Question

How do you simplify $2\dfrac{1}{2} \div 3\dfrac{1}{5}?$

Answer 1

First convert the mixed numbers into fractional form to easily perform the operation between them. To change a mixed number $a\dfrac{b}{c}$ into fractional form, use the following:

Formula Used
$a\dfrac{b}{c} = \dfrac{{a \times c + b}}{c}$
After converting the numbers into fractional form, invert the number which is after the division sign or invert the second number, inverting will cause the division sign to be replaced by multiplication sign. Then simply perform the multiplication.

Complete step by step solution: The given numbers in the question are in mixed fraction form and in order to perform algebraic operation between them we have to convert them into fractional form So converting the mixed fraction into fraction by following formula
$a\dfrac{b}{c} = \dfrac{{a \times c + b}}{c}$
Converting the first number which is $2\dfrac{1}{2}$
$2\dfrac{1}{2} = \dfrac{{2 \times 2 + 1}}{2} = \dfrac{{4 + 1}}{2} = \dfrac{5}{2}$
Now converting the second number which is $3\dfrac{1}{5}$
$3\dfrac{1}{5} = \dfrac{{3 \times 5 + 1}}{5} = \dfrac{{15 + 1}}{5} = \dfrac{{16}}{5}$
We have got the fractional form of the given mixed numbers, so now we can write them as
$= 2\dfrac{1}{2} \div 3\dfrac{1}{5} \\\ = \dfrac{5}{2} \div \dfrac{{16}}{5} \\\$
If we invert the second number that is the number after the division sign, then we will get the division sign being replaced by a multiplication sign, this is because division and multiplication operations are inverse function of each other.
If a fraction is $\dfrac{a}{b}$ then its inverted form can be written as $\dfrac{b}{a}$
Inverting the second number and replacing the division sign with multiplication sign,
$= \dfrac{5}{2} \div \dfrac{{16}}{5} \\\ = \dfrac{5}{2} \times \dfrac{5}{{16}} \\\ = \dfrac{{25}}{{32}} \\\$
Therefore we got the required answer $= \dfrac{{25}}{{32}}$

Note: We have seen mixed fraction to fraction conversion but if we want to convert a fraction $\dfrac{x}{y}$ into mixed fraction then we have to divide the numerator with denominator i.e. $x \div y$, let us consider we get $p\;{\text{and}}\;q$ as quotient and remainder respectively. Then mixed fraction will be given as $p\dfrac{q}{y}$