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# what interest rate would you have to earn if you wanted to double an investment in 3 years?

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### James

Guys, does anyone know the answer?

get what interest rate would you have to earn if you wanted to double an investment in 3 years? from EN Bilgi.

## Rule of 72 Calculator

The Rule of 72: Divide 72 by the interest rate to get the number of years to double your investment. A good estimate for how long it takes to double your money.

Calculators > Financial > Saving and Investing > Rule of 72 Calculator

## Rule of 72 Calculator

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## Calculator Use

Use 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 Rate

The annual nominal interest rate of your investment in percent.

Time Period in Years

The 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.

Compounding

This calculator assumes the frequency of compounding is once per period. It also assumes that accrued interest is compounded over time.

## Rule of 72 Formula

The 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

R = interest rate per period as a percentage

t = number of periods

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 Formula

The 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×ln(1+r)=ln(2) t×ln(1+r)=ln(2)

and isolating t on the left:

t= ln(2) ln(1+r) t=ln(2)ln(1+r)

We can rewrite this to an equivalent form:

t= ln(2) r × r ln(1+r) t=ln(2)r×rln(1+r)

Solving ln(2) = 0.69 rounded to 2 decimal places and solving the second term for 8% (r=0.08):*

t= 0.69 r × 0.08 ln(1.08) = 0.69 r (1.0395)

t=0.69r×0.08ln(1.08)=0.69r(1.0395)

Solving this equation for r times t:

rt=0.69×1.0395≈0.72 rt=0.69×1.0395≈0.72

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 Years

If 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 Months

If 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

## References

Vaaler, 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.

Cite this content, page or calculator as:

Furey, Edward "Rule of 72 Calculator" at https://www.calculatorsoup.com/calculators/financial/rule-of-72-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

Source : www.calculatorsoup.com

## Rule of 72

Have you always wanted to be able to do compound interest problems in your head? Perhaps not... but it's a very useful skill to have because it gives you a lightning fast benchmark to determine how good (or not so good) a potential investment is likely to be.

The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.

Y   =   72 / r   and   r   =   72 / Y

where Y and r are the years and interest rate, respectively.

## Compound Interest Curve

Suppose you invest $100 at a compound interest rate of 10%. The rule of 72 tells you that your money will double every seven years, approximately: Years Balance Now$100

7 $200 (doubles every 14$400   seven years)

### How Accurate Is the Rule of 72?

The Rule of 72 formula provides a reasonably accurate, but approximate, timeline—reflecting the fact that it's a simplification of a more complex logarithmic equation. To get the exact doubling time, you'd need to do the entire calculation.

The precise formula for calculating the exact doubling time for an investment earning a compounded interest rate of r% per period is:

\begin{aligned} &T = \frac{ \ln( 2 ) }{ \ln \left ( 1 + \frac{ r } { 100 } \right ) } \simeq \frac{ 72 }{ r } \\ &\textbf{where:}\\ &T = \text{Time to double} \\ &\ln = \text{Natural log function} \\ &r = \text{Compounded interest rate per period} \\ &\simeq = \text{Approximately equal to} \\ \end{aligned}

​ T= ln(1+ 100 r ​ ) ln(2) ​ ≃

Source : www.investopedia.com

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James 10 month ago

Guys, does anyone know the answer?