If you deposit $10,000 in an account with an annual rate of 9% compounded semiannually, how long will it take for you to have $2,000,000 in the account?

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BlackTom AI · Accepted Answer

Begin by restating the problem in my own words to ensure clarity: we start with a principal of $10{,}000 and want to know how long it takes for this amount to grow to $2{,}000{,}000 when the annual interest rate is 9% compounded semiannually.

First, convert the effective growth per compounding period. Since the rate is 9% per year compounded semiannually, each half-year period has a rate of 9%/2 = 4.5%, or 0.045 in decimal form. Over n such half-year periods, the future value is given by FV = PV × (1 + 0.045)^n.

Set up the equation using the given values: 2{,}000{,}000 = 10{,}000 × (1.045)^n. Divide both sides by 10{,}000 to simplify: 200 = (1.045)^n.

To solve for n (the number of half-year periods), take logarithms of both sides. Using natural logs (or common logs) yields n = log(200) / log(1.045).

Compute the logarithms approximately: log(200) ≈ 5.298317 and log(1.045) ≈ 0.044017. Therefore n ≈ 5.298317 / 0.044017 ≈ 120.36 half-year periods.

Convert half-year periods to years by dividing by 2: years ≈ 120.36 / 2 ≈ 60.18 years.

Now, check the logic: because interest compounds semiannually, we must use the per-period rate and the number of half-year periods to model growth. The setup assumes reinvestment at each compounding date with no withdrawals, which aligns with the standard compound interest model used here.

Common pitfalls to consider include mixing up the compounding frequency with the annual rate, which would lead to an incorrect per-period rate if not divided by 2. Another error would be using a simple interest model (FV = PV × (1 + r × t)) rather than the compound formula, which would grossly underestimate the time required. A third confusion might be miscounting the number of periods versus the number of years—remember n is the number of half-year periods, not years, in this setup.

In summary, applying the correct per-period growth factor and solving for the number of half-year periods yields a total time that translates to approximately 60.18 years for the balance to reach $2,000,000.

MSB-250-300-002 Topic 5 Quiz

If you deposit $10,000 in an account with an annual rate of 9% compounded semiannually, how long will it take for you to have $2,000,000 in the account?

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MSB-250-300-002 Topic 5 Quiz

If you deposit $10,000 in an account with an annual rate of 9% compounded semiannually, how long will it take for you to have $2,000,000 in the account?

查看解析

登录即可查看完整答案

类似问题

Part 1Consider a twominus−year ​investment: Given a constant and positive interest​ rate, the interest earned in the second year will be greater than the interest earned in the first year​ (assuming annual​ compounding). Part 1 True False

If Jack and Jill deposit $1,000 into a savings account that earns 4.5% annual interest (compounded monthly), how long will it take for their money to double to $2,000?

How much will you get back on a five year investment of $100 that pays 5% interest per year?

What is the effective yield (APY) if the Annual Percentage Rate (APR) is 7.64 percent and it is compounded quarterly?

In order to determine the future value of some lump sum, we must use the process of ________.

With compound interest, the investor earns interest on the principal amount invested plus interest on accumulated interest. Which of the following compounding frequencies would yield the investor the greatest ending balance assuming all else is equal?

You have located an investment that pays 9 percent per year. You invest $800. How much will you have in seven years? Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.Blank.xlsx

If you deposit $1'000 in a savings account that pays 4% compounded annually , compute the future value of your account after 4 years.

更多留学生实用工具

希望你的学习变得更简单

Part 1Consider a twominus−year investment: Given a constant and positive interest rate, the interest earned in the second year will be greater than the interest earned in the first year (assuming annual compounding). Part 1 True False