EXERCISE:2.7
Question 1
i) $1 \frac{1}{9}=1+\frac{1}{9}=\frac{9+1}{9}=\frac{10}{9}$
$3 \frac{1}{2}=3+\frac{1}{2}=\frac{6+1}{2}=\frac{7}{2}$
$\frac{3}{5}$ if $1 \frac{1}{9}=\frac{3}{5} \times \frac{10}{9}=\frac{2}{3}$
So $\frac{2}{3}+\frac{7}{2}=\frac{2 x^{2}+7 \times 3}{6}=\frac{4+21}{6}=\frac{25}{6}=4 \frac{1}{6}$
ii)$2 \frac{3}{8}=2+\frac{3}{8}=\frac{16+3}{8}=\frac{19}{8}$
$\frac{4}{5} \times 2 \frac{3}{8}=\frac{4}{5} \times \frac{19}{82}=\frac{19}{10}$
$2 \times \frac{3}{5}=\frac{6}{5} \times \frac{2}{2}=\frac{12}{10}$
$50, \frac{19}{10}-\frac{12}{10}=\frac{19-12}{10}=\frac{7}{10}$
iii) $\frac{4}{5}+2=\frac{4+2 \times 5}{5}=\frac{4+10}{5}=\frac{14}{5}$
$3-\frac{2}{3}=\frac{3 \times 3-2}{3}=\frac{9-2}{3}=\frac{7}{3}$
So, $\frac{14}{5} \times \frac{7}{3}=\frac{98}{15}=6 \frac{8}{15}$
Question 2
i) $2 \frac{2}{3}=2+\frac{2}{7}=\frac{2 x-2}{7}=\frac{16}{7} .$
$\frac{1}{4}$ of $2 \frac{2}{7}=\frac{1}{4} \times \frac{16}{7}=\frac{4}{7} .$
So, $\frac{4}{7} \div \frac{3}{5}=\frac{4 / 7}{3 / 5}$
$=\frac{4}{7} \times \frac{5}{3}=\frac{20}{21} .$
ii) $\frac{3}{7} \div \frac{1}{2}=\frac{3 / 7}{1 / 2}=\frac{3}{7} \times 2=\frac{6}{7}$
So $\frac{6}{7} \div \frac{7}{8}=\frac{6 / 7}{7 / 8}=\frac{6}{7} \times \frac{8}{7}=\frac{48}{49} .$
iii) $\frac{5}{8} \div \frac{3}{4}=\frac{5 / 8}{3 / 4}=\frac{5}{8} \times \frac{4}{3}=\frac{5}{6}$
so, $\frac{5}{6}+\frac{2}{5}=\frac{5 \times 5+2 \times 6}{30}=\frac{25+12}{30}=\frac{37}{30}=1$
$ \frac{7}{30} .$
Question 3
i) $4 \frac{1}{2}=4+\frac{1}{2}=\frac{4 \times 2+1}{2}=\frac{9}{2}$
$2 \frac{2}{3}=2+\frac{2}{3}=\frac{2 \times 3+2}{3}=\frac{8}{3}$
$5 \frac{1}{2}=5+\frac{1}{2}=\frac{5 \times 2+1}{2}=\frac{11}{2}$
$3 \frac{5}{6}=3+\frac{5}{6}=\frac{3 \times 6+5}{6}=\frac{18+5}{6}=\frac{23}{6}$
$\left(4 \frac{1}{2}-2 \frac{2}{3}\right)=\frac{9}{2}-\frac{8}{3}=\frac{9 \times 3-8 \times {2}}{6}=\frac{27-16}{6}=\frac{11}{6}$
$5 \frac{1}{2}$ of $3 \frac{5}{6}=\frac{11}{2} \times \frac{23}{6}=\frac{253}{12} .$
So, therefore $\frac{11}{6} \div \frac{7}{12}+\frac{253}{12}$
$=\frac{11}{6} \times \frac{12}{3}+\frac{253}{12}$
$=\frac{22}{7}+\frac{253}{12}$
$=\frac{22 \times 12+253 \times 7}{84}$
$=\frac{2075}{84}$
$=24 \frac{19}{84}$
ii)$5 \frac{1}{3}=5+\frac{1}{3}=\frac{5 x]+1}{3}=\frac{16}{3} .$
=$2 \frac{1}{2}=2+\frac{1}{2} \cdot \frac{2 x^{2}+1}{2}=\frac{5}{2} .$
=$\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{1}{4}-\frac{1}{6}\right)-\left[8-\left\{5 \frac{1}{3}-\left(3-2 \frac{1}{2}\right)\right\}\right]$
$=\left(\frac{3+2}{6}\right) \div\left(\frac{3-2}{12}\right)-\left[8-\left\{\frac{16}{3}-\left(3-\frac{5}{2}\right)\right\}\right]$
$=\left(\frac{5}{6}\right) \div\left(\frac{1}{12}\right)-\left[8-\left\{\frac{16}{3}-\left(\frac{6-5}{2}\right)\right\}\right]$
$\left(4 \frac{1}{2}-2 \frac{2}{3}\right)=\frac{9}{2}-\frac{8}{3}=\frac{9 \times 3-8 x^{2}}{6}=\frac{27-16}{6}=\frac{11}{6} .$
$5 \frac{1}{2}$ of $3 \frac{5}{6}=\frac{11}{2} \times \frac{23}{6}=\frac{253}{12} .$
So, therefore $\frac{11}{6} \div \frac{7}{12}+\frac{253}{12}$
$=\frac{11}{8} \times \frac{12}{7}+\frac{253}{12}$
$\frac{22}{7}+\frac{253}{12}$
$=\frac{22 \times 12+253 \times 7}{84}$
$=\frac{2075}{84}$
$=24 \frac{19}{84} .$
iii) $5 \frac{1}{3}=5+\frac{1}{3}=\frac{5 \times 3+1}{3}=\frac{16}{3}$
$2 \frac{1}{2}=2+\frac{1}{2} \cdot \frac{2 \times 2+1}{2}=\frac{5}{2}$
=$\left(\frac{1}{2}+\frac{1}{3}\right) \div\left(\frac{1}{4}-\frac{1}{6}\right)-\left[8-\left\{5 \frac{1}{3}-\left(3-2 \frac{1}{2}\right)\right\}\right] .$
$=\left(\frac{3+2}{6}\right) \div\left(\frac{3-2}{12}\right)-\left[8-\left\{\frac{16}{3}-\left(3-\frac{5}{2}\right)\right\}\right]$
$=\left(\frac{5}{6}\right) \div\left(\frac{1}{12}\right)-\left[8-\left\{\frac{16}{3}-\left(\frac{6-5}{2}\right)\right]\right]$
$=\frac{516}{1 / 12}-\left[8-\left(\frac{16}{3}-\frac{1}{2}\right)\right]$
$=\frac{5}{6} \times 1^{2}-\left[8-\left(\frac{16 \times 2-3 \times 1}{6}\right)\right]$
$=10-\left[8-\left(\frac{32-3}{6}\right)\right]$
$=10-\left[8-\frac{29}{6}\right]$
$10-\left[\frac{8 \times 6-29}{6}\right]$
$=10-\frac{48-29}{6}$
$=10-\frac{19}{6}$
$=\frac{60-19}{6}$
$=\frac{41}{6}=6 \frac{5}{6} .$
Question 4
(i)
$\begin{aligned} & 2.3-[1.89-\{3.6-(2.7-0.8-0.03)\}] \\ & 2.3-[1.89-\{3.6-(2.7-0.77)\}] \\ &=2.3-[1.89-\{3.6-1.93\}] \\=& 2.3-[1.89-1.67] \\=& 2.3-0.22 \\=& 2.08 \end{aligned}$
ii) $4.5-\frac{1}{2}$ of $(7.6-3.5)+2.3 \times 4.05$
$=4.5-\frac{1}{2} \times(4.1)+2.3 \times 4.05$
$=4.5-\frac{4.1}{2}+9.315$
$=4.5+9.315-2.05$
$=13.815-2.05$
$=11.765$
Question 5
i) $2 \frac{1}{2}=\frac{5}{2}$
$2 \frac{1}{2}+\frac{1}{5}=\frac{5}{2}+\frac{1}{5}$
$=\frac{-5 \times 5+1 \times 2}{10}=\frac{27}{10} .$
$2 \frac{1}{2} \div \frac{1}{5}=\frac{5}{2} \div \frac{1}{5}$
$\frac{5}{2} \times 5=\frac{25}{2}$
So, $\frac{2 \frac{1}{2}+\frac{1}{5}}{2 \frac{1}{2} \div \frac{1}{5}}=\frac{27 / 10}{25 / 2}$
$=\frac{27}{10} \times \frac{2}{25}$
$=\frac{27}{125} .$
(ii) $\frac{3.5 \times 0.24}{0.21}-0.037$
$3 \cdot 5=\frac{35}{10}=\frac{7}{2} .$
$0.24= \frac{24}{100}=\frac{6}{25}$
$0 .21=\frac{21}{100}$
$=\frac{\frac{7}{2} \times \frac{6}{25}}{\frac{21}{100}}-\frac{37}{1000}$
$=\frac{7}{2} \times \frac{6}{25} \times \frac{100}{21}-\frac{37}{1000}$
$=4-\frac{37}{1000}$
=4-0.037
=3.963
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