SOLUTION: A man whose end was approaching summoned his sons and said: "Divide my money as I shall prescribe." To his eldest son, he said, "You are to have 1 bezant and a seventh of what is

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Question 1029397: A man whose end was approaching summoned his sons and said: "Divide my
money as I shall prescribe." To his eldest son, he said, "You are to have 1 bezant and a seventh of what is left." To his second son he said, "Take 2 bezants and a seventh of what remains." To the third son, "You are to take 3 bezants and a seventh of what is left." Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left. After following their father's instructions with care, the sons found that they had shared their inheritance equally. How many sons were there, and how large was the estate ?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A man whose end was approaching summoned his sons and said: "Divide my
money as I shall prescribe."
To his eldest son, he said, "You are to have 1 bezant and a seventh of what is left."
To his second son he said, "Take 2 bezants and a seventh of what remains."
To the third son, "You are to take 3 bezants and a seventh of what is left."
Thus he gave each son 1 bezant more than the previous son and a seventh of what remained, and to the last son all that was left.
After following their father's instructions with care, the sons found that they had shared their inheritance equally.
How many sons were there, and how large was the estate ?
;
let a = the original amt of the estate (in Bezants)
:
1st son's share:
1 + 1%2F7(a-1)
then remaineth
a - (1 + 1%2F7(a-1)) = a - %287%2F7+%2B+a%2F7+-+1%2F7%29 = a - %28a%2F7+%2B+6%2F7%29 = %286a%29%2F7+-+6%2F7
2nd son's share;
2 + 1%2F7%28%286a%29%2F7+-+6%2F7+-+2%29 = 2 + 1%2F7%28%286a%29%2F7+-+6%2F7+-+14%2F7%29 = 2 + 1%2F7%28%286a%29%2F7+-+20%2F7%29 = 2 + %28%286a%29%2F49+-+20%2F49%29 =
98%2F49+%2B+%286a%29%2F49+-+20%2F49 = (%286a%29%2F49+%2B+78%2F49); 2nd son's share
:
1st son's share = 2nd son's share
1 + 1%2F7(a-1) = (%286a%29%2F49+%2B+78%2F49)
Multiply both sides by 49
49 + 7(a - 1) = 6a + 78
49 + 7a - 7 = 6a + 78
7a + 42 = 6a + 78
7a - 6a = 78 - 42
a = 36 Bezants was the amt of the Estate
:
Find the number of sons:
1st son: 1 + 1%2F7(36 - 1) = 6 Bezants;
Now we should be able to say their were 36/6 = 6 sons, but lets continue
Then 30 remaineth
2nd son: 2 + 1%2F7(30-2) = 6
Then 24 remaineth
3rd son: 3 + 1%2F7(24-3) = 6
then 18 remaineth
4th son: 4 + 1%2F7(18-4) = 6
then 12 remaineth
5th son: 5 + 1%2F7(12-5) = 6
then 6 remaineth for th 6th son