An amount doubles itself in 5 years with simple interest. what is the rate of interest

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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Solution

The correct option is B 15 years 15 वर्षMethod I: Let the initial amount and the compound rate of interest be P and R respectively. In 5 years the amount gets doubled, i.e. total amount = 2P In 10 year amount gets doubled again, i.e. total amount = 4 P Similarly, in 15 years the total amount = 2 × 4P = 8P Method II: Using the formula of compound interest, we get: Total amount after 5 years, 2 P = P (1 + R/100)5 Or (1+R/100)5 = 2 Or 1 + R/100 = 21/5 Let the amount becomes 8 times after n years. Hence, 8P = P (1+R/100)n = P (2n/5) Or 2n/5 = 23 Or n/5 = 3, i.e. n = 15 years विधि I : माना कि आरंभिक राशि और चक्रवृद्धि ब्याज दर क्रमशः P और R है। 5 वर्षों में राशि दोगुनी हो जाती है, अर्थात कुल राशि = 2P 10 वर्षों में राशि फिर से दोगुनी हो जाती है, अर्थात कुल राशि = 4 p इसी प्रकार, 15 वर्षों में कुल राशि = 2 × 4P = 8P विधि II: चक्रवृद्धि ब्याज़ के सूत्र का प्रयोग करते हुए, हम पाते हैं: 5 वर्षों के बाद कुल राशि, 2P = P (1+R/100)5 या (1 + R/100)5 = 2 या 1+R/100 = 21/5 माना कि राशि 7 वर्षों के बाद 8 गुना हो जाती है। इस प्रकार, 8P = P (1+R/100)n = P (2n/5) या 2n/5 = 23 या n/5 = 3, अर्थात n = 15 वर्ष

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