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**Hint:**The given question is to simplify the fractions. A fraction is a part of a whole. It has a numerator and a denominator.

For example: \[\dfrac{1}{2}\] here 1 is the numerator and 2 is the denominator. In maths, 'of' is also considered as one of the arithmetic operations meaning multiplication within the parenthesis.

So we want to replace ‘of’ with \[ \times \] to find out the solution. We must use the ‘BODMAS’ rule for simplification, It stands for

B - Brackets

O - Of Division,

M - Multiplication,

A - Addition,

S - Subtraction.

This specifies the priority. If there is no bracket then do multiplication or division, addition or subtraction.

**Complete step by step solution:**

We are given problem can be simplified as follows:

i) \[\dfrac{1}{4}\] of \[2\dfrac{2}{7} \div \dfrac{3}{5}\]

Let replace ‘of’ with multiply symbol \[( \times )\], then

\[\dfrac{1}{4} \times 2\dfrac{2}{7} \div \dfrac{3}{5}\]

\[2\dfrac{2}{7}\] is a mixed fraction want to convert into improper fraction then we will get \[\dfrac{{16}}{7}\] (multiply the whole number by the denominator and the result with the numerator now replaces the numerator by finding the answer).

\[\dfrac{1}{4} \times \dfrac{{16}}{7} \div \dfrac{3}{5}\]

Here no bracket so do multiplication first, we can cancel the numerator \[16\] by the denominator \[4\] (cross cancelling) then we get \[4\], \[16\] is \[4\] times in \[4\] table.

\[\dfrac{4}{7} \div \dfrac{3}{5}\] it means that \[\dfrac{{\dfrac{4}{7}}}{{\dfrac{3}{5}}}\] where \[\dfrac{4}{7}\] is the numerator and \[\dfrac{3}{5}\] is the denominator. When the fraction in the denominator brings up its reciprocal is associated with multiplication.

\[\dfrac{4}{7} \times \dfrac{5}{3}\]

We can’t simplify further when multiplies numerator with numerator and denominator with denominator we will get,

\[\dfrac{{20}}{{21}}\]

**Therefore, \[\dfrac{1}{4}\] of \[\;2\dfrac{2}{7} \div \dfrac{3}{5} = \dfrac{{20}}{{21}}\].**

ii) \[1\dfrac{1}{4}\] of \[\dfrac{1}{2} \div 1\dfrac{1}{3}\]

First, change the mixed fractions into improper fractions and replace ‘of’ with multiply symbol \[( \times )\], then

\[\dfrac{5}{4} \times \dfrac{1}{2} \div \dfrac{4}{3}\]

Here no brackets so first do multiplication,

\[\dfrac{5}{8} \div \dfrac{4}{3}\]

It is in the form \[\dfrac{{\dfrac{5}{8}}}{{\dfrac{4}{3}}}\], when we convert this by taking reciprocal we will get,

\[\dfrac{5}{8} \times \dfrac{3}{4}\]

We can’t simplify further so do multiplication we will get,

\[\dfrac{{15}}{{32}}\]

**Therefore, the final answer, \[1\dfrac{1}{4}\] of \[\dfrac{1}{2} \div 1\dfrac{1}{3} = \dfrac{{15}}{{32}}\].**

iii) \[6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8}\]

We can simply say the answer is \[0\]. Because “any number multiplied by \[0\] is \[0\] itself”.

**Therefore, \[6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8} = 0\].**

iv) \[\dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{3}{7}\] of \[2\dfrac{5}{8}\]

First of all change the mixed fractions into improper fractions and replace ‘of’ with multiply symbol \[( \times )\], then

\[\dfrac{3}{4} \times \dfrac{4}{3} \div \dfrac{3}{7} \times \dfrac{{21}}{8}\]

By doing multiplication, \[3\] and \[4\] will be cancelled by each other (cross cutting) and \[21\]is \[3\] by \[7\] tables.

\[1 \div \dfrac{3}{1} \times \dfrac{3}{8}\]

On performing multiplication, then

\[1 \div \dfrac{9}{8}\]

It is in the form, \[\dfrac{1}{{\dfrac{9}{8}}}\], when we convert this by taking reciprocal we will get,

\[1 \times \dfrac{8}{9}\]

Any number multiplied by \[1\] we will get the same number.

**Therefore, \[\dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{3}{7}\] of \[2\dfrac{5}{8} = \dfrac{8}{9}\].**

v) \[2\dfrac{1}{4} \div \dfrac{2}{7}\] of \[1\dfrac{1}{3} \times \dfrac{2}{3}\]

Primarily, change the mixed fractions into improper fractions and replace ‘of’ with multiply symbol (\[ \times \]),

\[\dfrac{9}{4} \div \left( {\dfrac{2}{7} \times \dfrac{4}{3}} \right) \times \dfrac{2}{3}\]

According to BODMAS solve the bracket first,

\[\dfrac{9}{4} \div \dfrac{8}{{21}} \times \dfrac{2}{3}\]

When a fraction is in division we can change it into multiplication by taking its reciprocal, therefore,

\[\dfrac{9}{4} \times \dfrac{{21}}{8} \times \dfrac{2}{3}\]

Doing cross cancellation finally we will get,

\[\dfrac{{63}}{{16}}\]

When representing this improper fraction into mixed fraction we will get,

\[3\dfrac{{15}}{{16}}\]

**Therefore, \[2\dfrac{1}{4} \div \dfrac{2}{7}\] of \[1\dfrac{1}{3} \times \dfrac{2}{3} = 3\dfrac{{15}}{{16}}\].**

**Note:**

When simplifying fractions we should remind certain things:

1. When the question is in a mixed fraction we should change it into an improper fraction and then only start to simplify.

2. If the answer we get is in an improper fraction finally we should change it into a mixed fraction.

3. Whenever we find ‘of’ we should replace it with \[ \times \] and it has the highest priority so we should solve it first.

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