My Favorite Math Problem
An Unusual Inheritance
A man dies and leaves his estate to his sons.
The estate is divided as follows:
The 1st son gets $100,000 from the estate + 10 % of what is left of that estate.
After the first son gets his share,
The 2nd son gets $200,000 dollars from what is remaining of the estate + 10 % of what is left after that.
So,
The nth son gets $100,000 × n dollars + 10% of what is left of the estate
If each son receives same amount, answer the following questions:
1) How much did each son receive?
2) What was the total amount of the estate?
3) How many sons were there?
Hints
1) We do not know the amount of the estate. So we have to find the 10% of the remainder of the estate in terms of a variable.
2) We need to know what the first son inherited in order to find what is left of the estate for the second son to inherit from, so think of how we would represent this in the formula for the second son’s inheritance.
3) You only need to find the formula for 2 sons because every son receives the same amount.
Solution
Let be each son. Let be the value of the entire estate.
The first son receives $100,000 + of the remainder of the estate.
So he receives:
S1 = 100,000 + 0.1(E - 100,000)
The second son receives $200,000 dollars + of what is left of the estate after the first son.
So he receives:
S2 = 200,000 + 0.1(E - S1 - 200,000)
Since all the sons receive the same amount, we can set these two equal to each other.
100,000 + 0.1(E - 100,000) = 200,000 + 0.1(E - S1 - 200,000)
1,000,000 + E - 100,000 = 2,000,000 + E - S1 - 200,000
900,000 + E = 1,800,000 + E - S1
900,000 = 1,800,000 - S1
S1 = 900,000
So each son receives $900,000
Next, we can plug 900,000 into our first equation and solve for E to find the total value of the estate.
S1 = 100,000 + 0.1(E - 100,000)
900,000 = 100,000 + 0.1(E - 100,000)
800,000 = 0.1(E - 100,000)
8,000,000 = E - 100,000
E = 8,100,000
Last, because every son receives the same amount, we can divide the estate total by the amount each son receives to find the total number of sons:
$8,100,000/$900,000 = 9 Sons
A man dies and leaves his estate to his sons.
The estate is divided as follows:
The 1st son gets $100,000 from the estate + 10 % of what is left of that estate.
After the first son gets his share,
The 2nd son gets $200,000 dollars from what is remaining of the estate + 10 % of what is left after that.
So,
The nth son gets $100,000 × n dollars + 10% of what is left of the estate
If each son receives same amount, answer the following questions:
1) How much did each son receive?
2) What was the total amount of the estate?
3) How many sons were there?
Hints
1) We do not know the amount of the estate. So we have to find the 10% of the remainder of the estate in terms of a variable.
2) We need to know what the first son inherited in order to find what is left of the estate for the second son to inherit from, so think of how we would represent this in the formula for the second son’s inheritance.
3) You only need to find the formula for 2 sons because every son receives the same amount.
Solution
Let be each son. Let be the value of the entire estate.
The first son receives $100,000 + of the remainder of the estate.
So he receives:
S1 = 100,000 + 0.1(E - 100,000)
The second son receives $200,000 dollars + of what is left of the estate after the first son.
So he receives:
S2 = 200,000 + 0.1(E - S1 - 200,000)
Since all the sons receive the same amount, we can set these two equal to each other.
100,000 + 0.1(E - 100,000) = 200,000 + 0.1(E - S1 - 200,000)
1,000,000 + E - 100,000 = 2,000,000 + E - S1 - 200,000
900,000 + E = 1,800,000 + E - S1
900,000 = 1,800,000 - S1
S1 = 900,000
So each son receives $900,000
Next, we can plug 900,000 into our first equation and solve for E to find the total value of the estate.
S1 = 100,000 + 0.1(E - 100,000)
900,000 = 100,000 + 0.1(E - 100,000)
800,000 = 0.1(E - 100,000)
8,000,000 = E - 100,000
E = 8,100,000
Last, because every son receives the same amount, we can divide the estate total by the amount each son receives to find the total number of sons:
$8,100,000/$900,000 = 9 Sons