Real-World Mathematics: Population, Demography, Growth and Dependency Ratios

Application of Mathematics in Real-World Usage · Guide 21 · BTT Mathematics Hub

A population can grow even when births fall. A city can become denser without gaining land. The dependency ratio can rise even if total population is stable. A percentage increase that looks small for one year can become large when compounded for decades. Demography is therefore not simply “counting people”; it is a system of stocks, flows, rates, age structures and carefully chosen denominators.

This guide uses fictional populations, age groups, migration flows and growth scenarios for Mathematics teaching. It does not predict any country, city or ethnic group, and it should not be used for political, immigration, pension or public-policy decisions. Published demographic definitions are linked where useful, but every worked case below is deliberately constructed so the arithmetic can be checked.

The core discipline is to keep stock and flow separate. Population at a point in time is a stock. Births, deaths, immigration and emigration are flows during an interval. Growth rates summarise how the stock changes relative to a chosen base. Age ratios compare subgroups inside the stock.

Population change begins with a conservation equation

For a defined region and time interval, a simple balance is:

Ending population = beginning population + births − deaths + immigration − emigration.

Suppose a fictional town begins with 100,000 residents. During one year it records 1,200 births, 800 deaths, 2,000 immigrants and 1,500 emigrants. Ending population is 100,000 + 1,200 − 800 + 2,000 − 1,500 = 100,900.

The change of 900 can be decomposed into natural increase of 400 and net migration of 500. That decomposition matters because the same net growth can arise from very different underlying flows.

Growth rate uses the starting population as a base

If the town rises from 100,000 to 100,900, the one-year growth rate is 900/100,000 = 0.9%.

If another town grows from 10,000 to 10,500, its absolute increase is only 500 but its rate is 5%. Absolute change and relative change answer different questions.

The United Nations glossary defines population growth rate as the increase or decrease during a period expressed relative to population size at the beginning of that period. The denominator belongs to the definition.

Natural increase is not total growth when migration exists

Suppose a region has 5,000 births and 4,000 deaths, giving natural increase of 1,000. If net migration is −1,400, total change is −400.

The population declines despite births exceeding deaths. Conversely, net migration can produce overall growth when deaths exceed births.

A report that uses only birth and death rates therefore cannot explain total population change unless migration is known to be negligible or explicitly excluded.

Crude rates use population exposure as a denominator

Suppose 900 births occur in a population of 60,000 during a year. A crude birth rate expressed per 1,000 population is 900/60,000×1,000 = 15 per 1,000.

If there are 480 deaths, crude death rate is 8 per 1,000. Their difference is 7 per 1,000, or 0.7%, before migration.

These are crude rates because the denominator is the whole population rather than the subgroup actually exposed to the event. They are useful summaries but should not be confused with age-specific or fertility measures.

Population density is population divided by area

A fictional district has 240,000 residents across 120 km². Density is 2,000 persons/km².

If population grows to 264,000 while area stays fixed, density becomes 2,200 persons/km², a 10% increase. If the administrative boundary expands to 132 km² at the same time, density returns to 2,000.

Density therefore depends on both numerator and geographic denominator. A change in boundary can alter density without any household moving.

Population share is different from population count

Suppose Group A has 30,000 people in a total population of 100,000, so its share is 30%. Later A grows to 33,000 while total population grows to 120,000. A increased by 10% in count, but its share fell to 27.5%.

A falling share does not necessarily mean a falling count. A rising count does not necessarily mean a rising share. The denominator is changing.

This distinction is central to age-structure analysis, migration shares and household composition.

Age structure is a distribution, not one average age

Consider a fictional population of 10,000 split into 2,000 aged 0–19, 5,500 aged 20–64 and 2,500 aged 65+. Those three counts sum to the full population.

The shares are 20%, 55% and 25%. A single mean age cannot reconstruct this structure. Two populations can have the same mean age while having very different numbers of children and older adults.

For many planning questions, the distribution across age bands is more informative than a single centre statistic.

Dependency ratios depend on the age convention chosen

The United Nations World Population Ageing glossary uses one convention in which total dependency ratio is the number under age 20 plus the number aged 65 or over per 100 people aged 20–64. Other statistical systems use different age boundaries. Therefore the age bands must be stated before comparing ratios.

Using the fictional 2,000 / 5,500 / 2,500 counts under the UN-style 0–19, 20–64 and 65+ grouping, total dependency ratio is (2,000+2,500)/5,500×100 ≈ 81.82 per 100 working-age persons.

Child dependency ratio is 2,000/5,500×100 ≈ 36.36. Old-age dependency ratio is 2,500/5,500×100 ≈ 45.45. Their sum gives the total 81.82.

A dependency ratio is not a literal count of economic dependants

Some people aged 20–64 are not employed. Some younger or older people are economically active. The age-based ratio is therefore a demographic structure measure, not a direct count of workers supporting non-workers.

This matters when interpreting the number. A ratio of 80 per 100 does not mean exactly eighty people are financially dependent on exactly one hundred workers.

The mathematical ratio is still useful as long as its operational definition remains visible.

Changing one age group can move the ratio nonlinearly

Keep 2,000 children and 2,500 older adults fixed, but reduce the 20–64 group from 5,500 to 5,000. Total dependency ratio becomes 4,500/5,000×100 = 90.

The dependent-age count did not change. The ratio rose because the denominator fell.

This is another reason to report underlying counts alongside ratios whenever possible.

Support ratios are reciprocals with a different interpretation

If total dependency ratio is 81.82 dependants-age persons per 100 working-age persons, the corresponding working-age-to-dependent-age count ratio is 5,500/4,500 ≈ 1.222.

This says there are about 1.22 working-age persons for each person in the two dependent-age bands under the defined grouping.

Reciprocals preserve the same counts but reverse the comparison direction. A large dependency ratio corresponds to a smaller support ratio.

Compound growth is exponential when the rate is constant

If population P grows by constant proportional rate r each period, the model is Pₙ=P₀(1+r)ⁿ.

Starting from 100,000 at 2% annual growth for 10 years gives 100,000×1.02¹⁰≈121,899. The total increase is about 21.9%, not 20%, because each year’s growth is applied to a changing base.

This is a hypothetical mathematical model. Real population growth rates change over time as fertility, mortality, migration and age structure change.

Linear and exponential projections answer different assumptions

Suppose a population increases by 2,000 people per year from 100,000. A linear projection after 10 years gives 120,000.

A constant 2% exponential model gives about 121,899. The gap appears because linear growth adds the same absolute number, while exponential growth adds the same percentage.

Neither model is automatically better. The correct model depends on the mechanism and time horizon. A projection is a conditional consequence of assumptions, not a guaranteed future.

Average annual growth over several years uses a root

A population rises from 100,000 to 121,000 over five years. The total increase is 21%, but the compound annual growth rate is (121,000/100,000)^(1/5)−1≈3.89%.

Dividing 21% by five gives 4.2%, which is a different arithmetic average and does not reproduce the final population under compounding.

The compound rate is the constant annual multiplier that would connect the endpoints exactly.

Doubling time follows logarithms in the constant-rate model

For constant annual rate r, doubling time n satisfies (1+r)ⁿ=2, so n=ln2/ln(1+r).

At 2% annual growth, n≈0.6931/ln1.02≈35.0 years. The familiar “rule of 70” gives roughly 70/2=35 years, close because the rate is small.

The approximation weakens as rates become larger, and real demographic growth seldom remains constant for an entire doubling period.

Cohort movement is a transition problem

Imagine 1,000 people aged 20–24 at the start of a five-year period. Under a fictional survival fraction of 0.995 and net migration of +50 for that cohort, the population entering age 25–29 at the end is approximately 1,000×0.995+50=1,045.

This is not the same as applying the whole-population growth rate to every age group. Cohorts move through age bands and can experience different mortality and migration patterns.

Advanced demographic models use age-specific transition structures for this reason. The simple example shows how a stock can move from one category into another over time.

A stable total population can hide large internal flows

Suppose a region begins and ends the year at exactly 1,000,000 people. During the year it has 12,000 births, 10,000 deaths, 40,000 immigrants and 42,000 emigrants.

Net change is zero, but 104,000 demographic events occurred across the four flow categories. “No population growth” does not mean “nothing changed.”

Stock stability can coexist with high turnover. This distinction appears in inventory, bank balances and reservoirs as well as demography.

Ageing can be represented by shifting shares

A fictional population keeps the same total of 10,000 but changes from 25% aged 65+ to 30%. The older group rises from 2,500 to 3,000.

If the 20–64 group falls from 5,500 to 5,000 while the child group remains 2,000, total dependency ratio rises from 81.82 to 100.

The total population did not change, but age structure changed enough to alter the ratio materially.

Population pyramids are area-like visual summaries of age shares

A population pyramid typically displays age bands along one axis and male/female counts or percentages across the other. Bar length represents the amount in each age band.

If one chart uses counts and another uses percentages, equal-looking bars do not necessarily represent equal numbers of people. The scale and denominator must be checked before comparison.

Visualisation is useful because age structure is multidimensional, but the graph inherits every definition used to construct it.

Median age and dependency ratio can move differently

Median age is the age that splits the population so half are younger and half older. Dependency ratio uses fixed age bands.

A shift concentrated within the 20–64 band can raise the median age without changing dependency ratio much. A shift from age 64 to 65 can alter the dependency ratio under the chosen boundary even if median age barely changes.

Different demographic indicators compress different aspects of the same age distribution.

Forecast ranges should not be mistaken for certainties

Suppose a teaching model starts at 500,000 and examines annual growth rates from 0.5% to 1.5% for 20 years. The low scenario gives 500,000×1.005²⁰≈552,448. The high gives 500,000×1.015²⁰≈673,428.

The interval comes from chosen scenario assumptions. It is not automatically a statistical confidence interval.

A projection should therefore say “under these rates” rather than “the population will be between these numbers.”

Per-capita quantities can rise while totals fall, or vice versa

Suppose total resource use falls from 1,000 units to 960 while population falls from 100 to 90. Per-capita use rises from 10 to 10.667.

Total use improved downward, but average per person increased. The two indicators answer different questions.

Demography is often the denominator of other systems: energy, housing, transport, schools and healthcare. A changing denominator can alter an apparently stable per-capita measure.

A complete demographic report states its boundaries

For every population ratio, report geography, date, age bands and denominator. For every growth rate, state the interval and whether migration is included. For every projection, state the assumed mechanism and whether the rate is fixed or changing.

Then distinguish observed counts from modelled estimates. A census-style count, survey estimate and projection can all produce population numbers, but they have different evidential status.

The mathematical habit is the same as elsewhere in the real-world series: preserve the identity of the quantity while transforming it.

Practice: twenty demography questions

1. A population starts at 80,000 with 1,000 births, 600 deaths, 1,200 immigrants and 900 emigrants. Find ending population.
2. Find the net change in question 1.
3. Find the annual growth rate relative to starting population.
4. A region of 50 km² has 125,000 people. Find density.
5. Population rises 10% while area stays constant. What happens to density?
6. A subgroup rises from 20,000 to 22,000 while total population rises from 50,000 to 60,000. Find old and new shares.
7. Using 2,000 aged 0–19, 5,500 aged 20–64 and 2,500 aged 65+, find total dependency ratio per 100 working-age people.
8. Find child dependency ratio for question 7.
9. Find old-age dependency ratio for question 7.
10. If the working-age group falls to 5,000 while the other groups stay fixed, find the new total dependency ratio.
11. Find population after 10 years from 100,000 at constant 2% annual growth.
12. Find the linear 10-year result if 2,000 people are added each year instead.
13. A population rises from 100,000 to 121,000 in five years. Find CAGR.
14. Estimate doubling time at 2% using ln2/ln1.02.
15. A cohort of 1,000 has survival fraction 0.995 and net migration +50. Find the next-cohort count.
16. Starting and ending population are equal, with births 12,000, deaths 10,000 and immigration 40,000. Find emigration required.
17. Find low 20-year scenario from 500,000 at 0.5% annual growth.
18. Find high 20-year scenario at 1.5%.
19. Total resource use falls from 1,000 to 960 while population falls from 100 to 90. Find old and new per-capita use.
20. Why must dependency-ratio age bands be stated explicitly?

Worked answers

1. 80,700. 80,000+1,000−600+1,200−900.
2. +700.
3. 0.875%. 700/80,000×100.
4. 2,500 persons/km².
5. It rises 10%. Density is proportional to population when area is fixed.
6. 40% then 36.67%. The count rose while share fell.
7. About 81.82 per 100.
8. About 36.36.
9. About 45.45.
10. 90 per 100.
11. About 121,899.
12. 120,000.
13. About 3.89% per year.
14. About 35.0 years.
15. 1,045.
16. 42,000. Net change zero requires 12,000−10,000+40,000−E=0.
17. About 552,448.
18. About 673,428.
19. 10 then about 10.667 units/person.
20. Because different statistical systems use different age boundaries. The numerical ratio depends directly on which ages enter the numerator and denominator.

Sources and connected applications

For demographic definitions used here, see the United Nations World Population Ageing glossary. Its dependency-ratio convention is stated explicitly on this page because other datasets may use different age boundaries.

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