2008 AMC 10B Problem 6

Attempt Problem 6 of the 2008 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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6.

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}?

\dfrac{1}{36}

\dfrac{1}{13}

\dfrac{1}{10}

\dfrac{5}{36}

\dfrac{1}{5}

Answer: C

Concepts:ratio and proportion fraction

Difficulty rating: 960

Small hint:

From

AB=4\,BD

and

AB+BD=AD,

find

BD

as a fraction of

AD

Big hint:

Similarly

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

and

BC=BD-CD

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.