2008 AMC 10B Problem 6

Problem 6 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 get

5\,BD=AD,

so

BD=\tfrac15 AD.

Since

AC=9\,CD

and

AC+CD=AD,

we get

10\,CD=AD,

so

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

Then

BC=BD-CD

=\tfrac15 AD-\tfrac{1}{10}AD

=\tfrac{1}{10}AD.

Thus, the correct answer is C.

More practice

Concepts: ratio and proportion · fraction

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