Question

Simplify $\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right)$.

Answer 1

Hint:We know that for a fraction has two parts a numerator and denominator. Also for multiplying fractions we have to multiply the numerator and denominator separately, and then represent the final answer as such as we get.Using the above property we can simplify the given question.

Complete step by step answer:
Given, $\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right)........................\left( i \right)$
Now we have to simplify the expression given in (i);
For that we have to consider the properties of fractions. We know that a fraction has two parts: a numerator and a denominator. Here we can see that the mathematical operation, multiplication is applied between the given terms.Also using the property of multiplication of fractions we know that during multiplication of two fractions we have to multiply the numerator and denominator separately and then we have to write the answer we get as such as.So multiplying the numerators and denominators separately in (i):
$\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right) = \dfrac{{1 \times 2 \times 3}}{{2 \times 3 \times 4}}................\left( {ii} \right)$
Solving (ii) we get:
$\dfrac{{1 \times 2 \times 3}}{{2 \times 3 \times 4}} = \dfrac{6}{{24}}...................\left( {iii} \right)$
No on observing (iii) we can see that we can simplify it.
Such that on simplifying (iii) we can write:
$
\dfrac{6}{{24}} = \dfrac{{1 \times 6}}{{4 \times 6}} \\
\Rightarrow \dfrac{{1 \times 6}}{{4 \times 6}} = \dfrac{1}{4}..................\left( {iv} \right) \\ $
Therefore on simplifying $\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right)$ we get: $\dfrac{1}{4}$.

Note:Alternative method: We know the property of multiplication of fractions we know that during multiplication of two fractions we have to multiply the numerator and denominator separately and then we have to write the answer we get as such as.
So on multiplying we get:
$\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right) = \dfrac{{1 \times 2 \times 3}}{{2 \times 3 \times 4}}................\left( v \right)$
On observing (v) we can say that the resultant fraction has a common factor of 2 and 3 in the numerator and denominator.Such that we can cancel the common factors from both the denominator and the numerator, and we get:
$\dfrac{{1 \times 2 \times 3}}{{2 \times 3 \times 4}} = \dfrac{1}{4}$
Therefore on simplifying $\left( {\dfrac{1}{2}} \right) \times \left( {\dfrac{2}{3}} \right) \times \left( {\dfrac{3}{4}} \right)$we get:$\dfrac{1}{4}$
On solving similar questions one should know the basic properties in general which can be applied to simplify a given expression easily. Also if in a fraction there exists a common factor for both the numerator and the denominator we can then cancel it and thus simplify it, and care must be taken while undertaking the mathematical process.