Rational Numbers as Decimals Brain Teasers
1. A. Tick the correct option.
(a) \( 0.225 \) expressed as a rational number is- (i) \( \frac{1}{4} \) (ii) \( \frac{45}{210} \) (iii) \( \frac{9}{40} \) (iv) \( \frac{225}{999} \)
Solution
\[ \begin{align*} &= 0.225 \\ \\ & = \frac{ {\cancel{225}^{\cancel{\color{green}45}}}^{\color{orange}9}}{ {\cancel{1000}^{\cancel{\color{green}200}}}^{\color{orange}40}} \\ \\ &= \frac{9}{40} \\ \\ \end{align*} \]
Answer \( \color{red} \frac{9}{40} \)
(b) A rational number \(\frac{p}{q}\) can be expressed as a terminating decimal if \( q \) has no prime factor other than- (i) 2, 3 (ii) 2, 5 (iii) 3, 5 (iv) 2, 3, 5
Answer \(\color{red} 2, 5 \)
(c) \( -7 \frac{8}{100} \) expressed as a decimal number is- (i) -7.800 (ii) 7.008 (iii) -7.008 (iv) -7.08
Solution
\[ \begin{align*} &=-7 \frac{8}{100}\\ \\ &= -\frac{708}{100} \\ \\ &= -7.08 \\ \\ \end{align*} \]
Answer \(\color{red} -7.08 \)
(d) \( 4.013\overline{25} \) is equal to- (i) \( 4.013252525... \) (ii) \( 4.0132555... \) (iii) \( 4.0132501325... \) (iv) \( 4.0130132525... \)
Answer \(\color{red} 4.013252525... \)
(e) The quotient when \( 0.00639 \) is divided by \( 0.213 \) is - (i) \( 3 \) (ii) \( 0.3 \) (iii) \( 0.03 \) (iv) \( 0.003 \)
Solution
\[ \begin{align*} &= \frac{0.00639 \times 10000 }{ 0.213 \times 10000} \\ \\ &= \frac{\cancelto{3}{639}}{ \cancelto{1}{213} \times 100} \\ \\ & = \frac{3}{100} \\ \\ & = 0.03 \end{align*} \]
Answer \(\color{red} 0.03 \)
B. Answer the following questions.
(a) Without actual division, determine if \( \frac{-28}{250} \) is a terminating or non-terminating decimal number.
\[ \begin{align*} &= \frac{-\cancelto{14}{28}}{\cancelto{125}{250}}\\ \\ &= \frac{-14}{125}\\ \\ &\text{Prime factorization of denominator (125)} \\ & 125 = 5 \times 5 \times 5 \\ &\text{Prime factorization of 250 contains only 2 and 5.} \\ \\ \frac{-28}{250} &\text{ has a terminating decimal representation} \\ \end{align*} \]
Answer \(\color{red} \frac{-28}{250} \text{ is a terminating decimal number} \)
(b) Convert \( \frac{-113}{7} \) to decimals.
Solution
\[ \begin{array}{l} \hspace{0.7cm}-16.142857142\ldots \\ 7 \enclose{longdiv}{113.000000000}\\ \hspace{0.3cm}-7\\ \hline \hspace{0.6cm}43\\ \hspace{0.3cm}-42\\ \hline \hspace{.8cm}10 \\ \hspace{.65cm}-7\\ \hline \hspace{1cm}30 \\ \hspace{.7cm}-28\\ \hline \hspace{1.2cm}20 \\ \hspace{.9cm}-14\\ \hline \hspace{1.5cm}60 \\ \hspace{1.2cm}-56\\ \hline \hspace{1.7cm}40 \\ \hspace{1.4cm}-35\\ \hline \hspace{2cm}50 \\ \hspace{1.7cm}-49\\ \hline \hspace{2.2cm}10 \\ \hspace{2.1cm}-7\\ \hline \hspace{2.5cm}30 \\ \hspace{2.2cm}-28\\ \hline \hspace{2.7cm}20 \\ \hspace{2.35cm}-14\\ \hline \hspace{2.95cm}6 \\ \hline \\ \end{array} \]\[ \begin{aligned} \frac{-113}{7} & = -16.142857142\ldots \\ \\ \frac{-113}{7} & = -16.\overline{142857} \\ \end{aligned} \]
Answer \( \frac{-113}{7} = \color{red} -16.\overline{142857} \)
(c) What should be subtracted from \( -15.834 \) to get \( 3.476 \)?
\[\text{Let the required number to } x \] \[ \begin{aligned} -15.834 - x &= 3.476 \\ -15.834 - 3.476 &= x \\ -19.310 &= x \\ x &= -19.31 \end{aligned} \]
Answer Required number \( = \color{red} -19.31 \)
(d) Express \( 4.82 \) as a rational number in standard form.
\[ \begin{aligned} & = 4.82 \\ \\ &= \frac{\cancelto{241}{482}}{\cancelto{50}{100}} \\ \\ &= \frac{241}{50} \\ \end{aligned} \]
Answer \(\color{red} \frac{241}{50} \)
(e) Find the value of \( 16.016 \div 0.4 \)
Answer \(\color{red} 40.04 \)
2. Convert the following rational numbers into decimals.
(i) \( \frac{259}{3} \)
Solution
\[ \begin{array}{l}\hspace{0.7cm}86.333\ldots \\ 3 \enclose{longdiv}{259.000}\\ \hspace{0.1cm}-24\\ \hline \hspace{0.6cm}19\\ \hspace{0.3cm}-18\\ \hline \hspace{0.8cm}10 \\ \hspace{.65cm}-9\\ \hline \hspace{1cm}10 \\ \hspace{.9cm}-9\\ \hline \hspace{1.2cm}10 \\ \hspace{1.1cm}-9\\ \hline \hspace{1.4cm}1 \\ \hline \\ \end{array} \] \[ \begin{aligned} \frac{259}{3} & = 86.333\ldots \\ \\ \frac{259}{3} & = 86.\overline{3} \\ \end{aligned} \]
Answer \( \frac{259}{3} = \color{red} 86.\overline{3} \)
(ii) \( \frac{19256}{11} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}1750.5454\ldots \\ 11 \enclose{longdiv}{19256.0000}\\ \hspace{0.3cm}-11\\ \hline \hspace{0.8cm}82\\ \hspace{0.5cm}-77\\ \hline \hspace{1cm}55 \\ \hspace{.7cm}-55\\ \hline\hspace{1.2cm}060 \\ \hspace{1.1cm}-55\\ \hline \hspace{1.6cm}50 \\ \hspace{1.3cm}-44\\ \hline \hspace{1.8cm}60 \\ \hspace{1.5cm}-55\\ \hline \hspace{2cm}50 \\ \hspace{1.7cm}-44\\ \hline \hspace{2.3cm}6 \\ \hline \\ \end{array} \]\[ \begin{aligned} \frac{19256}{11} & = 1750.5454\ldots \\ \\ \frac{19256}{11} & = 1750.\overline{54} \\ \end{aligned} \]
Answer \( \frac{19256}{11} = \color{red} 1750.\overline{54} \)
(iii) \( \frac{15735}{80} \)
Solution
\[ \color{green}Method - 1 \] \[ \begin{array}{l} \hspace{0.7cm}196.6875 \\ 80 \enclose{longdiv}{15735.0000}\\ \hspace{0.4cm}-80\\ \hline \hspace{0.8cm}773\\ \hspace{0.5cm}-720\\ \hline \hspace{1cm}535 \\ \hspace{.7cm}-480\\ \hline \hspace{1.2cm}550 \\ \hspace{.9cm}-480\\ \hline \hspace{1.4cm}700 \\ \hspace{1.1cm}-640\\ \hline \hspace{1.6cm}600 \\ \hspace{1.3cm}-560\\ \hline \hspace{1.8cm}400 \\ \hspace{1.5cm}-400\\ \hline \hspace{2cm}0 \\ \hline \\ \end{array} \]\[ \begin{aligned} \frac{15735}{80} & = \color{green} 196.6875 \\ \\ \end{aligned} \] \[ \begin{aligned} &\color{green}Method - 2 \\ \\ &= \frac{15735}{80} \\ \\ &= \frac{15735 \times 125}{80 \times 125} \\ \\ &= \frac{1966875}{10000} \\ \\ &= \color{green}196.6875 \\ \\ \end{aligned} \]
Answer \( \frac{15735}{80} = \color{red} 196.6875 \)
(iv) \( \frac{27}{7} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}3.85714285\ldots \\ 7 \enclose{longdiv}{27.000000}\\ \hspace{0.1cm}-21\\ \hline \hspace{0.6cm}60\\ \hspace{0.3cm}-56\\ \hline \hspace{.8cm}40 \\ \hspace{.5cm}-35\\ \hline \hspace{1cm}50 \\ \hspace{.7cm}-49\\ \hline \hspace{1.2cm}10 \\ \hspace{1.1cm}-7\\ \hline \hspace{1.4cm}30 \\ \hspace{1.1cm}-28\\ \hline \hspace{1.6cm}20 \\ \hspace{1.25cm}-14 \\ \hline \hspace{1.8cm}60 \\ \hspace{1.5cm}-56 \\ \hline \hspace{2cm}40 \\ \hspace{1.7cm}-35 \\ \hline \hspace{2.3cm}5 \\ \hline \\ \end{array} \]\[ \begin{aligned} \frac{27}{7} & = 3.85714285\ldots \\ \\ \frac{27}{7} & = 3.\overline{857142} \\ \end{aligned} \]
Answer \( \frac{27}{7} = \color{red} 3.\overline{857142} \)
(v) \( \frac{758}{1250} \)
Solution
\[ \begin{aligned} & = \frac{758}{1250}\\ \\ & = \frac{758 \times 8}{1250 \times 8}\\ \\ & = \frac{6064}{10000}\\ \\ & = 0.6064 \\ \\ \end{aligned} \]
Answer \( \frac{758}{1250} = \color{red} 0.6064 \)
(vi) \( \frac{15625}{12} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}1302.0833\ldots \\ 12 \enclose{longdiv}{15625.0000}\\ \hspace{0.3cm}-12\\ \hline \hspace{0.8cm}36\\ \hspace{0.5cm}-36\\ \hline \hspace{1cm}025 \\ \hspace{.9cm}-24\\ \hline \hspace{1.5cm}100 \\ \hspace{1.4cm}-96\\ \hline \hspace{1.9cm}40 \\ \hspace{1.6cm}-36\\ \hline \hspace{2.1cm}40 \\ \hspace{1.8cm}-36\\ \hline \hspace{2.3cm}4 \\ \hline \end{array} \] \[ \begin{aligned} \frac{15625}{12} & = 1302.0833\ldots \\ \\ \frac{15625}{12} & = 1302.083\overline{3} \\ \end{aligned} \]
Answer \( \frac{15625}{12} = \color{red} 1302.083\overline{3} \)
3. Find the decimal representation of the following rational numbers
(i) \( \frac{-12}{13} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}-0.92307692\ldots \\ 13 \enclose{longdiv}{12.000000000}\\ \hspace{0.3cm}-117\\ \hline \hspace{1.1cm}30\\ \hspace{0.8cm}-26\\ \hline \hspace{1.3cm}40 \\ \hspace{1cm}-39\\ \hline \hspace{1.5cm}100 \\ \hspace{1.4cm}-91\\ \hline \hspace{1.9cm}90 \\ \hspace{1.6cm}-78\\ \hline \hspace{1.9cm}120 \\ \hspace{1.55cm}-117\\ \hline \hspace{2.3cm}30 \\ \hspace{2cm}-26\\ \hline \hspace{2.5cm}4 \\ \hline \end{array} \] \[ \begin{aligned} \frac{-12}{13} & = -0.92307692\ldots \\ \\ \frac{-12}{13} & = -0.\overline{923076} \\ \end{aligned} \]
Answer \( \frac{-12}{13} = \color{red} -0.\overline{923076} \)
(ii) \( \frac{-1525}{50} \)
Solution
\[ \begin{aligned} & = \frac{-1525 \times 2}{50 \times 2} \\ \\ & = \frac{-3050}{100} \\ \\ & = -30.50 \\ \\ \end{aligned} \]
Answer \( \frac{-1525}{50} = \color{red} -30.50 \)
(iii) \( \frac{-127}{7} \)
Solution
\[ \begin{array}{l} \hspace{0.7cm}-18.14285714\ldots \\ 7 \enclose{longdiv}{127.000000000}\\ \hspace{0.3cm}-7\\ \hline \hspace{0.7cm}57\\ \hspace{0.35cm}-56\\ \hline \hspace{.9cm}10 \\ \hspace{.75cm}-7\\ \hline \hspace{1.2cm}30 \\ \hspace{.9cm}-28\\ \hline \hspace{1.5cm}20 \\ \hspace{1.15cm}-14\\ \hline \hspace{1.7cm}60 \\ \hspace{1.4cm}-56\\ \hline \hspace{1.9cm}40 \\ \hspace{1.6cm}-35\\ \hline \hspace{2.1cm}50 \\ \hspace{1.75cm}-49\\ \hline \hspace{2.3cm}10 \\ \hspace{2.2cm}-7\\ \hline \hspace{2.5cm}30 \\ \hspace{2.2cm}-28\\ \hline \hspace{2.8cm}2 \\ \hline \\ \end{array} \] \[ \begin{aligned} \frac{-127}{7} & = -18.14285714\ldots \\ \\ \frac{-127}{7} & = -18.\overline{142857} \\ \end{aligned} \]
Answer \( \frac{-127}{7} = \color{red} -18.\overline{142857} \)
(iv) \( \frac{-539}{80} \)
Solution
\[ \begin{aligned} & = \frac{-539}{80} \\ \\ & = \frac{-539 \times 125}{80 \times 125} \\ \\ & = \frac{-67375}{10000} \\ \\ & = -6.7375 \\ \\ \end{aligned} \]
Answer \( \frac{-539}{80} = \color{red} -6.7375 \)
4. Simplity the following expressions.
(i) \(3.2 + 16.09 + 26.305 - 1.232\)
Solution
\[ \begin{aligned} & = 3.2 + 16.09 + 26.305 - 1.232 \\ & = 19.29 + 26.305 - 1.232 \\ & = 45.595 - 1.232 \\ & = 44.363 \end{aligned} \]
Answer \(3.2 + 16.09 + 26.305 - 1.232 = \color{red}44.363 \)
(ii) \(-5.7 + 13.20 - 15.009 + 0.02\)
Solution
\[ \begin{aligned} & = -5.7 + 13.20 - 15.009 + 0.02 \\ & = 7.50 - 15.009 + 0.020 \\ & = -7.509 + 0.020 \\ & = -7.489 \end{aligned} \]
Answer \(-5.7 + 13.20 - 15.009 + 0.02 = \color{red}-7.489 \)
(iii) \((0.357 + 0.96) - (3.25 - 2.79)\)
Solution
\[ \begin{aligned} & = (0.357 + 0.96) - (3.25 - 2.79) \\ & = (1.317) - (0.46) \\ & = 1.317 - 0.46 \\ & = 0.857 \end{aligned} \]
Answer \((0.357 + 0.96) - (3.25 - 2.79) = \color{red}0.857 \)
(iv) \(15 + 2.57 - 23.07 - 5.003\)
Solution
\[ \begin{aligned} & = 15 + 2.57 - 23.07 - 5.003 \\ & = 17.570 - 23.07 - 5.003 \\ & = -5.500 - 5.003 \\ & = -10.503 \end{aligned} \]
Answer \(15 + 2.57 - 23.07 - 5.003 = \color{red}-10.503 \)
5. Without actual division, determine which of the following rational numbers have a terminating decimal representation.
(i) \( \frac{327}{125} \)
Solution
\[ \begin{aligned} & \text{Prime factorization of denominator } (125) \\ & 125 = 5 \times 5 \times 5 \\ & \text{Prime factorization of 125 contains only 5.} \\ \\ \frac{327}{125} & \text{ has a terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Terminating} \)
(ii) \( \frac{99}{800} \)
Solution
\[ \begin{aligned} & \text{Prime factorization of denominator } (800) \\ & 800 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \\ & \text{Prime factorization of 800 contains only 2 and 5.} \\ \\ \frac{99}{800} & \text{ has a terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Terminating} \)
(iii) \( \frac{17}{1250} \)
Solution
\[ \begin{aligned} & \text{Prime factorization of denominator } (1250) \\ & 1250 = 2 \times 5 \times 5 \times 5 \times 5 \\ & \text{Prime factorization of 1250 contains only 2 and 5.} \\ \\ \frac{17}{1250} & \text{ has a terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Terminating} \)
(iv) \( \frac{29}{200} \)
Solution
\[ \begin{aligned} & \text{Prime factorization of denominator } (200) \\ & 200 = 2 \times 2 \times 2 \times 5 \times 5 \\ & \text{Prime factorization of 200 contains only 2 and 5.} \\ \\ \frac{29}{200} & \text{ has a terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Terminating} \)
(v) \( \frac{135}{1625} \)
Solution
\[ \begin{aligned} &= \frac{\cancelto{27}{135}}{\cancelto{325}{1625}} \\ \\ &= \frac{27}{325} \\ \\ & \text{Prime factorization of denominator } (325) \\ & 325 = 5 \times 5 \times 13 \\ & \text{Prime factorization of 325 contains 13, which is neither 2 nor 5.} \\ \\ \frac{135}{1625} & \text{ has a non-terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Non-Terminating} \)
(vi) \( \frac{1276}{680} \)
Solution
\[ \begin{aligned} &= \frac{\cancelto{319}{1276}}{\cancelto{170}{680}} \\ \\ &= \frac{319}{170} \\ \\ & \text{Prime factorization of denominator } (170) \\ & 170 = 2 \times 5 \times 17 \\ & \text{Prime factorization of 170 contains 17, which is neither 2 nor 5.} \\ \\ \frac{1276}{680} & \text{ has a non-terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Non-Terminating} \)
(vii) \( \frac{22}{190} \)
Solution
\[ \begin{aligned} &= \frac{\cancelto{11}{22}}{\cancelto{95}{190}} \\ \\ &= \frac{11}{95} \\ \\ & \text{Prime factorization of denominator } (95) \\ & 95 = 5 \times 19 \\ & \text{Prime factorization of 95 contains 19, which is neither 2 nor 5.} \\ \\ \frac{22}{190} & \text{ has a non-terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Non-Terminating} \)
(viii) \( \frac{11}{750} \)
Solution
\[ \begin{aligned} & \text{Prime factorization of denominator } (750) \\ & 750 = 2 \times 3 \times 5 \times 5 \times 5 \\ & \text{Prime factorization of 750 contains 3, which is neither 2 nor 5.} \\ \\ \frac{11}{750} & \text{ has a non-terminating decimal representation} \\ \end{aligned} \]
Answer \( \color{red}\text{Non-Terminating} \)
6. Simplify the following and express the result as decimals
(i) \(2.7 \times 1.5 \times 2.1\)
Solution
\[ \begin{aligned} \begin{array}{r} & 27 \\ \times & 15 \\ \hline & 135 \\ + & 270 \\ \hline & 405 \\ \hline \\ & 405 \\ \times & 21 \\ \hline & 405 \\ + & 8100 \\ \hline & 8505 \\ \hline \end{array} \end{aligned} \] \[ \begin{aligned} 27 \times 15 \times 21 & = 8505 \\ 2.7 \times 1.5 \times 2.1 & = 8.505 \\ \end{aligned} \]
Answer \(2.7 \times 1.5 \times 2.1 = \color{red}8.505 \)
(ii) \(12 \times 13.6 \times 0.25\)
Solution
\[ \begin{aligned} \begin{array}{c|c} \begin{array}{r} & 136 \\ \times & 12 \\ \hline & 272 \\ + & 1360 \\ \hline & 1632 \\ \hline \end{array} & \begin{array}{r} & 1632 \\ \times & 25 \\ \hline & 8160 \\ + & 32640 \\ \hline & 40800 \\ \hline \end{array} \end{array} \end{aligned} \] \[ \begin{aligned} 12 \times 136 \times 25 & = 40800 \\ 12 \times 13.6 \times 0.25 & = 40.800 \\ \end{aligned} \]
Answer \(12 \times 13.6 \times 0.25 = \color{red}40.8 \)
(iii) \(3.25 \times 72.6\)
Solution
\[ \begin{aligned} \begin{array}{c|c} \begin{array}{r} & 325 \\ \times & 726 \\ \hline & 1950 \\ + & 6500 \\ + & 227500 \\ \hline & 235950 \\ \hline \end{array} \end{array} \end{aligned} \] \[ \begin{aligned} 325 \times 726 & = 235950 \\ 3.25 \times 72.6 & = 235.950 \\ \end{aligned} \]
Answer \(3.25 \times 72.6 = \color{red}235.95 \)
(iv) \( (156.25 \div 0.025) \times 0.02 - 5.2\)
Solution
\[ \begin{aligned} &= \left(\frac{156.25} {0.025} \right) \times 0.02 - 5.2 \\ \\ &= \left(\frac{156.25 \times 1000} {0.025 \times 1000} \right) \times 0.02 - 5.2 \\ \\ &= \left(\frac{\cancelto{6250}{156250}} {\cancelto{1}{25}} \right) \times 0.02 - 5.2 \\ \\ &= 6250 \times 0.02 - 5.2 \\ &= 125.00 - 5.2 \\ &= 119.8 \end{aligned} \]
Answer \( (156.25 \div 0.025) \times 0.02 - 5.2 = \color{red}119.8 \)
(v) \( (75.05 \div 0.05) \times 0.001 + 2.351 \)
Solution
\[ \begin{aligned} & = \left(\frac{75.05 \times 100 } {0.05 \times 100} \right) \times 0.001 + 2.351 \\ \\ & = \left(\frac{\cancel{7505}^{1501}} {\cancel5^{1}} \right) \times 0.001 + 2.351 \\ \\ & = 1501 \times 0.001 + 2.351 \\ & = 1.501 + 2.351 \\ & = 3.852 \\ \end{aligned} \]
Answer \( (75.05 \div 0.05) \times 0.001 + 2.351 = \color{red} 3.852 \)
7. Simplify and express the result as a rational number in its lowest form.
(i) \(3.125 \div 0.125 + 0.50 - 0.225\)
Solution
\[ \begin{aligned} & = \left(\frac{3.125}{0.125}\right) + 0.50 - 0.225 \\ \\ & = \left(\frac{\cancelto{625}{3125} \times \cancel{1000} }{\cancelto{25}{125} \times \cancel{1000} }\right) + 0. 50 - 0.225 \\ \\ & = \left(\frac{\cancelto{25}{625}}{\cancelto{1}{25}}\right) + 0.50 - 0.225 \\ \\ & = 25 + 0.50 - 0.225 \\ & = 25.50 - 0.225 \\ & = 25.275 \\ \\ & = \frac{ {\cancel{25275}^{\cancel{\color{green}5055}}}^{\color{orange}1011}}{ {\cancel{1000}^{\cancel{\color{green}200}}}^{\color{orange}40}} \\ \\ & = \frac{ 1011}{40} \end{aligned} \]
Answer \(3.125 \div 0.125 + 0.50 - 0.225 = \color{red}\frac{ 1011}{40} \)
(ii) \( \frac{0.4 \times 0.04 \times 0.005}{0.1 \times 10 \times 0.001} - \frac{1}{2} + \frac{1}{5} \)
Solution
\[ \begin{aligned} & = \frac{0.4 \times 0.04 \times 0.005}{0.1 \times 10 \times 0.001} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{ 0.016 \times 0.005}{1 \times 0.001} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{ 0.000080 \times {\color{green}100000}}{0.001 \times {\color{green}100000}} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{8}{100} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{\cancelto{2}8}{\cancelto{25}{100}} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{2}{25} - \frac{1}{2} + \frac{1}{5} \\ \\ & = \frac{4-25+10}{50} \\ \\ & = \frac{-21+10}{50} \\ \\ & = \frac{-11}{50} \\ \\ \end{aligned} \]
Answer \( \frac{0.4 \times 0.04 \times 0.005}{0.1 \times 10 \times 0.001} - \frac{1}{2} + \frac{1}{5} = \color{red}\frac{-11}{50} \)
(iii) \(\frac{0.144 \div 1.2}{0.016 \div 0.02} + \frac{7}{5} - \frac{21}{8}\)
Solution
\[ \begin{aligned} & = \frac{\frac{0.144}{1.2}}{\frac{0.016}{0.02}} + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{0.144}{1.2} \times {\frac{0.02}{0.016}} + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{0.144 \times 0.02 \times {\color{green}100000} }{1.2 \times 0.016 \times {\color{green}100000} } + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{\cancelto{12}{144} \times \cancelto{1}2}{\cancelto{1}{12} \times \cancelto{8}{16} \times 10} + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{\cancelto{3}{12}}{\cancelto{2}{8} \times 10} + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{3}{20} + \frac{7}{5} - \frac{21}{8} \\ \\ & = \frac{6 + 56 -105}{40} \\ \\ & = \frac{62 -105}{40} \\ \\ & = \frac{-43}{40} \\ \\ \end{aligned} \]
Answer \(\frac{0.144 \div 1.2}{0.016 \div 0.02} + \frac{7}{5} - \frac{21}{8} = \color{red} \frac{-43}{40} \)