1955 AMC 12 Problem 50

Question

In order to pass

B

going

40

mph on a two-lane highway

A,

going

50

mph, must gain

30

feet. Meantime,

C,

210

feet from

A,

is headed toward him at

50

mph. If

B

and

C

maintain their speeds, then, in order to pass safely,

A

must increase his speed by:

30

mph

10

mph

5

mph

15

mph

3

mph

Answer: C

Concepts:relative speed rate-time-distance passing

Difficulty rating: 1870

Small hint:

Let

v

be

A

’s new speed and equate the passing time to the time before

A

and

C

meet

Big hint:

Use relative distances and speeds:

\frac{30}{v-40}=\frac{210}{v+50}

Solution:

Let

v

be

A

’s new speed. Relative to

B,

A

gains at

v-40

mph, so the passing time is proportional to

\frac{30}{v-40}.

A

and

C

close their

210

-foot separation at

v+50

mph, so their meeting time is proportional to

\frac{210}{v+50}.

At the limiting safe time,

\frac{30}{v-40}=\frac{210}{v+50}.

Thus

30v+1500=210v-8400,

giving

v=55

mph.

A

must increase his original speed by

5