Question

Andy looks at his account and notices that if the current monthly interest rate stays constant he is expected to have $54,000 in 6 years (i.e. once 6 years have elapsed) and $67,000 in 8 years. a. How much money does he have now (at time 0)? b. If his predictions are correct, except after 7 years, the nominal rate halves and then stays at that value, how much money will he have in 8 years? Assume the interest rate is compounded monthly.

Answer 1

initial principal amount is $ 28272.36

after 7 year interest rate = 0.4514 %

amount after 8 year is = $63489.85

Step-by-step explanation:

given data

amount = $54000 for 6 year i.e 72 months

amount = $67000 for 8 year i.e 96 months

to find out

How much money does he have now and after 7 years, the nominal rate halves and then stays at that value, how much money will he have in 8 years

solution

we consider here principal amount P = x

then amount equation will be

amount = P ×
`( 1+r)^(t)` ......................1

here P is initial principal amount and r is rate and t is time

54000 = x ×
`( 1+r)^(72)`

x =
`(54000)/(( 1+r)^(72))` .......................3

67000 = x ×
`( 1+r)^(96)`

x =
`(67000)/(( 1+r)^(96))` .........................4

so from equation 3 and 4 we get

`(54000)/(( 1+r)^(72))` =
`(67000)/(( 1+r)^(96))`

solve it we get r

r = 0.9028 %

and x will be then from equation 3

x =
`(54000)/(( 1+0.009028)^(72))`

x = 28272.36

so initial principal amount is $ 28272.36

and

interest rate after 7 year is

interest rate =
`(0.009028)/(2)` = 0.4514 %

and

amount after 8 year will be

amount = 28272.36 ×
`(1+0.009028)^(84)` ×
`(1+0.004514)^(12)`

amount after 8 year is = $63489.85