Exercise 18

The value of the numeric expression   \( \cfrac{ 0.\hat{6} + 0.\hat{3} }{ 2.\hat{2} } \),   is:

The value of the numeric expression:

\[ \cfrac{ 0.\hat{6} + 0.\hat{3} }{ 2.\hat{2} }, \]

is:

  1. \( \cfrac{ 81 }{ 100 } \)

  2. \( \cfrac{ 20 }{ 9 } \)

  3. \( \cfrac{ 9 }{ 20 } \)

  4. \( \cfrac{ 2 }{ 9 } \)

  5. \( \cfrac{ 9 }{ 22 } \)

Try to solve it before checking the answer.

  1. \( \cfrac{ 9 }{ 20 } \)

First of all, we must find the generating fractions of the three periodic decimals:

\[ 0.\hat{6} = \cfrac{6 – 0}{9} = \cfrac{2}{3}, \]
\[ 0.\hat{3} = \cfrac{3 – 0}{9} = \cfrac{1}{3}, \]
\[ 2.\hat{2} = \cfrac{22 – 2}{9} = \cfrac{20}{9} \]

Now, we have:

\[ \begin{aligned} \frac{ 0.\hat{6} + 0.\hat{3} }{2.\hat{2}} &= \frac{ \frac{2}{3} + \frac{1}{3} }{ \frac{ 20 }{ 9 } } \\[1em] &= \frac{ \frac{2 + 1}{3} }{ \frac{20}{9} } \\[1em] &= \frac{ \frac{3}{3} }{ \frac{20}{9} } \\[1em] &= \frac{ 3 \times 9 }{ 3 \times 20 } \\[1em] &= \boldsymbol{ \frac{9}{20} } \end{aligned} \]

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