Question 1198416: An inheritance is split among 5 brothers. The first receives half of the inheritance plus $1. The second receives half of the remainder plus $2. The third receives half of the remainder plus $3. The fourth receives half of the remainder plus $4. The last brother receives $500. What is the total amount of the inheritance?
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website! .
An inheritance is split among 5 brothers.
The first receives half of the inheritance plus $1.
The second receives half of the remainder plus $2.
The third receives half of the remainder plus $3.
The fourth receives half of the remainder plus $4.
The last brother receives  $500.
What is the total amount of the inheritance?
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Let the brothers be B1, B2, B3, B4, and B5.
Let b[0] be the entire inheritance.
Brother B1 got half of inheritance + $1, or b[0]/2 + 1 dollars. The remainder is r[1] = b[0]/2 - 1. (1)
Brother B2 got half of the remainder r[1] plus $2, or r[1]/2 + 2. The remainder is r[2] = r[1]/2 - 2. (2)
Brother B3 got half of the remainder r[2] plus $3, or r[2]/2 + 3. The remainder is r[3] = r[2]/2 - 3. (3)
Brother B4 got half of the remainder r[3] plus $4, or r[3]/2 + 4. The remainder is r[4] = r[3]/2 - 4. (4)
r[4] is what brother B5 got: r[4] = 500 = r[3]/2 - 4.
From it, we have r[3]/2 = 500 + 4 = 504, which gives r[3] = 2*504 = 1008.
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| Now I will solve equations (3), (2), (1) in this backward order. |
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From (3), we have r[2]/2 - 3 = r[3] = 1008, or r[2]/2 = 1008+3 = 1011, which gives r[2] = 2*1011 = 2022.
From (2), we have r[1]/2 - 2 = r[2] = 2022, or r[1]/2 = 2022+2 = 2024, which gives r[1] = 2*2024 = 4048.
From (1), we have b[0]/2 - 1 = r[1] = 4048, or b[0]/2 = 4048+1 = 4049, which gives b[0] = 2*4049 = 8098.
ANSWER. The whole inheritance was 8098 dollars.
Solved.
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The method which I used is called a "backward solution method".
On the way forward I establish the equations.
On the way back I solve these equations, one after another.
Nice method, isn't it ?
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
An inheritance is split among 5 brothers. The first receives half of the inheritance plus $1. The second receives half of the remainder plus $2. The third receives half of the remainder plus $3. The fourth receives half of the remainder plus $4. The last brother receives $500. What is the total amount of the inheritance?
Let inheritance be i
After 1st gets ½ inheritance, plus $1, remaining amount is: 
After 2nd gets ½ remainder, plus $2, remaining amount is: 
After 3rd gets ½ remainder, plus $3, remaining amount is: 
After 4th gets ½ remainder, plus $4, remaining amount is: 
Although NOT stated, the 5th rec’d the remainder after the 4th got his.
So, as the 5th rec’d $500, we have: 
I - 98 = 16(500) ----- Multiplying by LCD, 16
Amount of inheritance, or 
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