John deposited $1000 on 1st January 2011 in an account paying interest of 12% per annum compounded quaterly. He also deposited $800 (on 1st January 2011) in another account which pays 15% per annum 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.

What is the compounding period for the second account?

To find the time when the two accounts will be of equal value, we need to compare the future value of both accounts at various time periods. Let's calculate the future value of each account separately and compare them.

1. Account A with compound interest:

Since the interest is compounded quarterly, we need to convert the annual interest rate to a quarterly interest rate:
Quarterly interest rate = (1 + annual interest rate)^(1/number of compounding periods) - 1
Quarterly interest rate = (1 + 0.12)^(1/4) - 1

Let's calculate the future value of Account A after n quarters:
Future value of Account A = Principal * (1 + Quarterly interest rate)^(number of compounding periods)
Future value of Account A = $1000 * (1 + Quarterly interest rate)^(4n)

2. Account B with effective interest:

To calculate the future value of Account B, we'll use the formula for future value with simple interest:
Future value of Account B = Principal * (1 + effective interest rate * number of years)
In this case, the effective interest rate is already given as 15%, so we can substitute it directly.

Now, we can set up an equation to compare the future values of both accounts and find the value of n.

Future value of Account A = Future value of Account B
$1000 * (1 + Quarterly interest rate)^(4n) = $800 * (1 + effective interest rate * number of years)

Simplifying the equation, we have:
(1 + Quarterly interest rate)^(4n) = (1 + effective interest rate * number of years) * (800/1000)
(1 + Quarterly interest rate)^(4n) = (1 + 0.15 * number of years) * 0.8

Now, we need to solve this equation for n. You can use numerical methods or trial and error to find the value of n that satisfies the equation.