Question

The expression $\dfrac{{5\dfrac{5}{8}}}{{6\dfrac{3}{7}}}$ of $\dfrac{{6\dfrac{7}{{11}}}}{{9\dfrac{1}{8}}} \div \dfrac{8}{9}\left( {2\dfrac{3}{{11}} + \dfrac{{13}}{{22}}} \right)$ of $\dfrac{3}{5}$ equals:

Answer 1

Hint: Here for this question, we need to use three conditions.
It $a\dfrac{b}{c}$ is given, then we can write it as $\dfrac{{ac + b}}{c}$ . That is, we are converting mixed fractions into improper fractions.
It $\dfrac{a}{b} \div \dfrac{c}{d}$ is given, it can be rewritten as $\dfrac{a}{b} \times \dfrac{d}{c}$ . That is, we have taken reciprocal for this expression.
If “of” is given, we need to just multiply the terms. That refers to multiplication.
Here, we need to apply the BODMAS rule in this question.
 That is, we need to calculate the brackets first and then orders, then division or multiplication, and finally we need to add or subtract.

Complete step by step answer:
The given expression is $\dfrac{{5\dfrac{5}{8}}}{{6\dfrac{3}{7}}}$ of$\dfrac{{6\dfrac{7}{{11}}}}{{9\dfrac{1}{8}}} \div \dfrac{8}{9}\left( {2\dfrac{3}{{11}} + \dfrac{{13}}{{22}}} \right)$ of $\dfrac{3}{5}$.
Let us convert all mixed fractions into improper fractions.
$5\dfrac{5}{8} = \dfrac{{5 \times 8 + 5}}{8}$
         $ = \dfrac{{45}}{8}$
$6\dfrac{3}{7} = \dfrac{{6 \times 7 + 3}}{7}$
        $ = \dfrac{{45}}{7}$
$6\dfrac{7}{{11}} = \dfrac{{6 \times 11 + 7}}{{11}}$
           $ = \dfrac{{73}}{{11}}$
$9\dfrac{1}{8} = \dfrac{{9 \times 8 + 1}}{8}$
         $ = \dfrac{{73}}{8}$
$2\dfrac{3}{{11}} = \dfrac{{2 \times 11 + 3}}{{11}}$
          $ = \dfrac{{25}}{{11}}$
Now, we shall rewrite the given expressions in improper fractions.
Hence, we get,
$\dfrac{{\dfrac{{45}}{8}}}{{\dfrac{{45}}{7}}}of\dfrac{{\dfrac{{73}}{{11}}}}{{\dfrac{{73}}{8}}} \div \dfrac{8}{9}\left( {\dfrac{{25}}{{11}} + \dfrac{{13}}{{22}}} \right)of\dfrac{3}{5}$
Now, we need to apply reciprocals for the above expression.
\[\dfrac{{45}}{8} \times \dfrac{7}{{45}}of\dfrac{{73}}{{11}} \times \dfrac{8}{{73}} \div \dfrac{8}{9}\left( {\dfrac{{25}}{{11}} + \dfrac{{13}}{{22}}} \right)of\dfrac{3}{5}\]
On canceling, we get,
\[\dfrac{7}{8}of\dfrac{8}{{11}} \div \dfrac{8}{9}\left( {\dfrac{{25}}{{11}} + \dfrac{{13}}{{22}}} \right)of\dfrac{3}{5}\]
Now, rewrite by multiplication symbol.
\[\dfrac{7}{8} \times \dfrac{8}{{11}} \div \dfrac{8}{9}\left( {\dfrac{{25}}{{11}} + \dfrac{{13}}{{22}}} \right) \times \dfrac{3}{5}\]
Here, we need to take LCM for the terms inside the brackets.
\[\dfrac{7}{{11}} \div \dfrac{8}{9}\left( {\dfrac{{50 + 13}}{{22}}} \right) \times \dfrac{3}{5}\]
\[\dfrac{7}{{11}} \div \dfrac{8}{9} \times \dfrac{{63}}{{22}} \times \dfrac{3}{5}\]
Now, we shall cancel the terms.
\[\dfrac{7}{{11}} \div \dfrac{4}{1} \times \dfrac{7}{{11}} \times \dfrac{3}{5}\]
\[\dfrac{7}{{11}} \div \dfrac{{84}}{{55}}\]
Taking reciprocal, we have
\[\dfrac{7}{{11}} \times \dfrac{{55}}{{84}}\]
On canceling, we get,
\[\dfrac{5}{{12}}\] and it is the required answer.
Hence, the given expression is simplified into,
$\dfrac{{5\dfrac{5}{8}}}{{6\dfrac{3}{7}}}$ of $\dfrac{{6\dfrac{7}{{11}}}}{{9\dfrac{1}{8}}} \div \dfrac{8}{9}\left( {2\dfrac{3}{{11}} + \dfrac{{13}}{{22}}} \right)$ of $\dfrac{3}{5}$\[ = \dfrac{5}{{12}}\]

Note: Simplification of an expression is the process of changing the expression effectively without changing the meaning of an expression. An algebraic expression is an expression where the variables and the constants are combined.
Moreover, various steps are involved to simplify an algebraic expression. Some of the steps are listed below:
If the given algebraic expression contains like terms, we need to combine them.
Example: $3x + 2x + 4 = 5x + 4$
We need to split an algebraic expression into factors (i.e) the process of finding the factors for the given expression.
Example: ${x^2} + 4x + 3 = (x + 3)(x + 1)$
We need to expand an algebraic expression (i.e) we have to remove the respective brackets of an expression.
Example: $3(a + b) = 3a + 3b$.
We need to cancel out the common terms in an algebraic expression.
Example: $\dfrac{{{x^2} + 4x + 3}}{{x + 1}} = \dfrac{{(x + 3)(x + 1)}}{{x + 1}}$
$ = x + 3$