Question

Using the appropriate properties of operations of rational numbers, evaluate the following
$\dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5}$.

Answer 1

Hint: Here, we are given an expression of rational numbers and we have to evaluate its value using the appropriate properties. First of all, we will write the terms that are being multiplied in brackets and then take out the common fraction as common. Then we will carry out addition in brackets and then carry out subtraction using LCM on the remaining terms.

Complete step by step solution:
In this question, we are given an expression of rational numbers and we need to solve it using the appropriate properties of rational numbers.
The given expression is: $\dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5}$
Here, we are having 5 terms in the expression.
Here, we need to write these terms in brackets. So, therefore,
$ \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \left( {\dfrac{2}{5} \times \dfrac{{ - 3}}{7}} \right) - \left( {\dfrac{3}{7} \times \dfrac{3}{5}} \right) - \dfrac{1}{{14}}$
We can take out the 2nd and 4th term as common as they are the same. So, taking $\dfrac{{ - 3}}{7}$ as common, we get
$ \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 3}}{7}\left( {\dfrac{2}{5} + \dfrac{3}{5}} \right) - \dfrac{1}{{14}}$
Now, in brackets, both the terms have the same denominator. Hence, we can add them by adding their numerators. Therefore,
$ \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 3}}{7}\left( {\dfrac{{3 + 2}}{5}} \right) - \dfrac{1}{{14}} = \dfrac{{ - 3}}{7}\left( {\dfrac{5}{5}} \right) - \dfrac{1}{{14}}$
Now, 5 divided by 5 is equal to 1. Therefore, the expression becomes
$ \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 3}}{7} - \dfrac{1}{{14}}$
Now, we have fractions with different denominators. So, to solve this we need to take the LCM of the denominators. Therefore,
$
   \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 3\left( 2 \right) - 1}}{{14}} \\
   \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 6 - 1}}{{14}} \\
   \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 7}}{{14}} \\
 $
Now, we can reduce the fraction and get
$ \Rightarrow \dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5} = \dfrac{{ - 1}}{2}$

Hence, the value of $\dfrac{2}{5} \times \dfrac{{ - 3}}{7} - \dfrac{1}{{14}} - \dfrac{3}{7} \times \dfrac{3}{5}$ using properties of operations of rational numbers is $\dfrac{{ - 1}}{2}$.

Note:
$\left( {\dfrac{2}{5} \times \dfrac{{ - 3}}{7}} \right) - \left( {\dfrac{3}{7} \times \dfrac{3}{5}} \right) - \dfrac{1}{{14}}$
 Here, the most important and tricky part in this question is changing the sign when we take out $\dfrac{{ - 3}}{7}$ out as is common. If we take out $\dfrac{{ - 3}}{7}$ as common, then we need to change the – sign between two brackets to + as we are taking that – sign out along with $\dfrac{3}{7}$.