AP Environmental Science

## Learning Outcomes

- Calculate **doubling time** using the Rule of 70
- Convert between growth rate and doubling time
- Explain why the Rule of 70 applies to **exponential growth**
- Distinguish the Rule of 70 from exact exponential formulas
- Connect doubling time to human population trends, demographic transition, and resource limits
- Avoid common APES calculation mistakes involving percent versus decimal form

## AP Environmental Science Syllabus

For the AP Environmental Science, you are expected to understand how the **Rule of 70** is used in human population dynamics, with a focus on the following syllabus points:

- Know that the **Rule of 70** estimates population doubling time under roughly constant exponential growth
- Use the formula **doubling time ≈ 70 ÷ annual growth rate (%)**
- Calculate doubling time from a stated growth rate and calculate growth rate from a stated doubling time
- Understand that the growth rate must be entered as a **percent**, not a decimal
- Explain why a higher growth rate produces a **shorter doubling time**
- Understand that real populations cannot double forever because of **carrying capacity** and limiting factors
- Connect changes in doubling time to changing birth rates, death rates, and stages of demographic transition
- Interpret AP-style examples involving projected future dates for population doubling

## Test Your Knowledge

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision._

1. What is the Rule of 70 used to estimate?
2. A population grows at **2%** per year. How long will it take to double?
3. If a country’s population is expected to double in **35 years**, what is its annual growth rate?
4. Why does the Rule of 70 work for exponential growth but not for linear growth?
5. Why can real human populations not continue doubling forever?

## Introduction

In AP Environmental Science, the Rule of 70 is a quick math tool used to estimate how long it takes a population to **double in size**. It appears in Unit 3 because human population change is not just about memorising birth rates or fertility rates. You also need to interpret what a growth rate means over time. A population growing by a small percentage each year can still become much larger surprisingly fast.

This matters because doubling time helps link math to environmental impact. If a human population doubles quickly, demand for **food, freshwater, land, energy, and waste disposal** can rise fast as well. The Rule of 70 therefore sits at the point where population biology meets sustainability.

> **Key Term: doubling time**
> The amount of time required for a population to double in size.

### What the Rule of 70 means and why it works

The Rule of 70 is an approximation used when a population is growing at a roughly constant **percentage rate**. The formula is:

$$
\text{Doubling time in years} \approx \frac{70}{\text{annual growth rate (\%)}}
$$

So if a population grows by **2% per year**, its doubling time is:

$$
\frac{70}{2} = 35 \text{ years}
$$

If the annual growth rate is **1%**, then:

$$
\frac{70}{1} = 70 \text{ years}
$$

This relationship is one of the most testable ideas in human population dynamics because it turns a percentage into a clear real-world consequence. A growth rate that sounds small may still produce a large increase over a few decades.

> **Key Term: Rule of 70**
> A shortcut used to estimate doubling time by dividing 70 by the annual growth rate expressed as a percent.
>
> **Key Term: exponential growth**
> Population growth in which the increase becomes larger over time because growth is based on a percentage of the current population size.

Why does this work? In true exponential growth, each year’s increase is based on the new, larger population. That means growth compounds. The number 70 is a rounded version of **69.3**, which comes from the mathematical constant **ln(2) ≈ 0.693**. AP Environmental Science does not expect you to derive this, but it helps explain that the Rule of 70 is not random. It is a convenient estimate based on exponential math.

The approximation works best when growth rates are relatively small and fairly stable. That is why it is useful in population studies, economics, and resource use projections.

A very common exam distinction is this:

- **Exponential growth**: grows by a percentage, so doubling time makes sense
- **Linear growth**: grows by the same fixed amount each time, so the Rule of 70 does not apply

For example, if a city gains **10,000 people each decade** no matter what, that is not exponential growth. The increase is fixed, not compounding. But if the city grows by **2% each year**, the number added each year gets bigger over time. That is the kind of pattern the Rule of 70 estimates.

> **Test Tip:** If a question asks for doubling time, first check whether the growth rate is given as a **percent per year**. That is your signal that the Rule of 70 probably applies.

### How to calculate doubling time, growth rate, and future year

APES questions often test the Rule of 70 in three ways:

1. Calculate doubling time from a growth rate
2. Calculate growth rate from doubling time
3. Estimate the year when doubling will occur

#### Worked example 1: growth rate to doubling time

A population grows at **1.4% per year**. How long will it take to double?

Step 1: Write the formula

$$
\text{Doubling time} = \frac{70}{\text{growth rate (\%)}}
$$

Step 2: Substitute the value

$$
\text{Doubling time} = \frac{70}{1.4}
$$

Step 3: Calculate

$$
\text{Doubling time} = 50 \text{ years}
$$

So the population will double in about **50 years**.

#### Worked example 2: doubling time to growth rate

A country is expected to double in **35 years**. What is its annual growth rate?

Rearrange the formula:

$$
\text{Growth rate (\%)} = \frac{70}{\text{doubling time}}
$$

Substitute:

$$
\text{Growth rate} = \frac{70}{35} = 2\%
$$

So the annual growth rate is **2%**.

#### Worked example 3: estimate a future year

A country’s growth rate is **1.88%** in **2019**. When will its population roughly double?

Step 1: Find doubling time

$$
\frac{70}{1.88} \approx 37 \text{ years}
$$

Step 2: Add the doubling time to the starting year

$$
2019 + 37 = 2056
$$

So the population would be expected to double around **2056**.

This type of question is very APES-friendly because it combines math with interpretation. You are not just finding a number; you are placing that number in a demographic timeline.

> **Test Tip:** When asked for the doubling **year**, do the math in two stages: first calculate **years to double**, then add that value to the starting year. This reduces careless errors.

### Common mistakes and what the Rule of 70 does not tell you

Students often lose easy marks on this topic because the math is simple but the setup is easy to mishandle.

#### Mistake 1: using the decimal form instead of the percent form

This is the biggest error.

If the growth rate is **2%**, the Rule of 70 uses:

$$
70 \div 2
$$

not

$$
70 \div 0.02
$$

Using **0.02** would give **3500**, which is wildly wrong.

The Rule of 70 expects the growth rate as a **whole-number percent**.

#### Mistake 2: applying it to zero or negative growth

If the population growth rate is **0%**, the population is not doubling. If growth is negative, the population is shrinking, not doubling. The Rule of 70 is not meant for those cases.

#### Mistake 3: thinking it predicts unlimited future growth

The Rule of 70 estimates doubling under constant exponential growth. But real populations do not keep doubling forever. As population size rises, limits begin to matter:

- food availability
- freshwater supply
- space
- disease spread
- waste accumulation
- social and economic pressures

These factors can slow growth and shift a population away from exponential increase toward slower or even stable growth.

> **Key Term: carrying capacity**
> The maximum population size that an environment can support sustainably with available resources.

This is why the Rule of 70 is powerful but limited. It is a quick estimate, not a guarantee. It tells you what would happen **if** the current percentage growth continued.

It also does **not** tell you:

- the exact future population number
- whether the growth rate will stay the same
- anything directly about **total fertility rate**
- whether a country is developed or developing
- whether the population is near carrying capacity

That last point matters. A country can have a high growth rate today and still slow sharply later if death rates, fertility, migration, or policy change.

There are also more exact mathematical formulas for doubling time. For continuous exponential growth:

$$
T = \frac{\ln(2)}{r}
$$

where $r$ is the growth rate as a decimal. For discrete growth:

$$
T = \frac{\log(2)}{\log(1+r)}
$$

You do not usually need these in APES. The important distinction is that the Rule of 70 is an **approximation**, while these are more exact.

### Why the Rule of 70 matters in human population dynamics

The Rule of 70 is not just a calculator trick. It helps explain why population growth can create environmental pressure so quickly.

Suppose Country A grows at **4%** per year and Country B grows at **1%** per year.

- Country A doubling time:

$$
70 \div 4 = 17.5 \text{ years}
$$

- Country B doubling time:

$$
70 \div 1 = 70 \text{ years}
$$

That means Country A doubles four times faster. Even if both countries begin with similar populations, the environmental consequences diverge very quickly.

A rapidly doubling human population may increase demand for:

- cropland
- irrigation water
- housing
- roads
- energy
- sewage treatment
- waste disposal

This can intensify deforestation, groundwater depletion, air pollution, eutrophication, and habitat loss. That is why APES connects human population growth to sustainability.

The Rule of 70 also fits closely with the **demographic transition model**. In transitional and early industrial phases, death rates often fall before birth rates do. That creates a high growth rate, which produces short doubling times. Later, as education for women, urbanization, and access to family planning increase, birth rates often decline. Growth rates fall, and doubling times get longer.

> **Key Term: demographic transition**
> The shift from high birth and death rates to lower birth and death rates as a country becomes more industrialized.

You should also connect the Rule of 70 to **population momentum**. Even if fertility falls, a country with many young people may continue growing because a large share of the population is entering reproductive ages. That can keep growth rates positive for decades, though usually at a slower pace.

Another useful interpretation is global growth trends. If the global human population growth rate declines from, say, **1.8%** to **1.2%**, the population may still be growing, but the doubling time becomes longer.

- At 1.8%:

$$
70 \div 1.8 \approx 39 \text{ years}
$$

- At 1.2%:

$$
70 \div 1.2 \approx 58 \text{ years}
$$

So a falling growth rate means slower doubling, not necessarily immediate population decline.

This is exactly the kind of reasoning APES rewards: not just doing the formula, but explaining what the answer means for environmental systems.

## Key Point Checklist

_This article has covered the following key knowledge points:_

- The **Rule of 70** estimates how long it takes a population to double under roughly constant exponential growth
- The formula is **doubling time ≈ 70 ÷ growth rate (%)**
- The growth rate must be entered as a **percent**, not a decimal
- A population growing at **2%** per year doubles in about **35 years**
- If a doubling time is **35 years**, the growth rate is about **2%**
- The Rule of 70 can be used to estimate a future doubling year by adding the doubling time to a starting year
- Higher growth rates produce **shorter doubling times**
- The Rule of 70 applies to **exponential**, not linear, growth
- It is an **approximation**, not an exact prediction
- Real populations cannot double forever because of **carrying capacity** and limiting factors
- Short doubling times are often associated with rapid growth in earlier stages of demographic transition
- Falling growth rates increase doubling time, even if population size is still rising

## Key Terms and Concepts

- doubling time
- Rule of 70
- exponential growth
- carrying capacity
- demographic transition


Human Population Dynamics - Rule of 70