1955 AMC 12 Problem 40

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

Cross-multiply the equality

\frac{ax+b}{cx+d}=\frac{b}{d}

After cancellation, equality is governed by

x(ad-bc)=0

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.

Problem 40 in other years