Dylan is saving for a trip to Sweden, which will cost him $52,000. At the moment, he has $36,000 saved. By depositing this amount in an account offering a 8.6% p.a. interest rate, compounded semi-annually, how many months will it take for Dylan's savings to reach the required amount for the trip to Sweden? (Round your answer to full integer)
Question
Dylan is saving for a trip to Sweden, which will cost him 36,000 saved. By depositing this amount in an account offering a 8.6% p.a. interest rate, compounded semi-annually, how many months will it take for Dylan's savings to reach the required amount for the trip to Sweden? (Round your answer to full integer)
Solution 1
To solve this problem, we can use the formula for compound interest, which is:
A = P (1 + r/n)^(nt)
Where: A = the amount of money accumulated after n years, including interest. P = the principal amount (the initial amount of money) r = annual interest rate (in decimal) n = number of times that interest is compounded per year t = time the money is invested for in years
We need to rearrange this formula to solve for t:
t = log(A/P) / (n * log(1 + r/n))
We know that: A = 36,000 (the initial amount Dylan has) r = 8.6% = 0.086 (annual interest rate in decimal) n = 2 (since the interest is compounded semi-annually)
Substituting these values into the formula gives:
t = log(52000/36000) / (2 * log(1 + 0.086/2))
Calculating this gives a time of approximately 3.5 years. However, the question asks for the time in months, so we multiply this by 12 to convert years to months:
t = 3.5 * 12 = 42 months
However, since we need to round to the nearest whole number, Dylan will need 42 months to save for his trip to Sweden.
Solution 2
To solve this problem, we can use the formula for compound interest, which is:
A = P (1 + r/n)^(
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