Calculus Calc Rate Of Interest
Annual Rate Of Interest Lara S. Follow • 1 Add
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Mark M. answered • 12/03/20 Tutor 5.0 (244) Mathematics Teacher - NCLB Highly Qualified About this tutor › About this tutor › 2 = (1 + r)12 12√2 = 1 + r 1.059463094 ≈ 1 + r 0.059463094 ≈ r Upvote • 1 Downvote Add comment More Report Still looking for help? Get the right answer, fast.Ask a question for free Get a free answer to a quick problem. ORFind an Online Tutor Now Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. Calculator UseUse the Rule of 72 to estimate how long it will take to double an investment at a given interest rate. Divide 72 by the interest rate to see how long it will take to double your money on an investment. Alternatively you can calculate what interest rate you need to double your investment within a certain time period. For example if you wanted to double an investment in 5 years, divide 72 by 5 to learn that you'll need to earn 14.4% interest annually on your investment for 5 years: 14.4 × 5 = 72. The Rule of 72 is a simplified version of the more involved compound interest calculation. It is a useful rule of thumb for estimating the doubling of an investment. This calculator provides both the Rule of 72 estimate as well as the precise answer resulting from the formal compound interest calculation. Interest RateThe annual nominal interest rate of your investment in percent.Time Period in YearsThe number of years the sum of money will remain invested. You can also input months or any period of time as long as the interest rate you input is compounded at the same frequency.CompoundingThis calculator assumes the frequency of compounding is once per period. It also assumes that accrued interest is compounded over time.Rule of 72 FormulaThe Rule of 72 is a simple way to estimate a compound interest calculation for doubling an investment. The formula is interest rate multiplied by the number of time periods = 72: R * t = 72 where
Commonly, periods are years so R is the interest rate per year and t is the number of years. You can calculate the number of years to double your investment at some known interest rate by solving for t: t = 72 ÷ R. You can also calculate the interest rate required to double your money within a known time frame by solving for R: R = 72 ÷ t. Derivation of the Rule of 72 FormulaThe basic compound interest formula is: A = P(1 + r)t, where A is the accrued amount, P is the principal investment, r is the interest rate per period in decimal form, and t is the number of periods. If we change this formula to show that the accrued amount is twice the principal investment, P, then we have A = 2P. Rewriting the formula: 2P = P(1 + r)t , and dividing by P on both sides gives us (1 + r)t = 2 We can solve this equation for t by taking the natural log, ln(), of both sides, \( t \times ln(1+r)=ln(2) \) and isolating t on the left: \( t = \dfrac{ln(2)}{ln(1+r)} \) We can rewrite this to an equivalent form: \( t = \dfrac{ln(2)}{r}\times\dfrac{r}{ln(1+r)} \) Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8% (r=0.08):* \( t = \dfrac{0.69}{r}\times\dfrac{0.08}{ln(1.08)}=\dfrac{0.69}{r}(1.0395) \) Solving this equation for r times t: \( rt=0.69\times1.0395\approx0.72 \) Finally, multiply both sides by 100 to put the decimal rate r into the percentage rate R: R*t = 72 *8% is used as a common average and makes this formula most accurate for interest rates from 6% to 10%. Example Calculations in YearsIf you invest a sum of money at 6% interest per year, how long will it take you to double your investment? t=72/R = 72/6 = 12 years What interest rate do you need to double your money in 10 years? R = 72/t = 72/10 = 7.2% Example Calculation in MonthsIf you invest a sum of money at 0.5% interest per month, how long will it take you to double your investment? t=72/R = 72/0.5 = 144 months (since R is a monthly rate the answer is in months rather than years) 144 months = 144 months / 12 months per years = 12 years ReferencesVaaler, Leslie Jane Federer; Daniel, James W. Mathematical Interest Theory (Second Edition), Washington DC: The Mathematical Association of America, 2009, page 75. Weisstein, Eric W. "Rule of 72." From MathWorld--A Wolfram Web Resource, Rule of 72. What rate of interest compounded annually is required to triple an investment in 8 years?That is, the rate of interest must be 14.72%.
What annual rate of interest compounded annually doubles an investment in 7 years?This means, at a 10% fixed annual rate of return, your money doubles every 7 years. Let's try another one: Given a 9% interest rate, how long will it take to double your money? Divide 72 by 9 and you'll get 8 years.
What rate of interest compounded annually is required to double an investment in 5 years?Calculator Use
For example if you wanted to double an investment in 5 years, divide 72 by 5 to learn that you'll need to earn 14.4% interest annually on your investment for 5 years: 14.4 × 5 = 72. The Rule of 72 is a simplified version of the more involved compound interest calculation.
What rate of interest compounded annually is required to double an investment in 6 years?You can also run it backwards: if you want to double your money in six years, just divide 6 into 72 to find that it will require an interest rate of about 12 percent.
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What rate of interest compounded annually is required to double an investment in 8 years?
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