Question

Convert the fraction 7 24 into a repeating decimal.

Answer 1

`\frac7{24}=\frac6{24}+\frac1{24}=\frac14+\frac1{24}`


`\frac1{24}=\frac1{8\time3}=\frac a8+\frac b3=(3a+8b)/(24)`

`\implies3a+8b=1`

We can choose
`a=-1` and
`b=\frac12`, so that


`\frac1{24}=-\frac18+\frac16`

Recalling that
`\frac13=0.333\ldots`, it follows that
`\frac16=0.166\ldots`. So,


`\frac1{24}=0.125+0.166\ldots=0.04166\ldots`

`\implies\frac7{24}=0.25+0.4166\ldots=0.29166\ldots`