Rational Numbers as Decimals Worksheet 3
1. Express the following decimals as rational numbers in standard form.
(i) \( 0.25 \)
Solution
\[ \begin{align*} & = \frac{\cancel{25}^{1}}{\cancel{100}^{4}} \\ \\ & = \frac{1}{4} \\ \\ \end{align*} \]
Answer \( 0.25 = \color{red}\frac{1}{4} \)
(ii) \( -0.052 \)
Solution
\[ \begin{align*} & = \frac{-\cancel{52}^{13}}{\cancel{1000}^{250}} \\ \\ & = \frac{-13}{250} \\ \\ \end{align*} \]
Answer \( -0.052 = \color{red}\frac{-13}{250} \)
(iii) \( 7.50 \)
Solution
\[ \begin{align*} & = \frac{75\cancel{0}}{10\cancel{0}} \\ \\ & = \frac{\cancel{75}^{15}}{\cancel{10}^{2}} \\ \\ & = \frac{15}{2} \\ \\ \end{align*} \]
Answer \( 7.50 = \color{red}\frac{15}{2} \)
(iv) \( -2.15 \)
Solution
\[ \begin{align*} & = \frac{-215}{100} \\ \\ & = \frac{-\cancelto{43}{215}}{\cancelto{20}{100}} \\ \\ & = \frac{-43}{20} \\ \\ \end{align*} \]
Answer \( -2.15 = \color{red}\frac{-43}{20} \)
(v) \( 0.036 \)
Solution
\[ \begin{align*} & = \frac{36}{1000} \\ \\ & = \frac{\cancelto{9}{36}}{\cancelto{250}{1000}} \\ \\ & = \frac{9}{250} \\ \\ \end{align*} \]
Answer \( 0.036 = \color{red}\frac{9}{250} \)
(vi) \( -9.6 \)
Solution
\[ \begin{align*} & = \frac{-96}{10} \\ \\ & = \frac{\cancelto{-48}{-96}}{\cancelto{5}{10}} \\ \\ & = \frac{-48}{5} \\ \\ \end{align*} \]
Answer \( -9.6 = \color{red}\frac{-48}{5} \)
(vii) \( 31.25 \)
Solution
\[ \begin{align*} & = \frac{3125}{100} \\ \\ & = \frac{{\cancel{3125}^{\cancel{\color{brown}625}}}^{\color{green}125}}{{\cancel{100}^{\cancel{\color{brown}20}}}^{\color{green}4}} \\ \\ & = \frac{125}{4} \\ \\ \end{align*} \]
Answer \( 31.25 = \color{red}\frac{125}{4} \)
(viii) \( 16.32 \)
Solution
\[ \begin{align*} & = \frac{1632}{100} \\ \\ & = \frac{\cancelto{408}{1632}}{\cancelto{25}{100}} \\ \\ & = \frac{408}{25} \\ \\ \end{align*} \]
Answer \( 16.32 = \color{red}\frac{408}{25} \)
(ix) \( 0.107 \)
Solution
\[ \begin{align*} & = \frac{107}{1000} \\ \\ \end{align*} \]
Answer \( 0.107 = \color{red}\frac{107}{1000} \)
2. Add the following decimals.
(i) \(2.5, 7.51\) and \(11.501\)
Solution
\[ \begin{array}{r} \hspace{0.7cm} 2.500 \\ + \hspace{0.5cm} 7.510 \\ + \hspace{0.3cm} 11.501 \\ \hline \hspace{0.7cm} 21.511 \\ \hline \end{array} \]
Answer \(2.5 + 7.51 + 11.501 = \color{red}21.511 \)
(ii) \(12, 5.96\) and \(3.076\)
Solution
\[ \begin{array}{r} \hspace{0.7cm} 12.000 \\ + \hspace{0.3cm} 5.960 \\ + \hspace{0.3cm} 3.076 \\ \hline \hspace{0.7cm} 21.036 \\ \hline \end{array} \]
Answer \(12 + 5.96 + 3.076 = \color{red}21.036 \)
(iii) \(9.08, 19.76\) and \(20.54\)
Solution
\[ \begin{array}{r} \hspace{0.7cm} 9.08 \\ + \hspace{0.3cm} 19.76 \\ + \hspace{0.3cm} 20.54 \\ \hline \hspace{0.7cm} 49.38 \\ \hline \end{array} \]
Answer \(9.08 + 19.76 + 20.54 = \color{red}49.38 \)
(iv) \(3.009, 0.592\) and \(14.745\)
Solution
\[ \begin{array}{r} \hspace{0.7cm} 3.009 \\ + \hspace{0.5cm} 0.592 \\ + \hspace{0.3cm} 14.745 \\ \hline \hspace{0.7cm} 18.346 \\ \hline \end{array} \]
Answer \(3.009 + 0.592 + 14.745 = \color{red}18.346 \)
(v) \(19, 9.5, 12.06\) and \(17.921\)
Solution
\[ \begin{array}{r} \hspace{0.7cm} 19.000 \\ + \hspace{0.53cm} 9.500 \\ + \hspace{0.3cm} 12.060 \\ + \hspace{0.3cm} 17.921 \\ \hline \hspace{0.7cm} 58.481 \\ \end{array} \]
Answer \(19 + 9.5 + 12.06 + 17.921 = \color{red}58.481 \)
3. Compute the following products of decimals.
(i) \(2.9 \times 3.5\)
Solution
\[ \begin{array}{r} & 29 \\ \times & 35 \\ \hline & 145 \\ + & 870 \\ \hline & 1015 \\ \hline \end{array} \] \[ 2.9 \times 3.5 = \color{green}10.15 \]
Answer \(2.9 \times 3.5 = \color{red}10.15 \)
(ii) \(37 \times 12.76\)
Solution
\[ \begin{array}{r} & 1276 \\ \times & 37 \\ \hline & 8932 \\ + & 38280 \\ \hline & 47212 \\ \hline \end{array} \] \[ 37 \times 12.76 = \color{green}472.12 \]
Answer \(37 \times 12.76 = \color{red}472.12 \)
(iii) \(0.84 \times 8.8\)
Solution
\[ \begin{array}{r} & 84 \\ \times & 88 \\ \hline & 672 \\ + & 6720 \\ \hline & 7392 \\ \hline \end{array} \] \[ 0.84 \times 8.8 = \color{green}7.392 \]
Answer \(0.84 \times 8.8 = \color{red}7.392 \)
(iv) \(2.56 \times 11.09\)
Solution
\[ \begin{array}{r} & 1109 \\ \times & 256 \\ \hline & 6654 \\ + & 55450 \\ + & 221800 \\ \hline & 283904 \\ \hline \end{array} \] \[ 2.56 \times 11.09 = \color{green}28.3904 \]
Answer \(2.56 \times 11.09 = \color{red}28.3904 \)
(v) \(12.4 \times 15.7 \times 13.2\)
Solution
\[ \color{green}12.4 \times 15.7 \] \[ \begin{array}{r} & 124 \\ \times & 157 \\ \hline & 868 \\ + & 6200 \\ + & 12400 \\ \hline & 19468 \\ \hline \end{array} \] \[ 12.4 \times 15.7 =\color{green} 194.68 \] \[ \text{Now multiply } 194.68 \text{ and } 13.2 \] \[ \begin{array}{r} & 19468 \\ \times & 132 \\ \hline & 38936 \\ + & 584040 \\ + & 1946800 \\ \hline & 25669776 \\ \hline \end{array} \] \[ 194.68 \times 13.2 = \color{green} 2569.776 \]
Answer \(12.4 \times 15.7 \times 13.2 = \color{red}2569.776 \)
4. Compute the following divisions.
(i) \( 59.049 \div 9 \)
Solution
\[ \begin{align*} & = \frac{59.049}{9} \\ \\ & = \frac{59.049 \times 1000}{9 \times 1000} \\ \\ & = \frac{\cancelto{6561}{59049}}{\cancelto{1}{9} \times 1000} \\ \\ & = \frac{6561}{1000} \\ \\ & = 6.561 \\ \\ \end{align*} \]
Answer \( 59.049 \div 9 = \color{red} 6.561 \)
(ii) \( 6.4 \div 0.2 \)
Solution
\[ \begin{align*} & = \frac{6.4}{0.2} \\ \\ & = \frac{6.4 \times 10}{0.2 \times 10} \\ \\ & = \frac{\cancelto{32}{64} }{\cancelto{1}{2} } \\ \\ & = 32 \\ \\ \end{align*} \]
Answer \( 6.4 \div 0.2 = \color{red} 32 \)
(iii) \( 0.015 \div 3 \)
Solution
\[ \begin{align*} & = \frac{0.015}{3} \\ \\ & = \frac{0.015 \times 1000}{3 \times 1000} \\ \\ & = \frac{\cancelto{5}{15}}{\cancelto{1}{3} \times 1000} \\ \\ & = \frac{5}{1000} \\ \\ & = 0.005 \\ \\ \end{align*} \]
Answer \( 0.015 \div 3 = \color{red}0.005 \)
(iv) \( 0.014 \div 12 \)
Solution
\[ \begin{align*} & = \frac{0.014}{12} \\ \\ & = \frac{0.014 \times 1000}{12 \times 1000} \\ \\ & = \frac{\cancelto{7}{14}}{\cancelto{6}{12} \times 1000} \\ \\ & = \frac{7}{6000} \\ \\ \end{align*} \] \[ \begin{array}{l} \hspace{1.1cm}0.0011666\ldots \\ 6000 \enclose{longdiv}{7.000000}\\ \hspace{1.1cm}7000\\ \hspace{.8cm}-6000\\ \hline \hspace{1.1cm}10000\\ \hspace{1cm}-6000\\ \hline \hspace{1.5cm}40000 \\ \hspace{1.2cm}-36000\\ \hline \hspace{1.7cm}40000 \\ \hspace{1.4cm}-36000\\ \hline \hspace{1.9cm}40000 \\ \hspace{1.6cm}-36000\\ \hline \hspace{2.2cm}4000 \\ \hline \end{array} \] \[ \begin{aligned} & = 0.0011666\ldots \\ \\ & = 0.0011\overline{6} \\ \\ \end{aligned} \]
Answer \( 0.014 \div 12 = \color{red}0.0011\overline{6} \)
(v) \( 0.02472 \div 0.008 \)
Solution
\[ \begin{align*} & = \frac{0.02472}{0.008} \\ \\ & = \frac{0.02472 \times 100000}{0.008 \times 100000} \\ \\ & = \frac{2472}{8\times 100} \\ \\ & = \frac{\cancelto{309}{2472}}{\cancelto{1}{8} \times 100} \\ \\ & = \frac{309}{100} \\ \\ & = 3.09 \\ \\ \end{align*} \]
Answer \( 0.02472 \div 0.008 = \color{red}3.09 \)
(vi) \( 51.51 \div 0.17 \)
Solution
\[ \begin{align*} & = \frac{51.51 }{0.17} \\ \\ & = \frac{51.51 \times 100}{0.17 \times 100} \\ \\ & = \frac{\cancel{5151}^{303}}{\cancel{17}^1} \\ \\ & = 303 \\ \\ \end{align*} \]
Answer \( 51.51 \div 0.17 = \color{red}303 \)
5. Evaluate the following:
(i) \(25.75 + 2.09 - 13.6\)
Solution
\[ \begin{array}{c|c} \begin{aligned} & =25.75 + 2.09 - 13.6 \\ \\ & = 27.84 - 13.6 \\ \\ & = 27.84 - 13.60 \\ \\ & = 14.24 \\ \\ \end{aligned} & \begin{aligned} \color{green}\hspace{0.7cm} 27.75 \\ \color{green}+ \hspace{0.3cm} 2.09 \\ \hline \color{green}\hspace{0.7cm} 27.84 \\ \hline \\ \color{green}\hspace{0.7cm} 27.84 \\ \color{green}- \hspace{0.3cm}13.60 \\ \hline \color{green}\hspace{0.7cm} 14.24 \\ \hline \end{aligned} \end{array} \]
Answer \(25.75 + 2.09 - 13.6 = \color{red}14.24 \)
(ii) \(37 - 16.58 + 12.25\)
Solution
\[ \begin{array}{c|c} \begin{aligned} & = 37 - 16.58 + 12.25 \\ \\ & = 20.42 + 12.25 \\ \\ & = 20.42 + 12.25 \\ \\ & = 32.67 \\ \\ \end{aligned} & \begin{aligned} \color{green}\hspace{0.7cm} 37.00 \\ \color{green}- \hspace{0.3cm} 16.58 \\ \hline \color{green}\hspace{0.7cm} 20.42 \\ \hline \\ \color{green}\hspace{0.7cm} 20.42 \\ \color{green}+ \hspace{0.3cm} 12.25 \\ \hline \color{green}\hspace{0.7cm} 32.67 \\ \hline \end{aligned} \end{array} \]
Answer \(37 - 16.58 + 12.25 = \color{red}32.67 \)
(iii) \(42.7 - 11 - 9.025 + 2.16\)
Solution
\[ \begin{array}{c|c} \begin{aligned} & = 42.7 - 11 - 9.025 + 2.16 \\ \\ & = 31.7 - 9.025 + 2.16 \\ \\ & = 22.675 + 2.16 \\ \\ & = 24.835 \\ \\ \end{aligned} & \begin{aligned} \color{green}\hspace{0.7cm} 42.7 \\ \color{green}- \hspace{0.3cm} 11.0 \\ \hline \color{green}\hspace{0.7cm} 31.7 \\ \hline \\ \color{green}\hspace{0.7cm} 31.700 \\ \color{green}- \hspace{0.3cm} 9.025 \\ \hline \color{green}\hspace{0.7cm} 22.675 \\ \hline \\ \color{green}\hspace{0.7cm} 22.675 \\ \color{green}+ \hspace{0.3cm} 2.160 \\ \hline \color{green}\hspace{0.7cm} 24.835 \\ \hline \end{aligned} \end{array} \]
Answer \(42.7 - 11 - 9.025 + 2.16 = \color{red}24.835 \)
(iv) \((6.05 + 5.01) - (12.5 - 0.09)\)
Solution
\[ \begin{array}{c|c} \begin{aligned} & = (6.05 + 5.01) - (12.5 - 0.09) \\ \\ & = (11.06) - (12.41) \\ \\ & = 11.06 - 12.41 \\ \\ & = -1.35 \\ \\ \end{aligned} & \begin{aligned} \color{green}\hspace{0.7cm} 6.05 \\ \color{green}+ \hspace{0.3cm} 5.01 \\ \hline \color{green}\hspace{0.7cm} 11.06 \\ \hline \\ \color{green}\hspace{0.7cm} 12.50 \\ \color{green}- \hspace{0.3cm} 0.09 \\ \hline \color{green}\hspace{0.7cm} 12.41 \\ \hline \\ \color{green}\hspace{0.7cm} 11.06 \\ \color{green}- \hspace{0.3cm} 12.41 \\ \hline \color{green}\hspace{0.7cm} -1.35 \\ \hline \end{aligned} \end{array} \]
Answer \((6.05 + 5.01) - (12.5 - 0.09) = \color{red}-1.35 \)
(v) \(182.3 + 12.65 - 0.23 - 10.71\)
Solution
\[ \begin{array}{c|c} \begin{aligned} & = 182.3 + 12.65 - 0.23 - 10.71 \\ \\ & = 194.95 - 0.23 - 10.71 \\ \\ & = 194.72 - 10.71 \\ \\ & = 184.01 \\ \\ \end{aligned} & \begin{aligned} \color{green}\hspace{0.7cm} 182.30 \\ \color{green}+ \hspace{0.3cm} 12.65 \\ \hline \color{green}\hspace{0.7cm} 194.95 \\ \hline \\ \color{green}\hspace{0.7cm} 194.95 \\ \color{green}- \hspace{0.3cm} 0.23 \\ \hline \color{green}\hspace{0.7cm} 194.72 \\ \hline \\ \color{green}\hspace{0.7cm} 194.72 \\ \color{green}- \hspace{0.3cm} 10.71 \\ \hline \color{green}\hspace{0.7cm} 184.01 \\ \hline \end{aligned} \end{array} \]
Answer \(182.3 + 12.65 - 0.23 - 10.71 = \color{red}184.01 \)
6. Simplify and express the result as a rational number in its lowest terms.
(i) \(\frac{1}{2} + \frac{1}{5} + 6.25 \div 0.25\)
Solution
\[ \begin{aligned} & = \frac{1}{2} + \frac{1}{5} + \color{green}{6.25 \div 0.25} \\ \\ & = \frac{1}{2} + \frac{1}{5} + \frac{6.25 \times 100}{0.25 \times 100} \\ \\ & = \frac{1}{2} + \frac{1}{5} + \frac{625}{25} \\ \\ & = \frac{1}{2} + \frac{1}{5} + \frac{\cancelto{25}{625}}{\cancelto{1}{25}} \\ \\ & = \frac{1}{2} + \frac{1}{5} + 25 \\ \\ & = \frac{5+2+250}{10} \\ \\ & = \frac{257}{10} \\ \\ \end{aligned} \]
Answer \( \frac{1}{2} + \frac{1}{5} + 6.25 \div 0.25 = \color{red} \frac{257}{10} \)
(ii) \(\frac{2}{5} - \frac{1}{4} + (8.1 \times 2.7) \div 0.09\)
Solution
\[ \begin{aligned} & = \frac{2}{5} - \frac{1}{4} + {\color{green}(8.1 \times 2.7)} \div 0.09 \\ \\ & = \frac{2}{5} - \frac{1}{4} + 21.87 \div 0.09 \\ \\ & = \frac{2}{5} - \frac{1}{4} + \frac{21.87}{0.09} \\ \\ & = \frac{2}{5} - \frac{1}{4} + \frac{21.87 \times 100}{0.09 \times 100} \\ \\ & = \frac{2}{5} - \frac{1}{4} + \frac{\cancelto{243}{2187}}{\cancelto{1}{9}} \\ \\ & = \frac{2}{5} - \frac{1}{4} + 243 \\ \\ & = \frac{8-5+4860}{20} \\ \\ & = \frac{3+4860}{20} \\ \\ & = \frac{4863}{20} \\ \\ \end{aligned} \]
Answer \( \frac{2}{5} - \frac{1}{4} + (8.1 \times 2.7) \div 0.09 = \color{red} \frac{4863}{20} \)
(iii) \(1.44 \times (144 \div 12) - 0.225 + 3.276\)
Solution
\[ \begin{aligned} & = 1.44 \times {\color{green}(144 \div 12)} - 0.225 + 3.276 \\ \\ & = 1.44 \times {\left(\frac{\cancel{144}^{12}}{\cancel{12}^1}\right)} - 0.225 + 3.276 \\ \\ & = {\color{green} 1.44 \times 12} - 0.225 + 3.276 \\ & = 17.28 - 0.225 + 3.276 \\ & = 17.055 + 3.276 \\ & = 20.331 \\ & = \frac{20331}{1000} \end{aligned} \]
Answer \(1.44 \times (144 \div 12) - 0.225 + 3.276 = \color{red}\frac{20331}{1000} \)
(iv) \(\frac{1}{7} \times 0.049 + \frac{3}{8} - \frac{7}{20}\)
Solution
\[ \begin{aligned} & = \frac{1}{7} \times 0.049 + \frac{3}{8} - \frac{7}{20} \\ \\ & = \frac{1}{\cancelto{1}{7}} \times \frac{\cancelto{7}{49}}{1000} + \frac{3}{8} - \frac{7}{20} \\ \\ & = \frac{7}{1000} + \frac{3}{8} - \frac{7}{20} \\ \\ & = \frac{7+ 375 - 350}{1000} \\ \\ & = \frac{382 - 35}{1000} \\ \\ & = \frac{\cancelto{4}{32}}{\cancelto{125}{1000}} \\ \\ & = \frac{4}{125} \\ \end{aligned} \]
Answer \(\frac{1}{7} \times 0.049 + \frac{3}{8} - \frac{7}{20} = \color{red} \frac{4}{125}\)
(v) \(5 \times 0.16 - 0.52 + 8.263\)
Solution
\[ \begin{aligned} & = 5 \times0.16 - 0.52 + 8.263 \\ & = 0.80 - 0.52 + 8.263 \\ & = 0.28 + 8.263 \\ & = 8.543 \\ \\ & = \frac{8543}{1000} \end{aligned} \]
Answer \(5 \times 0.16 - 0.52 + 8.263 = \color{red}\frac{8543}{1000} \)
(vi) \(\frac{2}{5} \times \frac{3}{4} + \frac{1}{25} \times \frac{1}{2} - \frac{2}{10} \times \frac{1}{5}\)
Solution
\[ \begin{aligned} & = \left(\frac{2}{5} \times \frac{3}{4} \right) + \left(\frac{1}{25} \times \frac{1}{2} \right) - \left(\frac{2}{10} \times \frac{1}{5} \right) \\ \\ & = \left(\frac{\cancelto{1}{2}}{5} \times \frac{3}{\cancelto{2}{4}} \right) + \left(\frac{1}{25} \times \frac{1}{2} \right) - \left(\frac{\cancelto{1}{2}}{\cancelto{5}{10}} \times \frac{1}{5} \right) \\ \\ & = \frac{3}{10} + \frac{1}{50} - \frac{1}{25} \\ \\ & = \frac{15 + 1 - 2}{50} \\ \\ & = \frac{\cancelto{7}{14}}{\cancelto{25}{50}} \\ \\ & = \frac{7}{25} \\ \\ \end{aligned} \]
Answer \(\frac{2}{5} \times \frac{3}{4} + \frac{1}{25} \times \frac{1}{2} - \frac{2}{10} \times \frac{1}{5} = \color{red}\frac{7}{25}\)