Question 1176019: King Francis decided to divide up his kingdom among his sons. He gave his first son one-eighth of the villages he ruled over and one more village. The king gave his second son one-eighth of the remaining villages and two more villages. He continued like this, until he gave his seventh and youngest son one-eighth of the remaining villages and seven more villages. After that, all the villages in the kingdom had been passed on to his children. How many total villages are there in the kingdom?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
If you try working this problem in "chronological order", starting with the first son, you will end up with a really horrendous equation to solve.
Work the problem backwards, starting with what the 7th son got.
When the 7th son got his share of the villages, he got 1/8 of the villages that were left, plus 7 more villages. So if x is the number of villages left just before he received his share, the number of villages remaining after he received his share was x - ((1/8)x+7) = (7/8)x-7. That number of villages remaining was 0. So
(7/8)x-7 = 0
(7/8)x = 7
x = 8
There were 8 villages left just before the 7th son got his share. After he got his share, there were no villages left, so he got 8 villages.
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When the 6th son got his share of the villages, he got 1/8 of the villages that were left, plus 6 more villages. So if x is the number of villages left just before he received his share, the number of villages remaining after he received his share was x-((1/8)x+6) = (7/8)x-6. That number of villages remaining was 8. So
(7/8)x-6 = 8
(7/8)x = 14
x = 16
There were 16 villages left just before the 6th son got his share. After he got his share, there were 8 villages left, so he got 8 villages.
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Continue working backwards through the problem; you will find that each son got 8 villages, so the total number of villages in the kingdom was 7*8 = 56.
ANSWER: There are 56 villages in the kingdom.
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CHECK:
56 villages; 1st son gets 1/8 of them (7) plus 1 more, a total of 8. 48 villages left.
48 villages; 2nd son gets 1/8 of them (6) plus 2 more, a total of 8. 40 villages left.
40 villages; 3rd son gets 1/8 of them (5) plus 3 more, a total of 8. 32 villages left.
32 villages; 4th son gets 1/8 of them (4) plus 4 more, a total of 8. 24 villages left.
24 villages; 5th son gets 1/8 of them (3) plus 5 more, a total of 8. 16 villages left.
16 villages; 6th son gets 1/8 of them (2) plus 6 more, a total of 8. 8 villages left.
8 villages; 7th son gets 1/8 of them (1) plus 7 more, a total of 8. 0 villages left.
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