Question

The numerical expression \[\dfrac{3}{7} + \dfrac{{\left( { - 7} \right)}}{8} = \dfrac{{25}}{{56}}\] shows that
A.rational numbers are closed under division
B.rational numbers are closed under addition
C.rational numbers are closed under subtraction
D.rational numbers are closed under multiplication

Answer 1

Hint: Here, we will first find the sum of the right hand side of the equation. Then we use the definition of a rational number is a number that can expressed as the quotient or fraction of two integers, that is, \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and \[q\] is not equal to zero.
Apply this property, and then use the given conditions to find the required value.

Complete step-by-step answer:
Given that the numerical expression \[\dfrac{3}{7} + \dfrac{{\left( { - 7} \right)}}{8} = \dfrac{{25}}{{56}}\].
We know that a rational number is a number that can expressed as the quotient or fraction of two integers, that is, \[\dfrac{p}{q}\] where \[p\] and \[q\] are integers and \[q\] is not equal to zero.
We also know that the rational numbers are closed under addition, multiplication and division, but not closed under subtraction.
We will now show that rational numbers are closed under which relation.
First, we will now find the sum of the two rational numbers on the left side of the given equation \[\dfrac{3}{7}\] and \[ - \dfrac{7}{8}\].
\[
  \dfrac{3}{7} + \dfrac{{\left( { - 7} \right)}}{8} = \dfrac{{24 - 49}}{{56}} \\
   = \dfrac{{ - 25}}{{56}} \\
\]
Since both the numbers \[\dfrac{3}{7}\] and \[ - \dfrac{7}{8}\] are rational numbers, the sum of these numbers is also a rational number.
Thus, the sum of any two rational numbers is always a rational number.
Therefore, the numerical expression shows that rational numbers are closed under addition.
Hence, the option B is correct.

Note: In solving these types of questions, you should be familiar with the concept of rational numbers and their closure. Then use the given conditions and values given in the question, and find the sum of given numbers, to find the required value. Also, we are supposed to write the values properly to avoid any miscalculation.