Joeli deposited $1000 on 1 January 2011 in an account paying interest of 12% p.a compounded quarterly. He also deposited $800 (on 1 January 2011) in another account which pays 15% p.a. effective interest. Find the time (n) when the two accounts will be equal value if the exact method is used for fractions of an interest period.
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To find the time when the two accounts will have equal value, we need to equate the future values of the two accounts.
Let's start by calculating the future value of the first account, which pays 12% p.a compounded quarterly.
The formula to calculate the future value (FV) with compound interest is:
FV = P(1 + r/n)^(nt)
Where:
FV = Future Value
P = Principal amount (initial deposit)
r = Annual interest rate (in decimal form)
n = Number of times that interest is compounded per year
t = Number of years
In this case, P = $1000, r = 12% = 0.12, n = 4 (compounded quarterly), and t = n (since we want to find when the two accounts are equal). Let's denote the future value of the first account as FV1.
FV1 = $1000(1 + 0.12/4)^(4n)
Now let's calculate the future value of the second account, which pays 15% p.a. effective interest.
The formula to calculate the future value (FV) with effective interest rate is:
FV = P(1 + r)^t
Where:
FV = Future Value
P = Principal amount (initial deposit)
r = Annual effective interest rate (in decimal form)
t = Number of years
In this case, P = $800, r = 15% = 0.15, and t = n. Let's denote the future value of the second account as FV2.
FV2 = $800(1 + 0.15)^t
Now, we can set up an equation to equate the future values of the two accounts:
FV1 = FV2
$1000(1 + 0.12/4)^(4n) = $800(1 + 0.15)^n
To solve this equation for n, we can take the logarithm of both sides. Let's take the natural logarithm (ln) of both sides:
ln($1000(1 + 0.12/4)^(4n)) = ln($800(1 + 0.15)^n)
Using the logarithmic properties, we can simplify the equation further:
ln($1000) + ln(1 + 0.12/4)^(4n) = ln($800) + ln(1 + 0.15)^n
ln($1000) + 4n * ln(1 + 0.12/4) = ln($800) + n * ln(1 + 0.15)
Now, we can isolate the variable n to one side of the equation:
4n * ln(1 + 0.12/4) - n * ln(1 + 0.15) = ln($800) - ln($1000)
We can divide both sides of the equation by n to simplify further:
4 * ln(1 + 0.12/4) - ln(1 + 0.15) = (ln($800) - ln($1000)) / n
Finally, we can solve for n by rearranging the equation and evaluating:
n = (ln($800) - ln($1000)) / (4 * ln(1 + 0.12/4) - ln(1 + 0.15))
Using a calculator or a software, substitute the values and compute the expression to find the value of n.