Question

If $b\ne d,$ the fractions $\dfrac{ax+b}{cx+d}$ and $\dfrac bd$ are unequal if:

$a=c=1$ and $x\ne0$

$a=b=0$

$a=c=0$

$x=0$

$ad=bc$

Answer: A

Concepts:rational expressions cross multiplication inequalities

Difficulty rating: 1590

Small Hint:

Cross-multiply the equality $\frac{ax+b}{cx+d}=\frac{b}{d}$

Big Hint:

After cancellation, equality is governed by $x(ad-bc)=0$

Solution:

Where the fractions are defined, equality would require $d(ax+b)=b(cx+d),$ which simplifies to $x(ad-bc)=0.$ Under choice A, $a=c=1$ gives $ad-bc=d-b\ne0,$ and $x\ne0,$ so equality is impossible. Each of B through E instead forces equality directly.

Thus, the fractions are unequal under choice A.

1955 AMC 12 Problem 40

Answer: A

Difficulty rating: 1590

Small Hint:

Big Hint:

Solution:

Problem 40 in Other Years