Question

On the set of positive rational, a binary operation is defined by
$a.b = \dfrac{{2ab}}{5}$
If $2.x = {3^{ - 1}}$ then $x$=

Answer 1

Hint: Here in this question only binary operation and inverse binary operation is needed. Start solving the question with the first equation given. Use the binary multiplication method.

Complete step-by-step answer:
Given in the question
$a.b = \dfrac{{2ab}}{5}$
Here if we put $b = e$,
according to the binary operation
$a.e = a$
Here e is the identify elements in the binary multiplication process
Put the values of a. b
$\dfrac{{2ae}}{5} = a$
We get
$e = \dfrac{{5a}}{{2a}}$
Cancel the a in the numerator and denominator
$e = \dfrac{5}{2}$
 We know that the inverse binary operation
$a.{e^{ - 1}} = e$
$ = \dfrac{{2a{a^{ - 1}}}}{5} = \dfrac{5}{2}$
We have given in the question that
$2.x = {3^{ - 1}}$
Put the value of x
$\dfrac{{2(2x)}}{5} = \dfrac{{25}}{{4(3)}}$
Do the multiplication
$\dfrac{{4x}}{5} = \dfrac{{25}}{{12}}$
For solving the above equation do the cross multiplication, we get
$x = \dfrac{{125}}{{48}}$
Hence, we have the value of x

Note: Here in type of question students mostly get confused between the inverse function. Use the binary operation to solve the question. Always remember that the sum and multiplication of two real numbers is always a real number. There are total four number of binary operations which is
A. Binary addition
B. Binary subtraction
C. Binary division
D. Binary multiplication
Cancel the denominator by numerator whenever possible. Separate and compare the variable and constant term with the same variable and constant term. Always solve the question with the help of a cross multiplication method whenever there are two equations combined with a sign of equal to.as the question mentions there is a set of positive rational numbers, so the answers should be in the positive.