.000} \\ \underline{-90\phantom{.000}} \\ 110\phantom{00} \\ \underline{-105\phantom{00}} \\ 50\phantom{0} \\ \underline{-45\phantom{0}} \\ 50 \\ \underline{-45} \\ 5... \end{array}$

∴ $\dfrac{101}{15} = 6.733... = 6.7\overline{3}$

(iv) $\dfrac{303}{11}$

By actual division, we have:

$\begin{array}{r} 27.5454... \\ 11 \overline{) 303.0000} \\ \underline{-22\phantom{.0000}} \\ 83\phantom{.0000} \\ \underline{-77\phantom{.0000}} \\ 60\phantom{000} \\ \underline{-55\phantom{000}} \\ 50\phantom{00} \\ \underline{-44\phantom{00}} \\ 60\phantom{0} \\ \underline{-55\phantom{0}} \\ 50 \\ \underline{-44} \\ 6... \end{array}$

∴ $\dfrac{303}{11} = 27.5454... = 27.\overline{54}$

(v) $\dfrac{212}{143}$

By actual division, we have:

$\begin{array}{r} 1.482517... \\ 143 \overline{) 212.000000} \\ \underline{-143\phantom{.000000}} \\ 690\phantom{00000} \\ \underline{-572\phantom{00000}} \\ 1180\phantom{0000} \\ \underline{-1144\phantom{0000}} \\ 360\phantom{000} \\ \underline{-286\phantom{000}} \\ 740\phantom{00} \\ \underline{-715\phantom{00}} \\ 250\phantom{0} \\ \underline{-143\phantom{0}} \\ 1070 \\ \underline{-1001} \\ 69... \end{array}$

∴ $\dfrac{212}{143} = 1.482517482517... = 1.\overline{482517}$

(vi) $\dfrac{16}{7}$

By actual division, we have:

$\begin{array}{r} 2.285714... \\ 7 \overline{) 16.000000} \\ \underline{-14\phantom{.000000}} \\ 20\phantom{00000} \\ \underline{-14\phantom{00000}} \\ 60\phantom{0000} \\ \underline{-56\phantom{0000}} \\ 40\phantom{000} \\ \underline{-35\phantom{000}} \\ 50\phantom{00} \\ \underline{-49\phantom{00}} \\ 10\phantom{0} \\ \underline{-7\phantom{0}} \\ 30 \\ \underline{-28} \\ 2... \end{array}$

∴ $\dfrac{16}{7} = 2.285714... = 2.\overline{285714}$

(vii) $\dfrac{227}{30}$

By actual division, we have:

$\begin{array}{r} 7.566... \\ 30 \overline{) 227.000} \\ \underline{-210\phantom{.000}} \\ 170\phantom{00} \\ \underline{-150\phantom{00}} \\ 200\phantom{0} \\ \underline{-180\phantom{0}} \\ 200 \\ \underline{-180} \\ 20... \end{array}$

∴ $\dfrac{227}{30} = 7.566... = 7.5\overline{6}$

(viii) $\dfrac{2000}{33}$

By actual division, we have:

$\begin{array}{r} 60.6060... \\ 33 \overline{) 2000.0000} \\ \underline{-198\phantom{0.0000}} \\ 20\phantom{.0000} \\ \underline{-0\phantom{.0000}} \\ 200\phantom{000} \\ \underline{-198\phantom{000}} \\ 20\phantom{00} \\ \underline{-0\phantom{00}} \\ 200\phantom{0} \\ \underline{-198\phantom{0}} \\ 20 \\ \underline{-0} \\ 20... \end{array}$

∴ $\dfrac{2000}{33} = 60.6060... = 60.\overline{60}$

Fill in the blanks :

(i) $\dfrac{2}{3} = ...............$

(ii) $\dfrac{11}{30} = ...............$

(iii) $\dfrac{13}{11} = ...............$

(iv) $\dfrac{23}{55} = ...............$

Answer

(i) $\dfrac{2}{3}$ = $0.\overline{6}$

(ii) $\dfrac{11}{30}$ = $0.3\overline{6}$

(iii) $\dfrac{13}{11}$ = $1.\overline{18}$

(iv) $\dfrac{23}{55}$ = $0.4\overline{18}$

Explanation

(i) $\dfrac{2}{3}$

By actual division:

$\begin{array}{r} 0.666... \\ 3 \overline{) 2.000} \\ \underline{-0\phantom{.000}} \\ 2.0\phantom{00} \\ \underline{-1.8\phantom{00}} \\ 20\phantom{0} \\ \underline{-18\phantom{0}} \\ 20 \\ \underline{-18} \\ 2... \end{array}$

∴ $\dfrac{2}{3} = 0.666... = 0.\overline{6}$

(ii) $\dfrac{11}{30}$

By actual division:

$\begin{array}{r} 0.366... \\ 30 \overline{) 11.000} \\ \underline{-0\phantom{.000}} \\ 110\phantom{00} \\ \underline{-90\phantom{00}} \\ 200\phantom{0} \\ \underline{-180\phantom{0}} \\ 200 \\ \underline{-180} \\ 20... \end{array}$

∴ $\dfrac{11}{30} = 0.366... = 0.3\overline{6}$

(iii) $\dfrac{13}{11}$

By actual division:

$\begin{array}{r} 1.1818... \\ 11 \overline{) 13.0000} \\ \underline{-11\phantom{.0000}} \\ 20\phantom{000} \\ \underline{-11\phantom{000}} \\ 90\phantom{00} \\ \underline{-88\phantom{00}} \\ 20\phantom{0} \\ \underline{-11\phantom{0}} \\ 90 \\ \underline{-88} \\ 2... \end{array}$

∴ $\dfrac{13}{11} = 1.1818... = 1.\overline{18}$

(iv) $\dfrac{23}{55}$

By actual division:

$\begin{array}{r} 0.41818... \\ 55 \overline{) 23.00000} \\ \underline{-0\phantom{.00000}} \\ 230\phantom{0000} \\ \underline{-220\phantom{0000}} \\ 100\phantom{000} \\ \underline{-55\phantom{000}} \\ 450\phantom{00} \\ \underline{-440\phantom{00}} \\ 100\phantom{0} \\ \underline{-55\phantom{0}} \\ 450 \\ \underline{-440} \\ 10... \end{array}$

∴ $\dfrac{23}{55} = 0.41818... = 0.4\overline{18}$