Rational Numbers as Decimals Worksheet 2
1. Express the following rational numbers as decimals by using long division method.
(i) \( \frac{21}{16} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}1.3125\\ 16 \enclose{longdiv}{21\phantom{0000}}\\ \hspace{0.3cm}-16\\ \hline \hspace{0.8cm}50\\ \hspace{0.45cm}-48\\ \hline \hspace{1cm}20 \\ \hspace{.65cm}-16\\ \hline \hspace{1.2cm}40 \\ \hspace{.9cm}-32\\ \hline \hspace{1.5cm}80 \\ \hspace{1.15cm}-80\\ \hline \hspace{1.6cm}0\\ \hline \end{array} \]
Answer \( \frac{21}{16} = \color{red}1.3125 \)
(ii) \( \frac{129}{25} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}5.16\\ 25 \enclose{longdiv}{129\phantom{00}}\\ \hspace{0.3cm}-125\\ \hline \hspace{0.8cm}40\\ \hspace{0.45cm}-25\\ \hline \hspace{1cm}150 \\ \hspace{.65cm}-150\\ \hline \hspace{1.2cm}0 \\ \hline \end{array} \]
Answer \( \frac{129}{25} = \color{red}5.16 \)
(iii) \( \frac{17}{200} \)
Solution
\[ \begin{array}{l} \hspace{1cm}0.085\\ 200 \enclose{longdiv}{17.00}\\ \hspace{0.5cm}-1600\\ \hline \hspace{1cm}1000 \\ \hspace{.65cm}-1000\\ \hline \hspace{1.5cm}0 \\ \hline\end{array} \]
Answer \( \frac{17}{200} = \color{red}0.085 \)
(iv) \( \frac{5}{11} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}0.4545\ldots \\ 11 \enclose{longdiv}{5.0000}\\ \hspace{0.3cm}-0\\ \hline \hspace{0.8cm}50\\ \hspace{0.45cm}-44\\ \hline \hspace{1cm}60 \\ \hspace{.65cm}-55\\ \hline \hspace{1.2cm}50 \\ \hspace{.9cm}-44\\ \hline \hspace{1.5cm}60 \\ \hspace{1.15cm}-55\\ \hline \hspace{1.6cm}50 \\ \hline \end{array} \] \[ \begin{aligned} \frac{5}{11} & = 0.4545\ldots \\ \\ \frac{5}{11} & = 0.\overline{45} \\ \\ \end{aligned} \]
Answer \( \frac{5}{11} = \color{red}0.\overline{45} \)
(v) \( \frac{22}{7} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}3.14285714\ldots \\ 7 \enclose{longdiv}{22.00000000}\\ \hspace{0.1cm}-21\\ \hline \hspace{0.6cm}10\\ \hspace{0.45cm}-7\\ \hline \hspace{.8cm}30 \\ \hspace{.5cm}-28\\ \hline \hspace{1cm}20 \\ \hspace{.7cm}-14\\ \hline \hspace{1.3cm}60 \\ \hspace{1cm}-56\\ \hline \hspace{1.5cm}40 \\ \hspace{1.2cm}-35\\ \hline \hspace{1.8cm}50 \\ \hspace{1.5cm}-49\\ \hline \hspace{2cm}10 \\ \hspace{1.9cm}-7\\ \hline \hspace{2.2cm}30 \\ \hspace{1.9cm}-28\\ \hline \hspace{2.4cm}2 \\ \hline \end{array} \] \[ \begin{aligned} \frac{22}{7} & = 3.14285714\ldots \\ \\ \frac{22}{7} & = 3.\overline{142857} \\ \\ \end{aligned} \]
Answer \( \frac{22}{7} = \color{red}3.\overline{142857} \)
(vi) \( \frac{31}{27} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}1.14814\ldots \\ 27 \enclose{longdiv}{31\phantom{000000}}\\ \hspace{0.3cm}-27\\ \hline \hspace{0.8cm}40\\ \hspace{0.45cm}-27\\ \hline \hspace{1cm}130 \\ \hspace{.65cm}-108\\ \hline \hspace{1.2cm}220 \\ \hspace{.9cm}-216\\ \hline \hspace{1.5cm}40 \\ \hspace{1.15cm}-27\\ \hline \hspace{1.8cm}130 \\ \hspace{1.45cm}-108\\ \hline \hspace{2.0cm}22 \\ \hline \end{array} \] \[ \begin{aligned} \frac{31}{27} & = 1.14814\ldots \\ \\ \frac{31}{27} & = 1.\overline{148} \\ \\ \end{aligned} \]
Answer \( \frac{31}{27} = \color{red}1.\overline{148} \)
(vii) \( \frac{2}{15} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}0.133\ldots \\ 15 \enclose{longdiv}{2.0000}\\ \hspace{0.3cm}-15\\ \hline \hspace{.91cm}50 \\ \hspace{.6cm}-45\\ \hline \hspace{1.2cm}50 \\ \hspace{.9cm}-45\\ \hline \hspace{1.5cm}5 \\ \hline \end{array} \] \[ \begin{aligned} \frac{2}{15} & = 0.133\ldots \\ \\ \frac{2}{15} & = 0.1\overline{3} \\ \\ \end{aligned} \]
Answer \( \frac{2}{15} = \color{red}0.1\overline{3} \)
(viii) \( \frac{63}{55} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}1.14545\ldots \\ 55 \enclose{longdiv}{63.0000}\\ \hspace{0.3cm}-55\\ \hline \hspace{0.8cm}80\\ \hspace{0.45cm}-55\\ \hline \hspace{1cm}250 \\ \hspace{.65cm}-220\\ \hline \hspace{1.2cm}300 \\ \hspace{.9cm}-275\\ \hline \hspace{1.5cm}250 \\ \hspace{1.15cm}-220\\ \hline \hspace{1.8cm}300 \\ \hspace{1.5cm}-275\\ \hline \hspace{2.1cm}25\\ \end{array} \] \[ \begin{aligned} \frac{63}{55} & = 1.14545\ldots \\ \\ \frac{63}{55} & = 1.1\overline{45} \\ \\ \end{aligned} \]
Answer \( \frac{63}{55} = \color{red}1.1\overline{45} \)
2. Without actual division, determine which of the following rational numbers have a terminating decimal representation and which have a non-terminating decimal representation.
(i) \( \frac{11}{4} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (4) \\ & 4 = 2 \times 2 \\ &\text{Prime factorization of 4 contains only 2.} \\ \\ \frac{11}{4} &\text{ has a terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Terminating} \)
(ii) \( \frac{13}{80} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (80) \\ & 80 = 2 \times 2 \times 2 \times 2 \times 5 \\ &\text{Prime factorization of 80 contains only 2 and 5.} \\ \\ \frac{13}{80} &\text{ has a terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Terminating} \)
(iii) \( \frac{15}{11} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (11) \\ & 11 = 11 \times 1 \\ &\text{It has 11 in its prime factorization. } \\ \\ \frac{15}{11} &\text{ has a non-terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Non-Terminating} \)
(iv) \( \frac{22}{7} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (7) \\ & 7 = 7 \times 1 \\ &\text{It has 7 in its prime factorization. } \\ \\ \frac{22}{7} &\text{ has a non-terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Non-Terminating} \)
(v) \( \frac{29}{250} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (250) \\ & 250 = 2 \times 5 \times 5 \times 5 \\ &\text{Prime factorization of 250 contains only 2 and 5.} \\ \\ \frac{29}{250} &\text{ has a terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Terminating} \)
(vi) \( \frac{37}{21} \)
Solution
\[ \begin{aligned} &\text{Prime factorization of denominator } (21) \\ & 21 = 3 \times 7 \\ &\text{It has 3 and 7 in its prime factorization. } \\ \\ \frac{37}{21} &\text{ has a non-terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Non-Terminating} \)
(vii) \( \frac{49}{14} \)
Solution
\[ \begin{aligned} &\text{Convert to lower form } \\ & = \frac{\cancel{49}^{7}}{\cancel{14}^{2}} \\ \\ & = \frac{7}{2}\\ \\ &\text{Prime factorization of denominator } (2) \\ & 2 = 2 \times 1 \\ &\text{Prime factorization of 2 contains only 2.} \\ \\ \frac{49}{14} &\text{ has a terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Terminating} \)
(viii) \( \frac{126}{45} \)
Solution
\[ \begin{aligned} &\text{Convert to lower form } \\ & = \frac{\cancel{126}^{14}}{\cancel{45}^{5}} \\ \\ & = \frac{14}{5}\\ \\ &\text{Prime factorization of denominator } (5) \\ & 5 = 5 \times 1 \\ &\text{Prime factorization of 5 contains only 5.} \\ \\ \frac{126}{45} &\text{ has a terminating decimal representation} \\ \end{aligned} \]Answer \( \color{red}\text{Terminating} \)
3. Find the decimal representation of the following rational numbers
(i) \( \frac{-27}{4} \)
Solution
\[ \begin{aligned} & = \frac{-27 \times 25}{4 \times 25} \\ \\ & = \frac{-675}{100} \\ \\ &= -6.75 \end{aligned} \]Answer \( \frac{-27}{4} = \color{red}-6.75 \)
(ii) \( \frac{-37}{60} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}-0.6166\ldots \\ 60 \enclose{longdiv}{37.000000}\\ \hspace{0.3cm}-0\\ \hline \hspace{0.8cm}370\\ \hspace{0.45cm}-360\\ \hline \hspace{1cm}100 \\ \hspace{.85cm}-60\\ \hline \hspace{1.2cm}400 \\ \hspace{.9cm}-360\\ \hline \hspace{1.5cm}400 \\ \hspace{1.2cm}-360\\ \hline \hspace{1.7cm}40 \\ \end{array} \] \[ \begin{aligned} \frac{-37}{60}& = -0.6166\ldots \\ \\ \frac{-37}{60}& = -0.61\overline{6} \\ \\ \end{aligned} \]
Answer \( -\frac{37}{60} = \color{red} -0.61\overline{6} \)
(iii) \( \frac{-18}{125} \)
Solution
\[ \begin{aligned} & = \frac{-18 \times 8}{125 \times 8} \\ \\ & = \frac{-144}{1000} \\ \\ & = -0.144 \end{aligned} \]Answer \( \frac{-18}{125} = \color{red}-0.144 \)
(iv) \( \frac{-15}{8} \)
Solution
\[ \begin{aligned} & = \frac{-15 \times 125}{8 \times 125} \\ \\ & = \frac{-1875}{1000} \\ \\ & = -1.875 \end{aligned} \]Answer \( \frac{-15}{8} = \color{red}-1.875 \)
4. If the number \( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} \) is expressed as a decimal, will it be terminating or non-terminating? Justify your answer.
Solution
\[ \begin{aligned} & = \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} \\ \\ \text{LCM} &= 60 \\ \\ &= \frac{30 +40 +45 + 48}{60} \\ \\ &= \frac{163}{60} \\ \\ & \text{Prime factorization of the denominator } (60) \\ & 60 = 2 \times 2 \times 3 \times 5 \\ & \text{Since the prime factorization of 60 contains 3,} \\ & \text{the decimal representation is non-terminating.} \end{aligned} \]Answer The number \( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} \color{red}\text{ is Non-Terminating} \)
5. Justify the following statements as True or False.
(i) \( \frac{22}{7} \) can be represented as a terminating decimal.
Answer \( \color{red} False \)
\[ \begin{aligned} & \text{Denominator has factor other than 2 and 5 } \\ & \text{This is non-terminating } \\ \end{aligned} \](ii) \( \frac{51}{512} \) can be represented as a terminating decimal.
Answer \( \color{red} True \)
\[ \begin{aligned} &\text{Prime factorization of } 512 \\ 512 & = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &\text{Prime factorization contains only 2.} \\ & \text{This is Terminating } \\ \end{aligned} \](iii) \( \frac{19}{45} \) can be represented as a non-terminating repeating decimal.
Answer \( \color{red} True \)
\[ \begin{aligned} &\text{Prime factorization of } 45 \\ 45 & = 3 \times 3 \times 5 \\ &\text{Prime factorization contains 3 which is other than 2 or 5.} \\ &\text{This is non-terminating repeating decimal.} \\ \end{aligned} \](iv) \( \frac{3}{17} \) cannot be represented as a non-terminating repeating decimal.
Answer \( \color{red} False \)
\[ \begin{aligned} &\text{Prime factorization of } 17 \\ 17 & = 17 \\ &\text{Prime factorization contains 17 which is other than 2 or 5.} \\ &\text{This is non-terminating repeating decimal.} \\ \end{aligned} \](v) If \( \frac{3}{2} \) and \( \frac{7}{5} \) are terminating decimals, then \( \frac{3}{2} + \frac{7}{5} \) is also a terminating decimal.
Answer \( \color{red} True \)
\[ \begin{aligned} &\text{Prime factorization of } 2 = 2 \\ &\text{Prime factorization of } 5 = 5 \\ &\text{Both denominators have only 2 and 5.} \\ &\text{So, } \frac{3}{2} + \frac{7}{5} \text{ is a terminating decimal.} \\ \end{aligned} \](vi) If \( \frac{1}{4} \) and \( \frac{1}{5} \) both have terminating decimal representations, then \( \frac{1}{4} \times \frac{1}{5} \) also has a terminating decimal representation.
Answer \( \color{red} True \)
\[ \begin{aligned} &\text{Prime factorization of } 4 = 2 \times 2 \\ &\text{Prime factorization of } 5 = 5 \\ &\text{Both denominators have only 2 and 5.} \\ &\text{So, } \frac{1}{4} \times \frac{1}{5} \text{ is a terminating decimal.} \\ \end{aligned} \]