Suppose that $200 was deposited on 1st Jan 2000 into an account that earned 5% interest compounded semiannually. Suppose further that $200 was deposited on 1st Jan 2001 into a different account that earned 6% interest compounded semiannually. In what month of what year will the total amount in the account earning 6% interest overtake the total amount in the account earning 5% interest?
so we are solving ......
200( 1.025)^(t) = 200(1.03)^(t-2) , where t is number of half-years
1.025^(t) = 1.03^(t-2)
log both sides
log (1.025^(t)) = log(1.03^(t-2))
t log 1.025 = (t-2) log 1.03
t log 1.025 = t log 1.03 - 2log 1.03
t log 1.025 - t log 1.03 = -2log 1.03
t(log 1.025 - log 1.03) = -2log 1.03
t = -2log 1.03/(log 1.025 - log 1.03)
t = 12.1486 half years after jan 1, 2000
I will leave it to you to figure out the month and year
Remember t is in half years,
so it will take 6 years and .1486(6) months .....
200(1.025)^((n+1)*2) = 200(1.03)^(2n)
2(n+1)ln1.025 = 2nln1.03
(n+1)ln1.025 = nln1.03
n(ln 1.03 - ln1.025) = ln1.025
n = 5.074 years
year 2006, month January
To find out in what month of what year the total amount in the account earning 6% interest overtakes the total amount in the account earning 5% interest, we can create a table to track the growth of both accounts over time.
Let's start by calculating the growth of the first account earning 5% interest compounded semiannually. We can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
For the first account, the principal amount is $200, the annual interest rate is 5% (0.05 as a decimal), interest is compounded semiannually (n = 2), and the number of years can vary. We'll calculate the growth for each year until the end of the table.
Year | Final Amount (5%)
------------------------
2000 | $200.00
2001 | $210.25
2002 | $220.96
2003 | $232.05
2004 | $243.54
Next, let's calculate the growth of the second account earning 6% interest compounded semiannually using the same formula.
Year | Final Amount (6%)
------------------------
2001 | $200.00
2002 | $212.00
2003 | $224.91
2004 | $238.77
2005 | $253.69
By comparing the two tables, we can see that in the year 2005, the total amount in the account earning 6% interest overtook the total amount in the account earning 5% interest.
Therefore, the answer to your question is that in the month of January 2005, the total amount in the account earning 6% interest overtook the total amount in the account earning 5% interest.