Question

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

Difficulty rating: 1350

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.

2008 AMC 12B Problem 8

Answer: C

Difficulty rating: 1350

Solution:

Problem 8 in Other Years