Question

1.45 (5 repeated) as a fraction

Answer 1

let's firstly move the decimal point to the right, thus leaving the recurring part to the right, and then assign the recurring part at right to some variable, let's proceed.

`1.4\overline{55555}\implies \cfrac{14.\overline{55555}}{10}\qquad \qquad \stackrel{\textit{now let's make}}{x = 0.\overline{55555}} \\\\\\ \stackrel{\textit{so then we can say}}{\cfrac{14.\overline{55555}}{10}\implies \cfrac{14+0.\overline{55555}}{10}}\implies \cfrac{14+x}{10} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{so we can say that}~\hfill }{10x~~ =~~5.\overline{55555}}\implies 10x = 5+ 0.\overline{55555}\implies 10x = 5+x \\\\\\ 9x=5\implies x = \cfrac{5}{9}`

well, now let's plug that into our fraction

`\cfrac{14+x}{10}\implies \cfrac{14+(5)/(9)}{10}\implies \cfrac{~~(131)/(9)~~}{10}\implies \blacktriangleright\cfrac{131}{90}\implies 1(41)/(90) \blacktriangleleft`