Problems from From decimal numbers to fractions

Define an exact decimal number $$a$$, a pure periodic decimal number $$b$$ and a ultimately periodic decimal number $$c$$. Find the equivalent fraction of each one.

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Development:

$$a=2,654$$

$$b=13,\widehat{187}$$

$$c=19,13\widehat{76}$$

  • The fraction equivalent to $$a$$:

$$\dfrac{2654}{1000}=\dfrac{1327}{500}$$

It is not possible to simplify it any more since $$500$$ only contains the factors $$2$$ and $$5$$, and $$1327$$ cannot be divided by either $$2$$ or by $$5$$.

  • Equivalent fraction to $$b$$:

$$\dfrac{13187-13}{999}=\dfrac{13174}{999}$$

And it is not possible to simplify it any more.

  • Fraction equivalent to $$c$$:

$$\dfrac{1}{100}\cdot 1913,\widehat{76}=\dfrac{1}{100}\cdot \dfrac{191376-1913}{99}=\dfrac{189463}{9900}$$

Solution:

$$a=\dfrac{1327}{500}$$

$$b=\dfrac{13174}{999}$$

$$c=\dfrac{189463}{9900}$$

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