Michaela, Gabriela and Susan were given the task of converting a decimal $$x=12.3\overline{45}$$ into a fraction.
Michaela solved the problem in the following steps:
(1) Firstly, Michaela created a second equation by multiplying the equation for $x$ by $1\,000$: $$\begin{aligned} x&=12.3\overline{45}\cr 1\,000x&=12\,345.\overline{45}\end{aligned}$$
(2) Then, Michaela subtracted the first equation from the second: $$1\,000x-x=12\,345.\overline{45}-12.3\overline{45}$$
(3) Michaela simplified the above equation and obtained the equation without decimals: $$999x=12\,333$$
(4) Finally, Michaela expressed $x$ as a fraction:
$$x=\frac{12\,333}{999}=\frac{4\,111}{333}$$
Gabriela followed these steps:
(1) Firstly, Gabriela created two equations. First one by multiplying the equation $x=12.3\overline{45}$ by $10$: $$10x=123.\overline{45}.$$ Second one by multiplying the equation $x=12.3\overline{45}$ by $1\,000$: $$1\,000x=12\,345.\overline{45}.$$
(2) Then, Gabriela subtracted the first equation from the second: $$1\,000x-10x=12\,345.\overline{45}-123.\overline{45}$$
(3) Simplifying, Gabriela derived the equation without decimals: $$990x=12\,222$$
(4) She expressed $x$ as a fraction: $$x=\frac{12\,222}{990}=\frac{679}{55}$$
Susan proceeded as follows:
(1) She also created two equations. First one by multiplying the equation $x=12.3\overline{45}$ by $10$: $$10x=123.\overline{45}.$$ Second one by multiplying the equation $x=12.3\overline{45}$ by $100$: $$100x=1\,234.\overline{54}.$$
(2) Next, Susan subtracted the first equation from the second: $$100x-10x=1\,234.\overline{54}-123.\overline{45}$$
(3) Then she simplified the above equation and obtained the equation without decimals: $$90x=1\,111.09$$
(4) From the final equation, Susan expressed $x$ as a fraction: $$x=\frac{1\,111.09}{90}=\frac{111\,109}{9\,000}$$
Which one solved the problem correctly?
Michaela
Gabriela
Susan
None of them
All of them
To convert a number $x$ with an infinite periodic expansion to a fraction, we need to determine two multiples of $x$ such that the digits after the decimal point in both multiples are the same. Michaela and Susan did not follow the recommended procedure.
Michaela made a mistake in step (3) when she subtracted numbers $12\,345.\overline{45}$ and $12.3\overline{45}$. The difference $12\,345.\overline{45}-12.3\overline{45}$ is not a number with a finite decimal expansion, and therefore, it cannot be used to convert $x$ to a fraction.
Susan made a mistake in step (1) when multiplying the equation $x=12.3\overline{45}$ for $x$ by $100$. Even if Susan had determined $100x$ correctly, the difference between $100x$ and $10x$ would still be a number with an infinite decimal expansion, and therefore, it again cannot be used to convert $x$ to a fraction.