Question

Without using a calculator, work out $3{\displaystyle \frac{1}{8}}÷{\displaystyle \frac{5}{12}}$.
You must show all your working and give your answer as a mixed number in its simplest form.

Answer 1

Hint: First convert the given mixed fraction into improper fraction by using the formula of conversion: $a{\displaystyle \frac{b}{c}}={\displaystyle \frac{a\times c+b}{c}}$. Now, to divide the obtained improper fraction with ${\displaystyle \frac{5}{12}}$, take the reciprocal of ${\displaystyle \frac{5}{12}}$ and multiply it with the previously obtained improper fraction. Now, to convert the obtained product into mixed fraction again, divide the numerator by the denominator. Write the remainder on the top and the quotient on the left hand side of the fraction while leaving the denominator as it is.

Complete step-by-step solution -
Let us come to the question. We have asked to calculate the value of $3{\displaystyle \frac{1}{8}}÷{\displaystyle \frac{5}{12}}$, without using a calculator.
Now, changing the mixed fraction given by $3{\displaystyle \frac{1}{8}}$ into improper fraction, we get,
$3{\displaystyle \frac{1}{8}}={\displaystyle \frac{3\times 8+1}{8}}={\displaystyle \frac{25}{8}}$.
Now, the expression becomes, ${\displaystyle \frac{25}{8}}÷{\displaystyle \frac{5}{12}}$. To divide a given number by another number means, we have to multiply the first number with the reciprocal of the second number.
Reciprocal of a number is obtained by simply inter-changing its numerator and denominator.
Therefore, the reciprocal of ${\displaystyle \frac{5}{12}}$ is ${\displaystyle \frac{12}{5}}$.
Therefore, multiplying the two fractions, we get,
${\displaystyle \frac{25}{8}}\times {\displaystyle \frac{12}{5}}$
Cancelling the common terms, we get, $\text{expression}={\displaystyle \frac{5\times 3}{2}}={\displaystyle \frac{15}{2}}$.
Now, dividing 15 by 2, we have, 1 as remainder and 7 as quotient, so write 1 as the numerator and 7 on the left side of the fraction. Therefore, changing ${\displaystyle \frac{15}{2}}$ into a mixed fraction, we get, ${\displaystyle \frac{15}{2}}=7{\displaystyle \frac{1}{2}}$.

Note: You have to be careful while changing an improper fraction into a mixed fraction. Always follow the rule to write a mixed number. Now, one important thing is that only an improper fraction can be changed into a mixed fraction. Improper fraction is a type of fraction that has its numerator greater than denominator.