Question 1169507
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Consider the first year for now only. One year is 4 quarters.
The 12% annual interest rate converts to the quarterly rate of (12%)/4 = 3%
We'll let i = 0.03 to reflect this.


Compute the future value (FV) of an annuity due with 
<ul><li>deposit of P = 20000</li><li>quarterly interest rate i = 0.03</li><li>n = 4 quarters</li></ul> 
We get the following
FV = (1+i)*P*( (1+i)^n - 1)/i
FV = (1+0.03)*20000*( (1+0.03)^4 - 1)/0.03
FV = 86182.7162
FV = 86182.72


After four quarters pass by, we have 86182.72 in the account.


Let x be the quarterly interest rate for the next 8 quarters (2 years). 
We'll deposit P = 86182.72 dollars at this rate and time. 
At the end of 8 quarters, we'll be left with 86182.72*(1+x)^8 dollars
Note I'm using the compound interest formula here.


Let S = 86182.72*(1+x)^8
We'll come back to this later.


Again keeping x the same, whatever it is, we'll compute the FV of an annuity due like we did earlier. However, this time we don't know the interest rate and the value of n is doubled (n = 8 instead of n = 4). 


So we have
FV = (1+i)*P*( (1+i)^n - 1)/i
FV = (1+x)*20000*( (1+x)^8 - 1)/x
Let's call this R
R = (1+x)*20000*( (1+x)^8 - 1)/x


Add the expressions of S and R
S+R = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x
call this sum T. We'll make it a function of x
T(x) = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x


The goal is to solve T(x) = 300000 which is the same as finding the roots of T(x)-300000 = 0


The variable x is buried under an exponent of 8, which suggests that solving this by hand is going to be daunting. Luckily we can use technology. 


You can use any technology you like. For me, I prefer GeoGebra since it's a very handy program in many ways. You should find the root on the interval 0 < x < 1 is approximately 
x = 0.0345028581


This represents the decimal form of the quarterly interest rate.
So the quarterly rate is about 3.45%


Multiplying this by 4 leads to the annual rate needed
4*x = 4*0.0345 = 0.138 = 13.8%


Answer: 13.8%
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