2008 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra

Points

B

and

C

lie on

\overline{AD}.

The length of

\overline{AB}

is

4

times the length of

\overline{BD},

and the length of

\overline{AC}

is

9

times the length of

\overline{CD}.

The length of

\overline{BC}

is what fraction of the length of

\overline{AD}?

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Solution

Since

AB = 4\,BD

and

AB + BD = AD,

we have

5\,BD = AD,

so

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.

More practice

Concepts: ratio and proportion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.