Question

Simplify each of the following to his lowest term:

(i) $\frac{75}{80}$
(ii) $\frac{52}{76}$
(iii) $\frac{84}{98}$
(iv) $\frac{68}{17}$
(v) $\frac{150}{50}$
(vi) $\frac{162}{108}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\frac{75}{80} \\ \mathrm{Factors}\mathrm{of}75\mathrm{are}1,3,5,15,25\mathrm{and}75. \\ \mathrm{Factors}\mathrm{of}80\mathrm{are}1,2,4,5,8,10,16,20,40\mathrm{and}80. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}75\&80\mathrm{are}1\&5. \\ \mathrm{HCF}\mathrm{of}75\&80\mathrm{is}5. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\&\mathrm{denominator}\mathrm{by}5,\mathrm{we}\mathrm{get}: \\ \frac{75÷5}{80÷5}=\frac{15}{16} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\frac{52}{76} \\ \mathrm{Factors}\mathrm{of}52\mathrm{are}1,2,4,13,26\mathrm{and}52. \\ \mathrm{Factors}\mathrm{of}76\mathrm{are}1,2,4,19,38\mathrm{and}76. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}52\&76\mathrm{are}1,2\mathrm{and}4. \\ \mathrm{HCF}\mathrm{of}52\mathrm{and}76\mathrm{is}4. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}4,\mathrm{we}\mathrm{get}: \\ \frac{52÷4}{76÷4}=\frac{13}{19} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\frac{84}{98} \\ \mathrm{Factors}\mathrm{of}84\mathrm{are}1,2,3,4,6,7,12,14,21,28,42\mathrm{and}84. \\ \mathrm{Factors}\mathrm{of}98\mathrm{are}1,2,7,14,49\mathrm{and}98. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}84\mathrm{and}98\mathrm{are}1,2,7\mathrm{and}14. \\ \mathrm{HCF}\mathrm{of}84\mathrm{and}98\mathrm{is}14. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}14,\mathrm{we}\mathrm{get}: \\ \frac{84÷14}{98÷14}=\frac{6}{7} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\frac{68}{17} \\ \mathrm{Factors}\mathrm{of}68\mathrm{are}1,2,4,17,34\mathrm{and}68. \\ \mathrm{Factors}\mathrm{of}17\mathrm{are}1\mathrm{and}17. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}68\mathrm{and}17\mathrm{are}1\mathrm{and}17. \\ \mathrm{HCF}\mathrm{of}68\mathrm{and}17\mathrm{is}17. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}17,\mathrm{we}\mathrm{get}: \\ \frac{68÷17}{17÷17}=\frac{4}{1} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\frac{150}{50} \\ \mathrm{Factors}\mathrm{of}150\mathrm{are}1,2,3,5,6,10,15,25,50\mathrm{and}150. \\ \mathrm{Factors}\mathrm{of}50\mathrm{are}1,2,5,10,25\mathrm{and}50. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}150\mathrm{and}50\mathrm{are}1,2,5,10,25\mathrm{and}50. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}50,\mathrm{we}\mathrm{get}: \\ \frac{150÷50}{50÷50}=\frac{3}{1} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{vi}\right)\frac{162}{108} \\ \mathrm{Factors}\mathrm{of}162\mathrm{are}1,2,3,6,9,18,27,54,81\mathrm{and}162. \\ \mathrm{Factors}\mathrm{of}108\mathrm{are}1,2,3,4,6,9,12,18,27\mathrm{and}54. \\ \mathrm{Common}\mathrm{factors}\mathrm{of}162\mathrm{and}108\mathrm{are}1,2,3,6,9,18,27\mathrm{and}54. \\ \mathrm{HCF}\mathrm{of}162\mathrm{and}108\mathrm{is}54. \\ \mathrm{Dividing}\mathrm{both}\mathrm{the}\mathrm{numerator}\mathrm{and}\mathrm{denominator}\mathrm{by}54,\mathrm{we}\mathrm{get}: \\ \frac{162÷54}{108÷54}=\frac{3}{2} \end{gathered}$$