Double Time Formula

Double time can be defined as the time in which any quantity growing at a certain rate becomes double the initial size/amount. Doubling time helps in making the calculations of simple interest or rate growth much easier when it is asked to find the time when the value of anything will be doubled.

Double Time Formula

In the below given double-time formula, we have taken the natural log and r is the rate of growth.

Double Time Formula = 

If the growth rate is given in percentage then double time can be calculated by modifying the formula so the new formula is,

Double time formula = 70/r

Here r is the percentage growth rate.

This new formula is also known as the Rule of 70 because in the above formula the double-time has been calculated by dividing 70 by r which represents the percentage rate of growth. 

Features of Double Time Formula

1. Double time can be easily found only with the growth rate.
2. It is used in different real-world aspects like population growth of a country, resource utilization, simple and compound interest, etc.
3. It provides a clear picture of profits received by an investment over some period of time.
4. Double time formula is a very old concept and it was used in Babylon to calculate interest on given loans.

Sample Problems

Question 1: How many years will it take to double the amount of growth rate is 10 % per annum.

Solution:

To find the time we will use double time formula

Double Time Formula = log2/log(1+ r)

Here r is given as 10% so r= 10/100 = 0.10

Double time = log2/log(1 + 0.10)

= 7.27 years

Hence it will take about 7.27 years to double the amount.

Question 2: Use the Rule of seventy to find the time in which the current population of a country will be doubled if the growth rate is 5% per annum.

Solution:

Given r = 5%

So using rule of 70

Double time = 70/r

= 70/5

= 14

Hence it will take 14 years for the population to get double.

Question 3: Find the rate of growth so that the given amount gets double in 10 years.

Solution:

We need to find the r and double time is given which is 10 years.

Now using the rule of 70

Double time = 70/r

10 = 70/r

r = 70/10

r = 7 

Hence rate is 7% per annum.

Question 4: How many years will it take to double the amount of growth rate is 18 % per annum.

Solution:

To find the time we will use double time formula

Double Time Formula = log2/log(1 + r)

Here r is given as 18% so r= 18/100 = 0.18

Double time = log2/log(1 + 0.18)

= 4.18 years

Hence it will take about 4.18 years to double the amount.

Question 5: In a pond bacteria is increasing at a rate of 7% find the time when it gets double.

Solution:

To find the time we will use double time formula

Double Time Formula = log2/log(1 +  r)

Here r is given as 7% so r= 7/100 = 0.07

Double time = log2/log(1 + 0.07)

= 10.24 years

Hence it will take about 10.24 years to double the bacteria in the pond.

Question 6:  Find the rate of growth so that the given amount gets double in 20 years.

Solution:

We need to find the r and double time is given which is 20 years.

Now using the rule of 70

Double time = 70/r

20 = 70/r

r = 70/20

r = 3.5

Hence rate is 3.5% per annum.

Question 7: Use Rule of seventy to find the time in which the current bacteria population of a pond will be doubled if the growth rate is 7% per annum.

Solution:

Given r = 7%

So using rule of 70

Double time = 70/7

= 70/7

= 10

Hence it will take 10 years for the population to get double.