Equating to each other?

Algebra Level 2

$\text{If }\ \dfrac AB = \dfrac CD\ \text{ and } \ A=C, \ \text{ then } \ B=D\, ?$

As shown above left, I have written up 2 fractions that are equal to each other.

If both the numerators are equal to each other, does this imply that both the denominators must be equal to each other as well?

Yes No

2 solutions

Achal Jain
Mar 13, 2017

$\large \dfrac{0}{5}=0$ and $\large \dfrac{0}{6}=0$ but $\large 5 \neq 6$

Chung Kevin
Mar 22, 2017

Multiplying throughout by $BD \neq 0$ , we obtain that $AD = BC$ .
Since $A = C$ , we have $AD = BA$ , so $A ( B - D ) = 0$ .

Hence, we can conclude that either $A = 0$ or $B - D = 0$ .
Thus, it need not always be true that $B = D$ , since we could have $A = 0$ .

For example, we have $\frac{0}{1} = \frac{0}{2}$ which satisfies $A = C ( = 0 )$ , but we have $B \neq D$ .