Question

if a/b and c/d are rational numbers then what is a×b/b×d​

Answer 1

ANSWER

If a=c, then a+

b

=c+

d

⇒

b

=

d

⇒b=d

So, let a



=c. Then, there exists a positive rational number x such that a=c+x.

Now,

⇒a+

b

=c+

d

⇒c+x+

b

=c+

d

[∵a=c+x]

⇒x+

b

=

d

⇒(x+

b

)

2

=(

d

)

2

⇒x

2

+2

b

x+b=d

⇒

b

=

2x

d−x

2

−b

⇒

b

is rational [∴d,x,b are rationals∴

2x

d−x 2 −b2

is rational]

⇒ b is the square of a rational number.

From(i), we have

d

=x+

b

⇒

d

is rational

⇒ d is the square of a rational number.

Hence, either a=c and b=d or b and d are the squares of rationals.