Question

Simplify and write the result in decimal form
 $ \left( {1 \div \dfrac{2}{9}} \right) + \left( {1 \div 3\dfrac{1}{5}} \right) + \left( {1 + 2\dfrac{2}{3}} \right) $

Answer 1

Hint: To solve this problem, we will first convert the mixed fraction in the normal fraction form. Then we will divide the fraction by multiplying the first fraction with the reciprocal of the other. Next, we will add the fraction by finding the LCM of the denominator. We will then convert the result which is in fraction into decimal by converting the denominator into power of 10 by multiplying the correct number with numerator and denominator.

Complete step-by-step answer:
The given expression is $ \left( {1 \div \dfrac{2}{9}} \right) + \left( {1 \div 3\dfrac{1}{5}} \right) + \left( {1 + 2\dfrac{2}{3}} \right) $ .
First of all, we will write the mixed fraction in the normal fraction form.
 $ = \left( {1 \div \dfrac{2}{9}} \right) + \left( {1 \div \dfrac{{16}}{5}} \right) + \left( {1 \div \dfrac{8}{3}} \right) $
To divide the fractions, we will find the reciprocal of the second term in the expression. Division of two fractions is similar to the multiplication of the first term with the reciprocal of the second term. hence, the above expression can be rewritten as:
 $ \begin{array}{l}
 = \left( {1 \times \dfrac{9}{2}} \right) + \left( {1 \times \dfrac{5}{{16}}} \right) + \left( {1 \times \dfrac{3}{8}} \right)\\
 = \left( {\dfrac{9}{2}} \right) + \left( {\dfrac{5}{{16}}} \right) + \left( {\dfrac{3}{8}} \right)
\end{array} $
Since, the LCM of 2, 16 and 8 is 16. We can add the above terms as,
 $ \begin{array}{l}
 = \dfrac{{\left( {9 \times 8} \right) + \left( {5 \times 1} \right) + \left( {3 \times 2} \right)}}{{16}}\\
 = \dfrac{{72 + 5 + 6}}{{16}}\\
 = \dfrac{{83}}{{16}}
\end{array} $
To convert the fraction into decimal form, we will find the factors of the term 16. This can be expressed as:
 $ 16 = 2 \times 2 \times 2 \times 2 $
If the factor of the denominator in the fraction is 2 or 5 then we can multiply the denominator with 5 or 2 respectively to convert it to a denominator in the power of 10.
Hence,
 $ \begin{array}{l}
 = \dfrac{{83 \times 5 \times 5 \times 5 \times 5}}{{2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5}}\\
 = \dfrac{{51875}}{{10000}}\\
 = 5.1875
\end{array} $
Therefor the decimal form of the expression $ \left( {1 \div \dfrac{2}{9}} \right) + \left( {1 \div 3\dfrac{1}{5}} \right) + \left( {1 + 2\dfrac{2}{3}} \right) $ is $ 5.1875 $ .

Note: The division of fraction is similar to multiplication of the fractions, such that the first term multiplies with the reciprocal of the other. Also, to convert a fraction into decimal form we can multiply the denominator with the relevant term which converts it into power of 10. We will have to multiply the same number with the numerator too.