Question

A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the other rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

$171$

$173$

$182$

$188$

$210$

Solution:

The top ring contributes its full outside diameter, $20$ cm. Because the rings are $1$ cm thick, each ring hangs $2$ cm below the top of the ring above it, so each lower ring adds its outside diameter minus $2.$

The outside diameters run $20, 19, \ldots, 3,$ so the added distances are $17, 16, \ldots, 1.$ The total is $20 + (17 + 16 + \cdots + 1) = 20 + \frac{17 \cdot 18}{2} = 20 + 153 = 173.$

Thus, the correct answer is B.

2006 AMC 10A Problem 14

Solution:

Problem 14 in Other Years