Problems from Decimal expression of rational numbers

Calculate the expression as quotient of integers of the following rational numbers:

  1. $$1,7\widehat{42}$$
  2. $$0,537\widehat{3}$$
  3. $$12,63\widehat{408}$$
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Development:

  1. According to our notation; $$a=17, b=1.742, m=1$$ and $$n=2$$. Then that corresponds to the quotient $$$\dfrac{b-a}{990}=\dfrac{1.742-17}{990}=\dfrac{115}{66}$$$

  2. According to our notation; $$a=537, b=5.373, m=3$$ and $$n=1$$. Then that corresponds to the quotient $$$\dfrac{b-a}{9.000}=\dfrac{5.373-537}{9.000}=\dfrac{403}{750}$$$

  3. According to our notation; $$a=1.263, b=1.263.408, m=2$$ and $$n=3$$. Then that corresponds to the quotient $$$\dfrac{b-a}{99.900}=\dfrac{1.263.408-1.263}{99.900}=\dfrac{84.143}{6.660}$$$

Solution:

  1. $$\dfrac{115}{66}$$
  2. $$\dfrac{403}{750}$$
  3. $$\dfrac{84.143}{6.660}$$
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Calculate the decimal expression and the repeating of the following rational numbers:

  1. $$\dfrac{7}{4}$$
  2. $$\dfrac{5}{11}$$
  3. $$\dfrac{5}{18}$$
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Development:

  1. If we do the division we obtain $$\dfrac{7}{4}=1,75$$, which is the decimal expression. There is no repeating.

  2. If we do the division we obtain $$\dfrac{5}{11}=0,454545\ldots$$

    So the repeating is $$45$$ and the decimal expression is $$0,\widehat{45}.$$

  3. If we do the division we obtain $$\dfrac{5}{18}=0,27777\ldots$$

    So the repeating is $$7$$ and the decimal expression is $$0,2\widehat{7}$$.

Solution:

  1. The decimal expression is $$\dfrac{7}{4}=1,75$$. There is no repeating.
  2. The repeating is $$45$$ and the decimal expression is $$0,\widehat{45}.$$
  3. The repeating is $$7$$ and the decimal expression is $$0,2\widehat{7}$$
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