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Writing decimals as fractions

Using a calculator, you can write a fraction as a decimal $$\frac{2}{25} = \Tred{0.08},\quad \frac{121}{12} = \Tred{10.0833\dot{3}} $$ How can you go from the decimal to the fraction?

  • If the decimal is terminating, multiply the number by ten to the power of the number of decimals. You get an integer that you can divide back and simplify. $$ \Tred{0.08} \times\Tblue{10} = \Torange{8},\quad \Tred{0.08} = \frac{\Torange{8}}{\Tblue{10}} = \frac{2}{25}$$
  • If the decimal is recurring, multiply the number by a power of ten to have the recurring part begin at the digit. Multiply by ten to the power of the recurring sequence and subtract. You have an integer. You can then work out the fraction and simplify. \begin{align*} \Tred{R} &= 10.08\Tgreen{33\dot{3}},\quad \Tblue{10} \Tred{R} = 108.\Tgreen{\dot{3}}, \quad \Tblue{10} \Tred{R} = 1083.\Tgreen{\dot{3}},\\ & \Tblue{90} \Tred{R} = 1083.\Tgreen{\dot{3}} - 108.\Tgreen{\dot{3}} = 1083 - 108 = \Torange{9075},\\ &\qquad\qquad \Tred{R} = \frac{\Torange{9075}}{\Tblue{90}} =\frac{121\times 75}{12\times 75} = \frac{121}{12} \end{align*}
  • If the decimal is infinite and not recurring, the number is not a fraction. It cannot be simplified.