At what rate of interest per annum will a sum of money double itself in 5 years?

Answer

Hint: To solve the problem, we should know the definition of annual simple interest. We have,
Simple Interest (I) = $\dfrac{P\times R\times t}{100}$
Where, P= principal amount
R = simple interest annual rate
t = time period of the annual simple interest
Here, we have R = 10% and have to calculate t for the sum of the money (that is P) to double.

Complete step-by-step answer:
In this question, we are left with two unknowns, P and t. However, we also have an additional condition. This condition tells that within the required time (which we have to calculate), the sum of money doubles itself. Thus, if originally, we had principal amount as P, finally, this amount would become 2P. Thus, simple interest (I) becomes 2P-P = P. Since, simple interest is basically the amount accumulated over the total principal amount. Further, for simplification, we can write,
$\dfrac{R}{100}=\dfrac{10}{100}=0.1$
Thus, we have,
I=$\dfrac{P\times R\times t}{100}$
Since, I = P (as calculated above), we have,
P = $\dfrac{P\times R\times t}{100}$
We can cancel P from both sides. Thus, we have,
1=$\dfrac{R\times t}{100}$
Plugging in the known values, we have,
1= 0.1$\times $t
Since, $\dfrac{R}{100}$=0.1
Now,
t=10 years
Hence, it will take 10 years for the sum of money to double itself with the rate of 10% per annum simple interest.

Note: While solving questions related to principal interest, it is important to keep in mind that simple interest calculated from the formula, Simple Interest (I) = $\dfrac{P\times R\times t}{100}$ , doesn’t represent the total amount of money. In fact, the total amount is the sum of Principal amount (P) and simple interest. Thus, in this case, when money was doubled, the total amount was 2P and simple interest was P.

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.

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,

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

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 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.

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