Study the schematic representation given above and answer the following questions : (a) Identify A in it. (b)

Question

Q. Study the schematic representation given above and answer the following questions :

(a) Identify A in it.

(b) Identify D in it.

(c) When the population density at time t is N as shown above, write the population density at time t + 1 in the form of an equation using appropriate symbols.

CBSECBSE Class XII Board 2020Subjective· 3mImportance★★★★★

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Concept understanding — Doubling Time

Doubling Time: A First Look

Imagine you put a small sum of money in a fixed deposit that earns compound interest. You don't touch it for years. One day, you check the balance and realise it has become exactly twice what you originally deposited. The time it took for that to happen — from the day you deposited to the day the amount doubled — is the doubling time.

That is the core idea. Doubling time is the period required for a quantity — money, population, price level, or any growing figure — to become double its original size, assuming it grows at a constant rate.

The everyday intuition

We all have a rough sense of this. If something grows slowly, it takes a long time to double. If it grows fast, it doubles quickly. A savings account earning 4% per year will take many years to double. A startup that doubles its revenue every year is growing explosively. Doubling time is simply a way to translate a growth rate into a concrete, easy-to-grasp time frame.

Why it matters in commerce and humanities

In your NCERT textbooks for subjects like Economics or Business Studies, doubling time appears in contexts where you study growth — of national income, population, or prices (inflation). It helps you compare different growth rates without staring at percentages.

Consider these examples:

Population growth: If a country's population grows at 2% per year, its doubling time tells you roughly how many years it will take for the population to become twice as large. This matters for planning schools, hospitals, and jobs.
Inflation: If prices rise at a steady rate, doubling time tells you how long it will take for the cost of living to double. That is crucial for understanding the erosion of purchasing power.
Economic growth: If a nation's GDP grows at a certain rate, doubling time shows how quickly the economy can double its output — a key measure of development.

Important

Doubling time is not a prediction. It assumes the growth rate stays constant, which rarely happens in real life. It is a tool for comparison and rough estimation, not a precise forecast.

The key insight without formulas

The relationship is simple: the higher the growth rate, the shorter the doubling time. A 1% growth rate gives a very long doubling time; a 10% growth rate gives a very short one. The two are inversely related — double the rate, and you roughly halve the time.

Your NCERT textbook will not ask you to calculate doubling time in a prose subject. Instead, it expects you to understand the concept: that a small difference in growth rates can lead to dramatically different outcomes over time, and that doubling time is a vivid way to see that difference.

A few bullet points to remember

Doubling time applies to any quantity that grows at a constant percentage rate — money, population, prices, output.
It is a time measure, not a rate measure. It answers "how long?" not "how fast?"
It is most useful for comparing growth scenarios: a country with a 3% growth rate will double its economy much sooner than one with a 1% growth rate. …

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This question is about a simple population growth model. Based on the standard schematic used for this topic, A most commonly represents the per-capita birth rate and D the per-capita death rate; the population density at time t+1 is N(t+1) = N + (B − D), where B and D are the total births and deaths in the interval. Honest gap: the original printed diagram for this specific paper was not available to us, so the exact identity of A and D could not be independently confirmed against the source figure — the interpretation below is the standard, widely-used one for this type of diagram.

Let's unpack this step by step. The schematic accompanying this question is a standard representation of population growth: a box labelled "Population density at time t" with value N, with arrows in and out representing the factors that change population size. This answer follows the standard version of that diagram (births and deaths only, ignoring migration), since the original printed figure for this exact paper is not available in our source material to confirm the precise letter-to-structure mapping.

The core idea is that population change over one time step is simply:

New population = Old population + Additions − Subtractions.

Identify A — the per capita birth rate

In population ecology, the symbol A in such diagrams most commonly stands for the per capita birth rate, often denoted b. This is the average number of offspring produced per individual per unit time. It is a rate, not a total count.

So, if the population density at time t is N, the total number of births in the interval from t to t+1 is bN.
Identify D — the per capita death rate

Similarly, D most commonly represents the per capita death rate, often denoted d. This is the probability (or average number) of deaths per individual per unit time. The total number of deaths in the interval is dN.

Watch out

A common mistake is to confuse A and D with total births and total deaths. They are per capita rates, not absolute numbers. The total births are bN, not just b.
Write the equation for population density at time t+1

Starting with N individuals at time t, we add the total births and subtract the total deaths (ignoring migration for this simple model).

So:

N(t+1) = N + bN − dN

Factorising:

N(t+1) = N(1 + b − d) …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2020Set 57/2/13 marks
$Q. Study the schematic representation given above and answer the following questions :$
(a) Identify A in it.
(b) Identify D in it.
(c) When the population density at time t is N as shown above, write the population density at time t + 1 in the form of an equation using appropriate symbols.
›Reveal solutionSolution
This question is about a simple population growth model. Based on the standard schematic used for this topic, A most commonly represents the per-capita birth rate and D the per-capita death rate; the population density at time t+1 is N(t+1) = N + (B − D), where B and D are the total births and deaths in the interval. Honest gap: the original printed diagram for this specific paper was not available to us, so the exact identity of A and D could not be independently confirmed against the source figure — the interpretation below is the standard, widely-used one for this type of diagram.

Let's unpack this step by step. The schematic accompanying this question is a standard representation of population growth: a box labelled "Population density at time t" with value N, with arrows in and out representing the factors that change population size. This answer follows the standard version of that diagram (births and deaths only, ignoring migration), since the original printed figure for this exact paper is not available in our source material to confirm the precise letter-to-structure mapping.

The core idea is that population change over one time step is simply:

New population = Old population + Additions − Subtractions.

Identify A — the per capita birth rate

In population ecology, the symbol A in such diagrams most commonly stands for the per capita birth rate, often denoted b. This is the average number of offspring produced per individual per unit time. It is a rate, not a total count.

So, if the population density at time t is N, the total number of births in the interval from t to t+1 is bN.

Identify D — the per capita death rate

Similarly, D most commonly represents the per capita death rate, often denoted d. This is the probability (or average number) of deaths per individual per unit time. The total number of deaths in the interval is dN.

Watch out

A common mistake is to confuse A and D with total births and total deaths. They are per capita rates, not absolute numbers. The total births are bN, not just b.

Write the equation for population density at time t+1

Starting with N individuals at time t, we add the total births and subtract the total deaths (ignoring migration for this simple model).

So:

N(t+1) = N + bN − dN

Factorising:

N(t+1) = N(1 + b − d) …
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Q.Study the schematic representation given above and answer the following questions :

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