Problem:
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 ?
Answer Choices:
A. 361​
B. 131​
C. 101​
D. 365​
E. 51​
Solution:
Because AB+BD=AD and AB=4BD, it follows that BD=51​⋅AD. By similar reasoning, CD=101​⋅AD. Thus
BC=BD−CD=51​⋅AD−101​⋅AD=101​⋅AD​
The problems on this page are the property of the MAA's American Mathematics Competitions