A mine is for sale for $240,000. It is believed the mine will produce a profit of $65,000 the first year, but the profit will decline $5,000 a year after that, eventually reaching zero, whereupon the mine will be worthless. What rate of return would this $240,000 investment produce for the purchaser of the mine

60.4%

Explanation:

Initial cost = $240,000

profit of first year = $65,000

this is reduced subsequently until it reaches zero

Note that this value reduces in an arithmetic progression from $65,000 , $60,000, ... , 0

the first term A1 = 65,000

the common difference d is 60,000 - 65,000 = -5000

the last term is An = 0

we calculate for number of terms

An = A1 +  (n - 1)d

0 = 65,000 + (n - 1)(-5000)

0 = 65,000 - 5000n +5000

5000n = 70,000

n = 14

using the equation for summation of terms in an arithmetic progression Sn, we solve as

Sn = [2A1 + (n - 1)d]

Sn = [2(60,000) + (14 - 1)(-5000)]

Sn = 7[120,000 - 65,000]

Sn = 7 x 55,000

Sn = $385,000.  This is the total profit on the mine

rate of return = (385,000 - 240,000)/240,000 = 145,000/240,000 = 0.604

i.e 60.4%

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