Convert rational number in $\frac {p}{q}$ form

Question

Convert rational number in $\frac {p}{q}$ form $0.40\bar 7$ (here bar is over $7$).

solution:

By solving I got the answer $367/900$ by multiplying by $10$

My friends are getting answer $4037/9900$ by multiplying by $100$

Are they correct $?$

Answer 1

Just divide and see: $$\frac{367}{900}=0.407777777\ldots=\frac{4037}{9900}$$

Answer 2

That's because $4037=367\times 11,$ so $$\frac{367}{900}=\frac{367\times 11}{900\times 11} = \frac{4037}{9900}.$$ Actually I would vote for your answer as it is the most reduced form.

Answer 3

$$900r=1000\cdot 0.40\bar7-100\cdot0.40\bar7=367$$ $$9900r=10000\cdot 0.40\bar7-100\cdot0.40\bar7=4037$$ $$99900r=100000\cdot 0.40\bar7-100\cdot0.40\bar7=40737$$ $$999900r=1000000\cdot 0.40\bar7-100\cdot0.40\bar7=407737$$ $$999\cdots900r=100000\cdots0\cdot 0.40\bar7-100\cdot0.40\bar7=407\cdots737$$

all yield correct answers, the first one being the most economical.

Answer 4

Example: just write it down: $$0.123\overline{4567}=\frac{1234567-123}{9999000}.$$

In your case $$0.40\overline{7}=\frac{407-40}{900}$$ or as well $$0.40\overline{77}=\frac{4077-40}{9900}$$ or even $$0.40\overline{77777}=\frac{4077777-40}{9999900}.$$