Adding fractions in base-$8$ as fractions and decimals.

Question

I would like to add the following fractions represented in base-$8$ as both fractions and decimals:

$(\frac{2}{3})_{\text{8}} + (\frac{5}{7})_{\text{8}}$

Please somebody explain to me how to do this problem? Thanks!

Answer 1

Properties of fractions do not change when it comes to the base:

So if you have $$\frac{2_8}{3_8}+\frac{5_8}{7_8}$$

Use the properties you already know. Get common denominators and add!

$$\frac{2_8}{3_8}+\frac{5_8}{7_8} = \frac{2_8*7_8}{3_8*7_8}+\frac{5_8*3_8}{7_8*3_8} = \frac{16_8}{25_8}+\frac{17_8}{25_8}=\frac{35_8}{25_8} = \frac{29_{10}}{21_{10}}$$

For decimals it is the same. Let's take $\frac{29}{10} = 1.380952381_{10}$... Using the recursive multiplication base method for decimals, first notice that $1.380952381>1$. Mark that down, and subtract 1. Our number now is $$1_8 + 0.380952381_{10}$$

Multiply $.380952381$ by our base, or 8. We get: $8*0.380952381 = 3.04739048$ Notice that $3.04739048>1$. Mark that down, and subtract 3. Our number now is $$1.3_8+0.04739048_{10}$$

Multiply $0.04739048$ by our base, or 8. We get: $8*0.04739048= 0.37912384$ Notice that $0.37912384<1$. Mark that down. Our number now is $$1.30_8+0.37912384_{10}$$

Keep repeating this on and on to your heart's content or until you reach a certian level of accuracy. After 7 iterations I got $$\frac{35_8}{25_8} = 1.30302071_8$$