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Hint- Use the concept that whenever the fraction $\dfrac{p}{q}$ is in lowest form then the fraction has no common factors except 1.

Whenever the numerator is greater than the denominator, the fraction is called an improper fraction. For example $\dfrac{5}{4}$

As we know that whenever the fraction $\dfrac{p}{q}$ is in lowest form then the fraction has no common factors except 1.

Now as we know mixed fraction$ = $quotient \[\dfrac{{{\text{remainder}}}}{{{\text{divisor}}}}\]

So, using this property of fraction we have,

$\left( i \right) = 7 \times \dfrac{3}{5} = \dfrac{{21}}{5}$ So, this is in lowest form as $\dfrac{{21}}{5}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{21}}{5} = 4\dfrac{1}{5}\], because \[\left( {\left( {\left( {5 \times 4} \right) + 1} \right) = 21} \right)\]

(Where 5 is divisor, 4 is quotient and 1 is remainder)

$\left( {ii} \right) = 4 \times \dfrac{1}{3} = \dfrac{4}{3}$ So, this is in lowest form as $\dfrac{4}{3}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{4}{3} = 1\dfrac{1}{3}\], because \[\left( {\left( {\left( {1 \times 3} \right) + 1} \right) = 4} \right)\]

(Where 3 is divisor, 1 is quotient and 1 is remainder)

$\left( {iii} \right) = 2 \times \dfrac{6}{7} = \dfrac{{12}}{7}$ So, this is in lowest form as $\dfrac{{12}}{7}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{12}}{7} = 1\dfrac{5}{7}\], because \[\left( {\left( {\left( {1 \times 7} \right) + 5} \right) = 12} \right)\]

(Where 7 is divisor, 1 is quotient and 5 is remainder)

$\left( {iv} \right) = 5 \times \dfrac{2}{9} = \dfrac{{10}}{9}$ So, this is in lowest form as $\dfrac{{10}}{9}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{10}}{9} = 1\dfrac{1}{9}\], because \[\left( {\left( {\left( {1 \times 9} \right) + 1} \right) = 10} \right)\]

(Where 9 is divisor, 1 is quotient and 1 is remainder)

$\left( v \right) = \dfrac{5}{2} \times 6 = 15$

Now, we have to convert 15 into fraction so multiply by any number in numerator and denominator but we have to write the lowest form of fraction so we multiply 1 in numerator and denominator.

$ \Rightarrow 15 = \dfrac{{15 \times 1}}{1} = \dfrac{{15}}{1}$ So, this is written in lowest form and has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{15}}{1} = 15\dfrac{0}{1}\], because \[\left( {\left( {\left( {15 \times 1} \right) + 0} \right) = 15} \right)\]

(Where 1 is divisor, 15 is quotient and 0 is remainder)

$\left( {vi} \right) = 11 \times \dfrac{4}{7} = \dfrac{{44}}{7}$ So, this is in lowest form as $\dfrac{{44}}{7}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{44}}{7} = 6\dfrac{2}{7}\], because \[\left( {\left( {\left( {7 \times 6} \right) + 2} \right) = 44} \right)\]

(Where 7 is divisor, 6 is quotient and 2 is remainder)

$\left( {vii} \right) = 20 \times \dfrac{4}{5} = \dfrac{{80}}{5} = 16$

Now, we have to convert 16 into fraction so multiply by any number in numerator and denominator but we have to write the lowest form of fraction so we multiply 1 in numerator and denominator.

$ \Rightarrow 16 = \dfrac{{16 \times 1}}{1} = \dfrac{{16}}{1}$ So, this is written in lowest form and has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{16}}{1} = 16\dfrac{0}{1}\], because \[\left( {\left( {\left( {16 \times 1} \right) + 0} \right) = 16} \right)\]

(Where 1 is divisor, 16 is quotient and 0 is remainder)

$\left( {viii} \right) = \dfrac{2}{3} \times 4 = \dfrac{8}{3}$ So, this is in lowest form as $\dfrac{8}{3}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{8}{3} = 2\dfrac{2}{3}\], because \[\left( {\left( {\left( {2 \times 3} \right) + 2} \right) = 8} \right)\]

(Where 3 is divisor, 2 is quotient and 2 is remainder)

So, these are the required answers.

Note- In such types of questions the key concept is that while converting the fraction into lowest form fraction i.e. in $\dfrac{p}{q}$ form where $q \ne 0$, then p and q must have no common factors except 1. So, if a fraction has some common factors simply cancel out, it will be converted into the lowest form fraction, then apply a mixed fraction formula to convert the fraction into mixed fraction, we will get the required answer.

Whenever the numerator is greater than the denominator, the fraction is called an improper fraction. For example $\dfrac{5}{4}$

As we know that whenever the fraction $\dfrac{p}{q}$ is in lowest form then the fraction has no common factors except 1.

Now as we know mixed fraction$ = $quotient \[\dfrac{{{\text{remainder}}}}{{{\text{divisor}}}}\]

So, using this property of fraction we have,

$\left( i \right) = 7 \times \dfrac{3}{5} = \dfrac{{21}}{5}$ So, this is in lowest form as $\dfrac{{21}}{5}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{21}}{5} = 4\dfrac{1}{5}\], because \[\left( {\left( {\left( {5 \times 4} \right) + 1} \right) = 21} \right)\]

(Where 5 is divisor, 4 is quotient and 1 is remainder)

$\left( {ii} \right) = 4 \times \dfrac{1}{3} = \dfrac{4}{3}$ So, this is in lowest form as $\dfrac{4}{3}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{4}{3} = 1\dfrac{1}{3}\], because \[\left( {\left( {\left( {1 \times 3} \right) + 1} \right) = 4} \right)\]

(Where 3 is divisor, 1 is quotient and 1 is remainder)

$\left( {iii} \right) = 2 \times \dfrac{6}{7} = \dfrac{{12}}{7}$ So, this is in lowest form as $\dfrac{{12}}{7}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{12}}{7} = 1\dfrac{5}{7}\], because \[\left( {\left( {\left( {1 \times 7} \right) + 5} \right) = 12} \right)\]

(Where 7 is divisor, 1 is quotient and 5 is remainder)

$\left( {iv} \right) = 5 \times \dfrac{2}{9} = \dfrac{{10}}{9}$ So, this is in lowest form as $\dfrac{{10}}{9}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{10}}{9} = 1\dfrac{1}{9}\], because \[\left( {\left( {\left( {1 \times 9} \right) + 1} \right) = 10} \right)\]

(Where 9 is divisor, 1 is quotient and 1 is remainder)

$\left( v \right) = \dfrac{5}{2} \times 6 = 15$

Now, we have to convert 15 into fraction so multiply by any number in numerator and denominator but we have to write the lowest form of fraction so we multiply 1 in numerator and denominator.

$ \Rightarrow 15 = \dfrac{{15 \times 1}}{1} = \dfrac{{15}}{1}$ So, this is written in lowest form and has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{15}}{1} = 15\dfrac{0}{1}\], because \[\left( {\left( {\left( {15 \times 1} \right) + 0} \right) = 15} \right)\]

(Where 1 is divisor, 15 is quotient and 0 is remainder)

$\left( {vi} \right) = 11 \times \dfrac{4}{7} = \dfrac{{44}}{7}$ So, this is in lowest form as $\dfrac{{44}}{7}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{44}}{7} = 6\dfrac{2}{7}\], because \[\left( {\left( {\left( {7 \times 6} \right) + 2} \right) = 44} \right)\]

(Where 7 is divisor, 6 is quotient and 2 is remainder)

$\left( {vii} \right) = 20 \times \dfrac{4}{5} = \dfrac{{80}}{5} = 16$

Now, we have to convert 16 into fraction so multiply by any number in numerator and denominator but we have to write the lowest form of fraction so we multiply 1 in numerator and denominator.

$ \Rightarrow 16 = \dfrac{{16 \times 1}}{1} = \dfrac{{16}}{1}$ So, this is written in lowest form and has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{{16}}{1} = 16\dfrac{0}{1}\], because \[\left( {\left( {\left( {16 \times 1} \right) + 0} \right) = 16} \right)\]

(Where 1 is divisor, 16 is quotient and 0 is remainder)

$\left( {viii} \right) = \dfrac{2}{3} \times 4 = \dfrac{8}{3}$ So, this is in lowest form as $\dfrac{8}{3}$ has no common factors except 1.

And this is an improper fraction according to the above condition.

Now, convert it into mixed fraction \[ \Rightarrow \dfrac{8}{3} = 2\dfrac{2}{3}\], because \[\left( {\left( {\left( {2 \times 3} \right) + 2} \right) = 8} \right)\]

(Where 3 is divisor, 2 is quotient and 2 is remainder)

So, these are the required answers.

Note- In such types of questions the key concept is that while converting the fraction into lowest form fraction i.e. in $\dfrac{p}{q}$ form where $q \ne 0$, then p and q must have no common factors except 1. So, if a fraction has some common factors simply cancel out, it will be converted into the lowest form fraction, then apply a mixed fraction formula to convert the fraction into mixed fraction, we will get the required answer.

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