Question

How do you simplify $ \dfrac{3}{4} \times \dfrac{5}{{12}} $ ?

Answer 1

Hint: Here, we have to multiply the two given fractions to simplify the expression. In order to multiply two fractions, we first multiply the top numbers that are numerators. Then, we multiply the bottom numbers that are the denominators. Lastly, we simplify the fraction by cancelling the common factors (if any).

Complete step-by-step answer:
(i)
In the question, as we have to multiply the two fractions, we should know that,
Product of fraction $ = $ Product of numerator \[ \div \] Product of denominator
i.e.,
  $ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{{a \times c}}{{b \times d}} $
Now, we are given $ \dfrac{3}{4} \times \dfrac{5}{{12}} $ to multiply. So, to multiply the fraction we need to multiply numerator with numerator and denominator with denominator. Therefore,
  $ \dfrac{3}{4} \times \dfrac{5}{{12}} = \dfrac{{3 \times 5}}{{4 \times 12}} $
(ii)
Now we have to look for a common factor in the numerator and the denominator. So, for that we will write all the numbers as a multiple of prime numbers. For example, $ 12 = 2 \times 2 \times 3 $ . Therefore, we will get:
  $ \dfrac{3}{4} \times \dfrac{5}{{12}} = \dfrac{{3 \times 5}}{{2 \times 2 \times 2 \times 2 \times 3}} $
(iii)
As we can see, $ 3 $ is a common factor present in both numerator and denominator, we will cancel it out. Therefore, we now have:
  $ \dfrac{3}{4} \times \dfrac{5}{{12}} = \dfrac{5}{{2 \times 2 \times 2 \times 2}} $
(iv)
Now, as we cannot see any common factor in the numerator and denominator, we will multiply the terms present in the numerator with themselves and the terms present in the denominator with themselves. Therefore, we will obtain:
  $ \dfrac{3}{4} \times \dfrac{5}{{12}} = \dfrac{5}{{16}} $ [Since, $ 2 \times 2 \times 2 \times 2 = 16 $ ]
Hence, after simplification of $ \dfrac{3}{4} \times \dfrac{5}{{12}} $ , we obtain $ \dfrac{5}{{16}} $ .
So, the correct answer is “ $ \dfrac{5}{{16}} $ ”.

Note: When we multiply fractions to get the required result, we should always cancel out the common factors from the numerator and the denominator. Otherwise, the result will not be in the simplest form. Also, if we obtain the result as an improper fraction, it is advised to convert the improper fraction into mixed fraction.