SOLUTION: Ellen and Angela went on a dinosaur dig in Mongolia. Each day the dug for fossils, and at the end of the day they checked in with David, the chief paleontologist. He helped them se

Question 876961: Ellen and Angela went on a dinosaur dig in Mongolia. Each day the dug for fossils, and at the end of the day they checked in with David, the chief paleontologist. He helped them separate the fossils, those that went to the museum and those they could keep. At the end of the first day 3/5 of the total went to the museum. On the second day, they found 8 new fossils and then 5/8 of the total went to the museum. On the third day, they found 26 new fossils and the museum kept 2/7 of the total. On the fourth day, they added 29 fossils and gave 5/9 to the museum. On the fifth day, they found 6 more fossils and the museum kept 5/6 of the total. They found an additional 10 fossils on the sixth day and the museum kept 2/3 of their collection. At the end of their expedition they had 5 fossils to take home. How many total fossils did they donate to the museum?
* I don't know how to approach this problem... Any help would be greatly appreciated. Thanks in advance.*
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Maybe the people at the artofproblemsolving website think this is an easy question.
For me, it is hard enough just to try to interpret the problem.
Here is my interpretation in table form:

Ellen and Angela found fossils the first day.
At the end of the day they checked in with David, the chief paleontologist.
He helped them separate the fossils and they ended having to give 3/5 of the total they had to the museum.
They could keep the remaining of the 40 fossils,
and that was .

On the second day, Ellen and Angela found fossils.
Added to the fossils thy had kept, they had fossils.
when they took those fossils to David,
he selected of them to give to the museum.
That meant that Ellen and Angela could keep the other of the fossils.
So they kept fossils.

On the third day, Ellen and Angela found fossils.
Added to the fossils they had kept the night before, they had a total of fossils.
When they took those fossils to David,
he selected of them to give to the museum.
That meant that Ellen and Angela could keep the other of the fossils.
So they kept fossils.

On the fourth day, Ellen and Angela found fossils.
Added to the fossils they had kept the night before, they had a total of fossils.
When they took those fossils to David,
he selected of them to give to the museum.
That meant that Ellen and Angela could keep the other of the fossils.
So they kept fossils.

On the fifth day, Ellen and Angela found fossils.
Added to the fossils they had kept the night before, they had a total of fossils.
When they took those fossils to David,
he selected of them to give to the museum.
That meant that Ellen and Angela could keep the other of the fossils.
So they kept fossils.

On the sixth day, Ellen and Angela found fossils.
Added to the fossils they had kept the night before, they had a total of fossils.
When they took those fossils to David,
he selected of them to give to the museum.
That meant that Ellen and Angela could keep the other of the fossils.
So they kept fossils.

The problem gives you some of the numbers for that table.
You start with

I see three options to build that table:

1) The table can be constructed from the bottom up.
Knowing that they ended up with fossils,
and that those fossils were of what Ellen and Angela had took to David on the 6th evening,
you figure that on the evening of the sixth day they had taken fossils to David.
Since they had collected fossils on the sixth day,
that means that they had started the sixth day with fossils that David had allowed them to keep on the night of the fifth day.
Those fossils that David had allowed them to keep on the night of the fifth day were of what they had shown David on the 5th evening.
That means that on the 5th evening, they had brought fossils to David.

You can keep back-calculating that way to fill the entire table.

2) Since I get easily mixed up, I chose to start filling the table from the top, by smart trial and error.
I used a spreadsheet program on the computer to make it easier for me.
I realized that to get integers on the first row, the first had to be a multiple of 5.
Using for the first , I would get integers on the twotop rows,
but I do not get integers for all spots on the other rows.
Using for the first , I get integers on the top row,
but the second row ends up in .
Using for the first , the second row ends up in .
Using for the first , the second row ends up in ,
and using for the first , the second row ends up in .
Unfortunately, I do not get integers for all spots on the other rows.
However, I see a pattern:
, , , , etc
will results in integers all across the first and second rows.
I tried for the first , and I did get integers for all spots on all rows.
Better yet, the last number on the table was so I knew I had a result.

3) The table can be constructed from the top down, using a little algebra and a lot of arithmetic:

I could try to calculate

Then , and is of that, so

At best I could say that
needs to be an integer, and that the smallest natural number that would work is .
I know I would make mistakes going on with calculations like that.
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