1. Doubling Time and Half-Life
Unit 8B
Doubling Time and Half-Life
Ms. Young

2. Doubling and Halving Times
The time required for each doubling in exponential growth is called doubling time.
The time required for each halving in exponential decay is called halving time.
Ms. Young

3. Doubling Time
After a time t, an exponentially growing quantity with a doubling time of Tdouble increases in size by a factor of The new value of the growing quantity is related to its initial value (at t = 0) by
New value = initial value x 2t/Tdouble
Ms. Young

4. Example
World Population Growth: World population doubled from 3 billion in 1960 to 6 billion in Suppose that the world population continued to grow (from 2000 on) with a doubling time of 40 years. What would be the population in 2050?
The doubling time is Tdouble 40 years. Let t = 0 represent 2000 and the year 2050 represent t = 50 years later. Use the 2000 population of 6 billion as the initial value.
new value = 6 billion x 250 yr/40 yr
= 6 billion x ≈ 14.3 billion
Ms. Young

5. Approximate Double Time Formula (The Rule of 70)
For a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately
This approximation works best for small growth rates and breaks down for growth rates over about 15%.
Work through several examples comparing perhaps two different countries and their growth rates or two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.”
Ms. Young

6. Exponential Decay and Half-Life
After a time t, an exponentially decaying quantity with a half-life time of Thalf decreases in size by a
factor of The new value of the decaying quantity is related to its initial value (at t = 0) by
New value = initial value x (1/2)t/Thalf
Ms. Young

7. The Approximate Half-Life Formula
For a quantity decaying exponentially at a rate of P% per time period, the half-life is approximately
This approximation works best for small decay rates and breaks down for decay rates over about 15%.
Work through several examples comparing perhaps two different radioactive decay percentages. As a challenge, have students try to find the rule of thumb for “tripling time.”
Ms. Young

8. Exact Doubling Time and Half-Life Formulas
For more precise work, use the exact formulas. These use the fractional growth rate, r = P/100.
For an exponentially growing quantity, the doubling time is
For an exponentially decaying quantity, use a negative value for r. The halving time is
Ms. Young