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?

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BlackTom AI · Accepted Answer

To determine how long it takes for the money to double with monthly compounding, we use the compound interest formula A = P(1 + r/12)^{12t}, where P = 1000, A = 2000, r = 0.045, and t is in years.

First, set up the doubling condition: 2000 = 1000(1 + 0.045/12)^{12t}, which simplifies to 2 = (1 + 0.045/12)^{12t}.

Next, take natural logarithms on both sides to bring down the exponent: ln(2) = 12t · ln(1 + 0.045/12).

Compute the inside terms approximately: 0.045/12 = 0.00375, so ln(1 + 0.00375) ≈ 0.003743 (using the Taylor approximation or a calculator for a more precise value).

Now solve for t: t = ln(2) / [12 · ln(1 + 0.00375)] ≈ 0.6931 / (12 × 0.003743) ≈ 0.6931 / 0.044916 ≈ 15.4 years.

With the calculation in mind, evaluate each option:

- Option: 15.4 years. This aligns with the computation above, since the exponential growth with monthly compounding at 4.5% indeed takes about 15.4 years to double, given the effective annual yield is slightly higher than 4.5% due to compounding.

- Option: 20.1 years. This would imply a much slower growth rate than 4.5% with monthly compounding; it would correspond to a smaller effective rate or longer time, which contradicts the given rate.

- Option: 13.7 years. This is shorter than the calculated time; if you plug t = 13.7 into the formula, the accumulated amount would be less than double, indicating an underestimation of growth or an overstatement of time savings.

- Option: 9.8 years. This would require a substantially higher effective return or more frequent compounding than monthly at 4.5%, making it inconsistent with the given parameters and not supported by the math.

In summary, comparing the derived value t ≈ 15.4 years to the options shows that 15.4 years is the one that matches the calculation for monthly compounding at 4.5%.

MSB-250-300-002 Proctored Final Exam

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?

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MSB-250-300-002 Proctored Final Exam

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?

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

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?

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?

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

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更多留学生实用工具

希望你的学习变得更简单

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