Question

Convert the fraction 7 24 into a repeating decimal.

Answer 1

$\dfrac7{24}=\dfrac6{24}+\dfrac1{24}=\dfrac14+\dfrac1{24}$

$\dfrac1{24}=\dfrac1{8\time3}=\dfrac a8+\dfrac b3=\dfrac{3a+8b}{24}$
$\implies3a+8b=1$

We can choose $a=-1$ and $b=\dfrac12$, so that

$\dfrac1{24}=-\dfrac18+\dfrac16$

Recalling that $\dfrac13=0.333\ldots$, it follows that $\dfrac16=0.166\ldots$. So,

$\dfrac1{24}=0.125+0.166\ldots=0.04166\ldots$
$\implies\dfrac7{24}=0.25+0.4166\ldots=0.29166\ldots$