Points B and C lie on AD. The length of AB is 4 times the length of BD, and the length of AC is 9 times the length of CD. The length of BC is what fraction of the length of AD?

2008 AMC 12B Problem 8

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Points

B

and

C

lie on

\overline{AD}.

The length of

\overline{AB}

4

times the length of

\overline{BD},

and the length of

\overline{AC}

9

times the length of

\overline{CD}.

The length of

\overline{BC}

is what fraction of the length of

\overline{AD}?

\dfrac{1}{36}

\dfrac{1}{13}

\dfrac{1}{10}

\dfrac{5}{36}

\dfrac{1}{5}

From

AB = 4\,BD

and

AB + BD = AD,

get

BD = \tfrac{1}{5}AD

Similarly

CD = \tfrac{1}{10}AD,

and

BC = BD - CD

Since

AB = 4\,BD

and

AB + BD = AD,

we have

5\,BD = AD,

BD = \tfrac{1}{5}AD.

Likewise

AC = 9\,CD

with

AC + CD = AD

gives

CD = \tfrac{1}{10}AD.

Because

B

and

C

both measure from

A,

\;BC = BD - CD

= \tfrac{1}{5}AD - \tfrac{1}{10}AD

= \tfrac{1}{10}AD.

Thus, the correct answer is C.