How do I represent the following fraction in decimal?

Question

I would like to know how do I represent the following fraction represented in base-$8$ in decimal form:

$(\frac{35}{25})_{\text{8}}$

It seems very hard. Please somebody help me.

Answer 1

Perform the division algorithm in base $8$ in the same way as in base $10$:

\begin{alignat}{2} & & &1.3030\dots\\ &&&----- \\ 35\enspace&(&\;&25\\ &&&10\mathbf{0}\\ &&&\phantom{1}77\\[-1em] &&&--- \\ &&&\phantom{17}1\mathbf{00}\\ &&&\phantom{001}77\\[-1em] &&&--- \\ &&&\phantom{0117}1\mathbf{00} \end{alignat} Some explanations:

In base $8$, $3\times 25=77$ and $77+1=100$. Thus $$\Bigl[\frac{35}{25}\Bigr]_8=[1.\overline{30}]_8.$$

Answer 2

Do just what you would normally do with a base $10$ number: divide the numerator by the denominator. First: $$\frac{35_8}{25_8}=1_8+\frac{10_8}{25_8}$$ Now, add a zero to the end of $10_8$ to get $100_8$. Make sure that you realize that $100_8÷25_8\neq4$, but $3$ with a remainder of $1$. So far, we have $1.3_8+\frac{1_8}{25_8}$ Continue until you get a repeating decimal.

If you can also find some number $n_8$ such that $25_8*n_8=7777...7_8$, then you can multiply both the numerator and the denominator by $n_8$ to get some number over a number made entirely of sevens. The numerator is your repeating decimal. For instance, $25_8*3_8=77_8$, so your repeating decimal is $10_8*3_8=30_8$, i.e. $1.\bar{30}$

Answer 3

You can do the operation in base $8$ if you can do math there. All the below is in base $8$ even though I do not show the subscript. $$\begin {align} \frac {35}{25}&=1+\frac {10}{25}\end {align}\\=1+\frac 1{10}\cdot\frac {100}{25}=1+\frac 3{10}+\frac 1{10}\cdot \frac 1{25}\\=1+\frac 3{10}+\frac 1{100}\cdot \frac {10}{25}\\=1.303\overline{03}$$ where we recognized the repeat.