# Problem of the Week Problem A and Solution Archaeological Ages

## Problem

You are an archaeologist, and have made a startling series of discoveries right around the corner from your office.

You have taken the pieces back to the lab, and are trying to figure out how old your objects are. Your rival, Indianapolis Jane, has tampered with your artifact dating machine. The ages it is computing are correct but are hard to compare. Here are the results from the machine:

Artifact #1 is 50 decades, 100 weeks, and 48 hours old

Artifact #2 is 3 centuries, 4 decades, and 6 years old

Artifact #3 is 2 centuries, 28 decades, and 52 weeks old

You want to present these artifacts to your colleagues in order from oldest to newest. In which order should you present them? Justify your answer.

Note: A decade is equal to 10 years, and a century is equal to 100 years.

## Solution

One way to solve this problem is to estimate the age of each artifact. To compare the ages, we need a common unit. We know that 1 decade is equal to 10 years, and that 1 century is equal to 100 years. We know that each year has approximately 52 weeks and each day has 24 hours. So we can approximate the age of each artifact in years.

• Approximate age of Artifact #1:

We see that 50 decades is equal to $$50 \times 10 = 500$$ years, and 100 weeks is approximately 2 years, and 48 hours is equal to 2 days which is insignificant in the calculation of the number of years in the age. We can estimate the age of Artifact #1 at approximately $$500 + 2 = 502$$ years old.

• Approximate age of Artifact #2:

Since 3 centuries is equal to $$3 \times 100 = 300$$ years, and 4 decades is equal to $$4 \times 10 = 40$$ years, we can estimate the age of Artifact #2 at approximately $$300 + 40 + 6 = 346$$ years old.

• Approximate age of Artifact #3:

Since 2 centuries is equal to $$2 \times 100 = 200$$ years, and 28 decades is equal to $$28 \times 10 = 280$$ years, and 52 weeks is approximately 1 year, we can estimate the age of Artifact #3 at approximately $$200 + 280 + 1 = 481$$ years old.

So the oldest item is Artifact #1, the next oldest is Artifact #3, and the newest is Artifact #2.