Question 39835
I'm not told how many fossils they collect at the end of the first day.
All I'm told is that 2/5 of them go to the museum.
At he end of 6 days, I'm told they get to keep 10 fossils, so I
will try to count how many they have kept at the end of each day,
and keep a running total that will equal 10 when I'm done.

Let x equal the number of fossils they collect at the end of the 1st day
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They keep
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1) (3/5)x  end of 1st day (2/5 went to museum)
2) ((3/5)x + 6)5/8  end of 2nd day (added 6 more and 3/8 went to museum)
3) (((3/5)x + 6)5/8 + 13)5/7 end of 3rd day (added 13, 2/7 to museum)
4) ((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 end of 4th day (added 7, 5/9 to museum)
5) (((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3)1/6 5th day (add 3, 5/6 to museum)
6) ((((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3)1/6 + 12)2/3 (add 12, 1/3 museum)
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This last expression is the number of fossils they keep after 6 days
and the problems states that it is 10.
((((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3)1/6 + 12)2/3 = 10
multiply each side by 3/2
(((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3)1/6 + 12 = 15
subtract 12 from both sides
(((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3)1/6 = 3
multiply both sides by 6
((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 + 3 = 18
subtract 3 from both sides
((((3/5)x + 6)5/8 + 13)5/7 + 7)5/9 = 15
multiply both sides by 9/5
(((3/5)x + 6)5/8 + 13)5/7 + 7 = 27
subtract 7 from both sides
(((3/5)x + 6)5/8 + 13)5/7 = 20
multiply both sides by 7/5
((3/5)x + 6)5/8 + 13 = 28
subtract 13 from both sides
((3/5)x + 6)5/8 = 15
multiply both sides by 8/5
(3/5)x + 6 = 24
subtract 6 from both sides
(3/5)x = 18
multiply both sides by 5/3
x = 30
x was the amount of fossils collected on the 1st day
going back to 1)
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they keep-----------------------to museum
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1) 3/5(30) = 18 ----------------2/5(30) = 12
2) 5/8(18 +6) = 15 -------------3/8 (18 + 6) = 9
3) 5/7(15 + 13) = 20 -----------2/7(15 + 13) = 8
4) 5/9(20 + 7) = 15-------------4/9(20 + 7) = 12
5) 1/6(15 + 3) = 3--------------5/6(15 + 3) = 15
6) 2/3(12 + 3) = 10-------------1/3(12 + 3) = 5
They have 10 fossils after day 6 as they should.
To find how many fossils the museum has, add up the
"to museum" column
12 + 9 + 8 + 12 + 15 + 5 = 61 answer