Question

Multiply and reduce to lowest form and convert into a mixed fraction.
$
  \left( i \right){\text{ 7}} \times \dfrac{3}{5}{\text{ }}\left( {ii} \right){\text{ 4}} \times \dfrac{1}{3}{\text{ }} \\
  \left( {iii} \right){\text{ 2}} \times \dfrac{6}{7}{\text{ }}\left( {iv} \right){\text{ 5}} \times \dfrac{2}{9}{\text{ }} \\
  \left( v \right){\text{ }}\dfrac{5}{2} \times 6{\text{ }}\left( {vi} \right){\text{ }}11 \times \dfrac{4}{7}{\text{ }} \\
  \left( {vii} \right){\text{ 20}} \times \dfrac{4}{5}{\text{ }}\left( {viii} \right){\text{ }}\dfrac{2}{3} \times 4{\text{ }} \\
$

Answer 1

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.