Kyoko has $10,000 that she wants to invest. Her bank has several accounts to choose from. Her goal is to have $20,000 by the time she finishes graduate school in 5 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal assuming they compound daily? (Hint: solve the compound interest formula for the intrerest rate. Also, assume there are 365 days in a year)
Question
Kyoko has 20,000 by the time she finishes graduate school in 5 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal assuming they compound daily? (Hint: solve the compound interest formula for the intrerest rate. Also, assume there are 365 days in a year)
Solution
The compound interest formula 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 = the time the money is invested for in years
In this case, we know that:
A = 10,000 n = 365 (since the interest is compounded daily) t = 5 years
We need to find r.
First, divide A by P to get:
20000/10000 = (1 + r/365)^(365*5)
2 = (1 + r/365)^(1825)
Take the 1825th root of both sides to get:
2^(1/1825) = 1 + r/365
Subtract 1 from both sides to get:
2^(1/1825) - 1 = r/365
Finally, multiply both sides by 365 to solve for r:
365 * (2^(1/1825) - 1) = r
This gives us the annual interest rate in decimal form. To convert it to a percentage, we multiply by 100.
So, the minimum annual interest rate Kyoko needs is approximately 12.25%.
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