Doubling Time, Rule of 70 — AP® Environmental Science Notes

Some quantities grow by the same amount each time: a savings jar with $10 added every month. Others grow by the same percent each year: a population, energy use, or money earning interest. Percent growth speeds up, because each year's percent is taken of a bigger number. The question the exam asks most about it is "how long until it doubles?", and the rule of 70 answers it with one division.

Growth rate

A growth rate is a percent change per unit of time. A population growing 2% a year adds 2 people for every 100 each year. Start with 1,000: after one year there are 1,020; after two, $1{,}020 \times 1.02 = 1{,}040.4$ ; the yearly increase keeps getting larger.

The rule of 70

For anything growing at a steady percent rate, the doubling time is about

$\text{doubling time} \approx \frac{70}{\text{growth rate in \%}}$

Use the rate as a percent: for 2%, divide by 2, not by 0.02. (The 70 comes from the math of compound growth; it is accurate for the small rates you will see.)

A curve from invented data: a population of 1 million in 2000 grows 2 percent a year. By the rule of 70 the doubling time is 70 divided by 2 = 35 years, so it reaches about 2 million in 2035 and about 4 million in 2070. The curve gets steeper as the population grows. — **Figure 1.** At 2% a year, 1 million becomes about 2 million in 35 years and about 4 million in 70. LevlPrep original diagram.

Worked example: the doubling time. A population grows 2% a year.

Setup: $70 / 2$

Answer: 35 years.

"In what year?"

Readers report that many students find the doubling time and stop, when the question asked for a year. The doubling time is a length of time; the year is the start year plus that time.

Worked example. A town of 50,000 people in 2024 grows 2.5% a year. In what year will it reach 100,000?

Step 1, doubling time: $70 / 2.5 = 28$ years.

Step 2, the year: $2024 + 28 = 2052$ .

Projecting several doublings

Each doubling multiplies by 2. After 2 doublings the population is 4 times as large; after 3, 8 times; after 4, 16 times.

Worked example. A population of 6 million in 2020 grows 3.5% a year. About how large will it be in 2060?

Doubling time: $70 / 3.5 = 20$ years. From 2020 to 2060 is 40 years, which is $40 / 20 = 2$ doublings.

Size: $6 \times 2 \times 2 = 24$ million (2.4 × 10⁷ = 24,000,000).

Working backward

The rule also runs the other way: growth rate ≈ 70 ÷ doubling time. Algae that double every 14 days are growing about $70 / 14 = 5\%$ per day.

Growth rate (% per year)	0.5	1	2	3.5	7
Doubling time (years)	140	70	35	20	10

Notice two things. Doubling the rate halves the doubling time. And the starting size does not matter: a city of 90,000 growing 7% doubles faster than a city of 2 million growing 0.5%.

Exam habits

Write the setup ( $70 / 2.5$ ), then the answer with "years".
Read the question to the end: a year, a number of years, or a population size?
Keep the scale: "24 million", not "24".