Class 8: Decimals – Exercise 6A

$\displaystyle \textbf{Question 1: }\text{Convert the following decimals into like decimals:}$
$\displaystyle \text{(i) }13,\ 2.212,\ 9.0,\ 5.0017,\ 0.44$

$\displaystyle \text{(ii) }1,\ 1.11,\ 0.1,\ 1.111,\ 1.1,\ 11$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) The greatest number of decimal places is }4.$
$\displaystyle 13.0000,\quad2.2120,\quad9.0000,\quad5.0017,\quad0.4400.$

$\displaystyle \text{(ii) The greatest number of decimal places is }3.$
$\displaystyle 1.000,\quad1.110,\quad0.100,\quad1.111,\quad1.100,\quad11.000.$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Compare the following decimals:}$
$\displaystyle \text{(i) }0.1\text{ and }0.01\qquad\text{(ii) }4.001\text{ and }3.987$

$\displaystyle \text{(iii) }0.1\text{ and }0.0999\qquad\text{(iv) }88.123\text{ and }88.08765$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }0.1=0.10>0.01.$

$\displaystyle \text{(ii) }4.001>3.987.$

$\displaystyle \text{(iii) }0.1=0.1000>0.0999.$

$\displaystyle \text{(iv) }88.12300>88.08765.$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Write each of the following decimals in expanded form:}$
$\displaystyle \text{(i) }3.57\qquad\text{(ii) }0.864\qquad\text{(iii) }1.002\qquad\text{(iv) }153.6074$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }3.57=3\times1+5\times\frac{1}{10}+7\times\frac{1}{10^2}.$

$\displaystyle \text{(ii) }0.864=8\times\frac{1}{10}+6\times\frac{1}{10^2}$
$\displaystyle \qquad\qquad\quad+4\times\frac{1}{10^3}.$

$\displaystyle \text{(iii) }1.002=1\times1+2\times\frac{1}{10^3}.$

$\displaystyle \text{(iv) }153.6074=1\times10^2+5\times10^1+3\times1$
$\displaystyle \qquad\qquad\quad+6\times\frac{1}{10}+7\times\frac{1}{10^3}$
$\displaystyle \qquad\qquad\quad+4\times\frac{1}{10^4}.$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Express each of the following as a decimal:}$
$\displaystyle \text{(i) }4\times10^1+7\times1+5\times\frac{1}{10}$

$\displaystyle \text{(ii) }1\times10^2+9\times10^1+4\times1+5\times\frac{1}{10}$
$\displaystyle \qquad+6\times\frac{1}{10^2}+8\times\frac{1}{10^3}$

$\displaystyle \text{(iii) }5\times10^1+9\times1+4\times\frac{1}{10^2}$
$\displaystyle \qquad+1\times\frac{1}{10^3}+6\times\frac{1}{10^4}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }4\times10^1+7\times1+5\times\frac{1}{10}=47.5.$

$\displaystyle \text{(ii) }1\times10^2+9\times10^1+4\times1+5\times\frac{1}{10}$
$\displaystyle \qquad+6\times\frac{1}{10^2}+8\times\frac{1}{10^3}=194.568.$

$\displaystyle \text{(iii) }5\times10^1+9\times1+4\times\frac{1}{10^2}$
$\displaystyle \qquad+1\times\frac{1}{10^3}+6\times\frac{1}{10^4}=59.0416.$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Write the following decimals in ascending order:}$
$\displaystyle \text{(i) }0.13,\ 1.056,\ 0.394,\ 1.13,\ 0.625,\ 2.004$

$\displaystyle \text{(ii) }0.01,\ 0.007,\ 0.7,\ 7.7,\ 7.007,\ 7.07$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }0.13<0.394<0.625<1.056<1.13<2.004.$

$\displaystyle \text{(ii) }0.007<0.01<0.7<7.007<7.07<7.7.$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Write the following decimals in descending order:}$
$\displaystyle \text{(i) }6.1,\ 7.3,\ 3.7,\ 6.07,\ 7.37,\ 3.71$

$\displaystyle \text{(ii) }12.5,\ 9.8,\ 11.31,\ 10.734,\ 9.79,\ 10.385$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }7.37>7.3>6.1>6.07>3.71>3.7.$

$\displaystyle \text{(ii) }12.5>11.31>10.734>10.385>9.8>9.79.$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Add:}$
$\displaystyle \text{(i) }0.351+2.96+19.6+0.005+137.09+0.0809$

$\displaystyle \text{(ii) }9.89+0.989+98.09+989.098+98.0989$

$\displaystyle \text{(iii) }66.666+6.6+6.066+666.006$

$\displaystyle \text{(iv) }2.38+23.8+238.28+28.383+0.382$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Converting the decimals into like decimals and adding:}$
$\displaystyle \begin{array}{r} \phantom{1}0.3510\\ \phantom{1}2.9600\\ 19.6000\\ \phantom{1}0.0050\\ 137.0900\\ +\phantom{1}0.0809\\ \hline 160.0869\\ \hline \end{array}$

$\displaystyle \text{(ii) Converting the decimals into like decimals and adding:}$
$\displaystyle \begin{array}{r} \phantom{1}\phantom{1}9.8900\\ \phantom{1}\phantom{1}0.9890\\ \phantom{1}98.0900\\ 989.0980\\ +\phantom{1}98.0989\\ \hline 1196.1659\\ \hline \end{array}$

$\displaystyle \text{(iii) Converting the decimals into like decimals and adding:}$
$\displaystyle \begin{array}{r} \phantom{1}66.666\\ \phantom{1}\phantom{1}6.600\\ \phantom{1}\phantom{1}6.066\\ +666.006\\ \hline 745.338\\ \hline \end{array}$

$\displaystyle \text{(iv) Converting the decimals into like decimals and adding:}$
$\displaystyle \begin{array}{r} \phantom{1}\phantom{1}2.380\\ \phantom{1}23.800\\ 238.280\\ \phantom{1}28.383\\ +\phantom{1}\phantom{1}0.382\\ \hline 293.225\\ \hline \end{array}$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Evaluate:}$
$\displaystyle \text{(i) Increase }37.984\text{ by }7.089$

$\displaystyle \text{(ii) Increase }99.9809\text{ by }0.198$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }37.984+7.089$
$\displaystyle \begin{array}{r} 37.984\\ +\phantom{0}7.089\\ \hline 45.073\\ \hline \end{array}$
$\displaystyle \therefore 37.984\text{ increased by }7.089\text{ is }45.073.$

$\displaystyle \text{(ii) }99.9809+0.1980$
$\displaystyle \begin{array}{r} 99.9809\\ +\phantom{0}0.1980\\ \hline 100.1789\\ \hline \end{array}$
$\displaystyle \therefore 99.9809\text{ increased by }0.198\text{ is }100.1789.$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Subtract:}$
$\displaystyle \text{(i) }16.48\text{ from }20.3\qquad\text{(ii) }0.0051\text{ from }0.01$

$\displaystyle \text{(iii) }9.841\text{ from }11.001\qquad\text{(iv) }0.0838\text{ from }0.1$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) Converting the decimals into like decimals and subtracting:}$
$\displaystyle \begin{array}{r} 20.30\\ -16.48\\ \hline 3.82\\ \hline \end{array}$
$\displaystyle \therefore 20.3-16.48=3.82.$

$\displaystyle \text{(ii) Converting the decimals into like decimals and subtracting:}$
$\displaystyle \begin{array}{r} 0.0100\\ -0.0051\\ \hline 0.0049\\ \hline \end{array}$
$\displaystyle \therefore 0.01-0.0051=0.0049.$

$\displaystyle \text{(iii) Converting the decimals into like decimals and subtracting:}$
$\displaystyle \begin{array}{r} 11.001\\ -\phantom{0}9.841\\ \hline 1.160\\ \hline \end{array}$
$\displaystyle \therefore 11.001-9.841=1.160.$

$\displaystyle \text{(iv) Converting the decimals into like decimals and subtracting:}$
$\displaystyle \begin{array}{r} 0.1000\\ -0.0838\\ \hline 0.0162\\ \hline \end{array}$
$\displaystyle \therefore 0.1-0.0838=0.0162.$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Evaluate:}$
$\displaystyle \text{(i) Decrease }10\text{ by }0.854\qquad\text{(ii) Decrease }11\text{ by }10.9091$

$\displaystyle \text{(iii) Take }2.53\text{ from }25.3$

$\displaystyle \text{(iv) Find the difference between }0.1\text{ and }0.0001$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }10-0.854$
$\displaystyle \begin{array}{r} 10.000\\ -\phantom{0}0.854\\ \hline 9.146\\ \hline \end{array}$
$\displaystyle \therefore 10\text{ decreased by }0.854\text{ is }9.146.$

$\displaystyle \text{(ii) }11-10.9091$
$\displaystyle \begin{array}{r} 11.0000\\ -10.9091\\ \hline 0.0909\\ \hline \end{array}$
$\displaystyle \therefore 11\text{ decreased by }10.9091\text{ is }0.0909.$

$\displaystyle \text{(iii) }25.3-2.53$
$\displaystyle \begin{array}{r} 25.30\\ -\phantom{0}2.53\\ \hline 22.77\\ \hline \end{array}$
$\displaystyle \therefore 2.53\text{ taken from }25.3\text{ gives }22.77.$

$\displaystyle \text{(iv) }0.1-0.0001$
$\displaystyle \begin{array}{r} 0.1000\\ -0.0001\\ \hline 0.0999\\ \hline \end{array}$
$\displaystyle \therefore \text{The difference between }0.1\text{ and }0.0001\text{ is }0.0999.$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Find the following products:}$
$\displaystyle \text{(i) }8.019\times10\qquad\text{(ii) }0.054\times1000$

$\displaystyle \text{(iii) }0.076\times0.09\qquad\text{(iv) }0.0567\times13$

$\displaystyle \text{(v) }2.57\times34\qquad\text{(vi) }0.319\times0.021$

$\displaystyle \text{(vii) }1.11\times0.11\times1.1\times1.01\times1000$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }8.019\times10=80.19.$

$\displaystyle \text{(ii) }0.054\times1000=54.$

$\displaystyle \text{(iii) }0.076\times0.09=0.00684.$

$\displaystyle \text{(iv) }0.0567\times13=0.7371.$

$\displaystyle \text{(v) }2.57\times34=87.38.$

$\displaystyle \text{(vi) }0.319\times0.021=0.006699.$

$\displaystyle \text{(vii) }1.11\times0.11\times1.1\times1.01\times1000=135.6531.$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Divide:}$
$\displaystyle \text{(i) }23.5\div10\qquad\text{(ii) }47.19\div1000$

$\displaystyle \text{(iii) }204.6\div20\qquad\text{(iv) }31.49\div47$

$\displaystyle \text{(v) }0.8085\div35\qquad\text{(vi) }0.00216\div27$

$\displaystyle \text{(vii) }0.169\div1.3\qquad\text{(viii) }1.21663\div0.89$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }23.5\div10=2.35.$

$\displaystyle \text{(ii) }47.19\div1000=0.04719.$

$\displaystyle \text{(iii) }204.6\div20=10.23.$

$\displaystyle \text{(iv) }31.49\div47=0.67.$

$\displaystyle \text{(v) }0.8085\div35=0.0231.$

$\displaystyle \text{(vi) }0.00216\div27=0.00008.$

$\displaystyle \text{(vii) }0.169\div1.3=0.13.$

$\displaystyle \text{(viii) }1.21663\div0.89=1.367.$
$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{Simplify:}$
$\displaystyle \text{(i) }3.5\div0.5\text{ of }5+0.5\times0.3-0.1$

$\displaystyle \text{(ii) }3.6\div0.4\text{ of }4+0.5\times0.2-0.003$

$\displaystyle \text{(iii) }3.1-2.4\div0.5\text{ of }3\times0.75+0.05$

$\displaystyle \text{(iv) }36.4-3.6\div1.2\times2.5\text{ of }4.6$

$\displaystyle \text{(v) }6.5-0.5\text{ of }(9.8-3.7)+4.3\times6.05$

$\displaystyle \text{(vi) }8.1-[2+3.51\text{ of }(7.1-3.4\times2.05)]$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }3.5\div0.5\text{ of }5+0.5\times0.3-0.1$
$\displaystyle =3.5\div(0.5\times5)+0.15-0.1$
$\displaystyle =3.5\div2.5+0.05=1.4+0.05=1.45.$

$\displaystyle \text{(ii) }3.6\div0.4\text{ of }4+0.5\times0.2-0.003$
$\displaystyle =3.6\div(0.4\times4)+0.1-0.003$
$\displaystyle =3.6\div1.6+0.097=2.25+0.097=2.347.$

$\displaystyle \text{(iii) }3.1-2.4\div0.5\text{ of }3\times0.75+0.05$
$\displaystyle =3.1-2.4\div(0.5\times3)\times0.75+0.05$
$\displaystyle =3.1-2.4\div1.5\times0.75+0.05$
$\displaystyle =3.1-1.6\times0.75+0.05$
$\displaystyle =3.1-1.2+0.05=1.95.$

$\displaystyle \text{(iv) }36.4-3.6\div1.2\times2.5\text{ of }4.6$
$\displaystyle =36.4-3.6\div1.2\times(2.5\times4.6)$
$\displaystyle =36.4-3.6\div1.2\times11.5$
$\displaystyle =36.4-3\times11.5=36.4-34.5=1.9.$

$\displaystyle \text{(v) }6.5-0.5\text{ of }(9.8-3.7)+4.3\times6.05$
$\displaystyle =6.5-0.5\times6.1+4.3\times6.05$
$\displaystyle =6.5-3.05+26.015$
$\displaystyle =29.465.$

$\displaystyle \text{(vi) }8.1-[2+3.51\text{ of }(7.1-3.4\times2.05)]$
$\displaystyle =8.1-[2+3.51\times(7.1-6.97)]$
$\displaystyle =8.1-[2+3.51\times0.13]$
$\displaystyle =8.1-[2+0.4563]$
$\displaystyle =8.1-2.4563=5.6437.$
$\displaystyle =3.1-4.8\times3\times0.75+0.05$
$\displaystyle =3.1-10.8+0.05=-7.65.$

$\displaystyle \text{(iv) }36.4-3.6\div1.2\times2.5\text{ of }4.6$
$\displaystyle =36.4-3\times2.5\times4.6=36.4-34.5=1.9.$

$\displaystyle \text{(v) }6.5-0.5\text{ of }(9.8-3.7)+4.3\times6.05$
$\displaystyle =6.5-0.5\times6.1+26.015$
$\displaystyle =6.5-3.05+26.015=29.465.$

$\displaystyle \textbf{Question 14: }\text{Simplify:}$
$\displaystyle \text{(i) }\frac{4.8\times4.8-3.8\times3.8}{4.8+3.8}$

$\displaystyle \text{(ii) }\frac{6.3\times6.3-3.7\times3.7} {6.3\times6.3+2\times6.3\times3.7+3.7\times3.7}$

$\displaystyle \text{(iii) }\frac{3.2\times3.2-2\times3.2\times2.7+2.7\times2.7} {3.2\times3.2+1.8\times1.8+2\times3.2\times1.8}$

$\displaystyle \text{(iv) }\frac{4.1\times4.1-2\times4.1\times3.7+3.7\times3.7} {3.7\times3.7-2\times3.7\times3.5+3.5\times3.5}$

$\displaystyle \text{(v) }\frac{1.7\times1.7\times1.7-0.9\times0.9\times0.9} {1.7\times1.7+1.7\times0.9+0.9\times0.9}$

$\displaystyle \text{(vi) }\frac{2.3\times2.3\times2.3+1.8\times1.8\times1.8} {2.3\times2.3-2.3\times1.8+1.8\times1.8}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\frac{4.8^2-3.8^2}{4.8+3.8}$
$\displaystyle =\frac{(4.8-3.8)(4.8+3.8)}{4.8+3.8}=4.8-3.8=1.$

$\displaystyle \text{(ii) }\frac{6.3^2-3.7^2}{6.3^2+2(6.3)(3.7)+3.7^2}$
$\displaystyle =\frac{(6.3-3.7)(6.3+3.7)}{(6.3+3.7)^2}$
$\displaystyle =\frac{6.3-3.7}{6.3+3.7}=\frac{2.6}{10}=0.26.$

$\displaystyle \text{(iii) }\frac{3.2^2-2(3.2)(2.7)+2.7^2} {3.2^2+1.8^2+2(3.2)(1.8)}$
$\displaystyle =\frac{(3.2-2.7)^2}{(3.2+1.8)^2} =\frac{0.5^2}{5^2}=\frac{0.25}{25}=0.01.$

$\displaystyle \text{(iv) }\frac{4.1^2-2(4.1)(3.7)+3.7^2} {3.7^2-2(3.7)(3.5)+3.5^2}$
$\displaystyle =\frac{(4.1-3.7)^2}{(3.7-3.5)^2} =\frac{0.4^2}{0.2^2}=\frac{0.16}{0.04}=4.$

$\displaystyle \text{(v) }\frac{1.7^3-0.9^3}{1.7^2+(1.7)(0.9)+0.9^2}$
$\displaystyle =\frac{(1.7-0.9)(1.7^2+(1.7)(0.9)+0.9^2)} {1.7^2+(1.7)(0.9)+0.9^2}$
$\displaystyle =1.7-0.9=0.8.$

$\displaystyle \text{(vi) }\frac{2.3^3+1.8^3}{2.3^2-(2.3)(1.8)+1.8^2}$
$\displaystyle =\frac{(2.3+1.8)(2.3^2-(2.3)(1.8)+1.8^2)} {2.3^2-(2.3)(1.8)+1.8^2}$
$\displaystyle =2.3+1.8=4.1.$
$\displaystyle \\$

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ICSE / ISC / CBSE Mathematics Portal for K12 Students

Ultimate Math Study Guide for Students

Class 8: Decimals – Exercise 6A

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