How long will it take the amount a principal to double itself with a simple interest of 10%?

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.

Principal: The money which we deposit in or the lower from the bank or the money learned called the principal.

Rate of interest: The interest paid on $ 100 for one year is called the rate per cent per year or rate per cent per annum.

Time: The period of time for which the money is lent or invested.

Interest: Additional money paid by the borrowed to the lender for using the money is called interest.

Simple Interest: If the interest is calculated uniformly on the original principal throughout the lone period, it is called simple interest.

Amount: The total money paid back to the lender is called the amount.

Calculate Simple Interest

Formula to calculate Simple Interest?

If P denotes the principal ($), R denotes the rate (percentage p.a.) and T denotes time (years), then:-

S.I = (P × R × T)/100

R = (S.I × 100)/(P × T)

P = (S.I × 100)/(R × T)

T = (S.I × 100)/(P × R)

If the denotes the amount, then A = P + S.I

Note:

● When we calculated the time period between two dates, we do not could the day on which money is deposited but we count the day on which money is retuned.

● Time is always taken according to the per cent rat.

● For converting time in days into years, divide th number of days by 365 (for ordering or lap year.)

● For converting time in month into years, divide th number of month by 12 (for ordering or lap year.)

Examples to find or calculate simple interest when principal, rate and time are known

Calculate Simple Interest

Find the simple interest on:

(a) $ 900 for 3 years 4 months at 5% per annum. Find the amount also.

Solution:

R = 5% p.a.

T = 3 years 4 months = 40/12 years = 10/3 years Therefore, S.I = (P × R × T)/100 = (900 × 5 × 10)/(100 × 3) = $ 150

Amount = P + S.I = $ 900 + $ 150 = $ 1050

(b) $ 1000 for 6 months at 4% per annum. Find the amount also.

Solution:

R = 4% p.a.

T = 6 months = 6/12 years S.I = (P × R × T)/100 = (1000 × 4 × 1)/(100 × 2) = $ 20 Therefore, A = P + I = $( 1000 + 20) = $ 1020

(c) $ 5000 for 146 days at 15¹/₂% per annum.

Solution:

(d) $ 1200 from 9ᵗʰ April to 21ˢᵗ June at 10% per annum.

Solution:

= 73 days [April = 21, May = 31, Jun = 21, 73 days]

= 73/365 years

S.I = (1200 × 10 × 73)/(100 × 365) = $ 24

Examples to find or calculate Time when Principal, S.I and Rate are known

Calculate Simple Interest

1. In how much time dose $ 500 invested at the rate of 8% p.a. simple interest amounts to $ 580.

Solution:

Therefore S.I = A - P = $ (580 - 500) = $ 80

Therefore T = (100 × S.I)/(P × R) = (100 × 80)/(500 × 3) = 2 years

2. In how many years will a sum of $ 400 yield an interest of $ 132 at 11% per annum?

Solution:

Calculate Simple Interest

3. In how many years will a sum double itself at 8 % per annum?

Solution:

So , S.I. = A - P = 2P – P = P

T = (100 × S.I)/(P × R) = ( 100 × P)/(P × 8) = 25/2 = 121/2 years

Calculate Simple Interest

4. In how many years will simple interest on certain sum of money at 6 1/4% Per annum be 5/8 of itself?

Solution:

T = 10 years

Examples to find or calculate Rate per cent when Principal, S.I. and Time are known

1. Find at what rate of interest per annum will $ 600 amount to $ 708 in 3 years.

Solution:

Calculate Simple Interest

2. Simple interest on a certain sum is 36/25 of the sum. Find the Rate per cent and time if they are both numerically equal.

Solution:

Calculate Simple Interest

3. At what rate per cent per annum will $ 6000 produce $ 300 as S.I. in 1 years?

Solution:

4. At what rate per cent per annum will a sum triple itself in 12 years ?

Solution:

Examples to find or calculate Principal when Rate, Time and S.I. are known

Calculate Simple Interest

1. What sum will yield $ 144 as S.I. in 21/2 years at 16% per annum?

Solution:

2. A some amount to $ 2040 in 21/2 years at , P = ?

Solution:

51x = 2040 × 40 ---- x = (2040 × 40)/51 = $ 1600

Calculate Simple Interest

3. A certain sum amounts to $ 6500in 2 years and to $ 8750 in 5 years at S.I. Find the sum and rate per cent per annum.

Solution:

4. Divide $ 6500 in to two parts , such that if one part is lent out at 9% per annum and other at 10% per annum, the total yearly income is $ 605

Solution:

But given that total S.I.= $ 605

So, (6500 - x)/100 =605 -----65000 - x = 60500

----- 65000 – 60500 = x ---- x = $ 4500

Now, second part = 6500 – x = 6500 – 4500 = $ 2000

Hence, first part = $ 2000 and second part = $ 4500

5. When the rate of interest in a bank is increased from 9% to 10% per annum; A person deposits $ 500 more into his account. If the annual interest now Received by him is $ 150more then before, find his original deposit.

Solution:

Then, S.I. on $ x for 1 year at (10 - 9 )% = 1% per annum + S.I. on $ 500

For I year at 10% per annum = $ 15

----- ( x × 1 × 1)/100 + ( 500 × 10 × 1)/100 = 150

----- x/(100 ) + 50 = 150 ---- x/(100 ) + 150 – 50 ----- x/(100 ) + 100

----- x = 100 × 100 = $ 10,000

Therefore, the original deposit is $ 10,000.

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