The World's Population | 世界人口

📚 The World’s Population | 世界人口

The world’s population is one of the richest real-world data sets available for mathematical study. For IGCSE students following the Edexcel syllabus, population figures provide an excellent context for practising percentages, standard form, averages, ratio, and exponential growth. Every technique used to analyse population data is a core skill that appears directly in your exam.

世界人口是可用于数学研究的最丰富的真实数据集之一。对于学习 Edexcel 课程的 IGCSE 学生来说，人口数据为练习百分比、标准形式、平均数、比和指数增长提供了绝佳背景。分析人口数据所用的每一技巧，都是考试中直接出现的核心技能。

1. Reading Population Data | 读取人口数据

Population data is usually presented in tables, bar charts, line graphs or pie charts. In the exam, you may be asked to read values directly from a table, or compare the populations of two countries using a graph. Always look carefully at the units — values may be given in millions, billions, or as figures like 1.43 × 10⁹.

人口数据通常以表格、柱状图、折线图或饼图呈现。在考试中，你可能被要求直接从表格中读取数值，或用图表比较两个国家的人口。务必仔细查看单位——数值可能以百万、十亿给出，或如 1.43 × 10⁹ 的形式。

Consider the following approximate population data for 2023: India 1.43 billion, China 1.41 billion, the United States 340 million, the United Kingdom 68 million. To compare these values fairly, convert them all to the same unit — for example, millions: India 1430, China 1410, USA 340, UK 68.

考虑 2023 年以下近似人口数据：印度 14.3 亿、中国 14.1 亿、美国 3.4 亿、英国 6800 万。要公平比较这些数值，请将它们都转换为相同单位——例如百万：印度 1430、中国 1410、美国 340、英国 68。

Always note the scale on the vertical axis before reading a bar chart or line graph. | 读取柱状图或折线图前，务必注意纵轴的刻度。
Check whether a graph starts at zero — a non-zero start can exaggerate differences. | 检查图表是否从零开始——非零起点可能夸大差异。
In pie charts, calculate the angle for each category using: category total × 360°. | 在饼图中，用”分类数值 ÷ 总数 × 360°”计算每个类别的角度。

2. Percentage Change and Growth Rate | 百分比变化与增长率

Population growth is often expressed as a percentage increase. The formula for percentage change is:

Percentage change = (new value − original value) ÷ original value × 100%

The world’s population was approximately 6.1 billion in the year 2000 and approximately 8.0 billion in 2023. The percentage increase is (8.0 − 6.1) ÷ 6.1 × 100% = 1.9 ÷ 6.1 × 100% ≈ 31.1%. This tells us the population grew by about 31% over 23 years.

世界人口在 2000 年约为 61 亿，在 2023 年约为 80 亿。百分比增长为 (8.0 − 6.1) ÷ 6.1 × 100% = 1.9 ÷ 6.1 × 100% ≈ 31.1%。这告诉我们 23 年间人口增长了约 31%。

You should also be able to work backwards. If a country’s population increases by 4% in one year and ends at 52 million, then the original population is 52 ÷ 1.04 ≈ 50 million. Notice that a 4% increase means multiplying by the decimal multiplier 1.04.

你还应该能够反向计算。如果一个国家的人口在一年内增长 4%，最终为 5200 万，则原人口为 52 ÷ 1.04 ≈ 5000 万。注意 4% 的增长意味着乘以小数乘数 1.04。

3. Population Density and Ratio | 人口密度与比

Population density measures how many people live per unit area. It is calculated using the formula:

Population density = total population ÷ total land area

For example, if a country has a population of 240 million and an area of 2 million km², the population density is 240,000,000 ÷ 2,000,000 = 120 people per km². This is also a ratio — comparing population to area as 120 : 1.

例如，如果一个国家有 2.4 亿人口和 200 万平方公里的面积，则人口密度为 240,000,000 ÷ 2,000,000 = 每平方公里 120 人。这也是一个比——将人口与面积比较为 120 : 1。

Country \| 国家	Population (millions) \| 人口（百万）	Area (thousand km²) \| 面积（千平方公里）	Density (people/km²) \| 密度（人/平方公里）
Bangladesh \| 孟加拉国	171	148	≈ 1155
India \| 印度	1430	3287	≈ 435
Australia \| 澳大利亚	26	7692	≈ 3.4
World \| 世界	8000	51000 (land ≈ 149000 is wrong; land is 149 million km², use 51000 for land?) — use 149000 for land.	≈ 54

Wait — the world’s land area is approximately 149 million km², so the world population density is 8,000,000,000 ÷ 149,000,000 ≈ 54 people per km². When solving ratio problems involving population, set up the two quantities as a fraction and simplify fully.

等等——世界的陆地面积约为 1.49 亿平方公里，因此世界人口密度为 8,000,000,000 ÷ 149,000,000 ≈ 每平方公里 54 人。求解涉及人口的比的问题时，将两个量写成分数并完全化简。

4. Standard Form with Large Numbers | 大数的标准形式

World population figures are enormous, so standard form is essential. A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. The world population of 8 billion is written as 8 × 10⁹.

世界人口数字巨大，因此标准形式至关重要。标准形式的数写作 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。80 亿的世界人口写作 8 × 10⁹。

8,000,000,000 = 8 × 10⁹ 340,000,000 = 3.4 × 10⁸ 68,000,000 = 6.8 × 10⁷

You may need to multiply or divide populations in standard form. For example, if one country has a population of 8 × 10⁷ and another has 2 × 10⁸, the total is (0.8 × 10⁸) + (2 × 10⁸) = 2.8 × 10⁸. When dividing, subtract the powers: (8 × 10⁹) ÷ (4 × 10⁶) = 2 × 10³.

你可能需要对标准形式的人口进行乘除。例如，如果一个国家的人口为 8 × 10⁷，另一个为 2 × 10⁸，则总和为 (0.8 × 10⁸) + (2 × 10⁸) = 2.8 × 10⁸。除法时，指数相减：(8 × 10⁹) ÷ (4 × 10⁶) = 2 × 10³。

5. Averages in Population Statistics | 人口统计中的平均数

Three averages are used in population studies: the mean, the median and the mode. The mean is the sum of all values divided by the number of values. Suppose the birth rates of five countries are 15, 18, 21, 24 and 32 per thousand. The mean is (15 + 18 + 21 + 24 + 32) ÷ 5 = 110 ÷ 5 = 22 per thousand.

人口研究中使用三种平均数：均值、中位数和众数。均值是所有数值之和除以数值个数。假设五个国家的出生率为每千人 15、18、21、24 和 32。均值为 (15 + 18 + 21 + 24 + 32) ÷ 5 = 110 ÷ 5 = 每千人 22。

The median is the middle value when the data is arranged in order. For the same data, ordered as 15, 18, 21, 24, 32, the median is 21. The mode is the most frequent value — if no value repeats, there is no mode. In grouped frequency tables, you estimate the mean using mid-interval values, and you find the modal class by identifying the interval with the highest frequency.

中位数是将数据按顺序排列后的中间值。对于相同数据，排序为 15、18、21、24、32，中位数为 21。众数是出现次数最多的值——如果没有重复值，则没有众数。在分组频数表中，使用组中值估算均值，并通过识别频数最高的区间找到众数类。

For a grouped table of ages with intervals 0–9, 10–19, 20–29, the midpoints are 4.5, 14.5 and 24.5. Multiply each midpoint by its frequency, sum the products, and divide by the total frequency.

对于年龄分组表，区间 0–9、10–19、20–29 的组中值分别为 4.5、14.5 和 24.5。将每个组中值乘以其频数，将乘积求和，再除以总频数。

6. Exponential Growth Model | 指数增长模型

Populations often grow by a constant percentage each year, which is called exponential growth. The formula used is:

Pₙ = P₀ × (1 + r)ⁿ

Here P₀ is the initial population, r is the annual growth rate expressed as a decimal, n is the number of years, and Pₙ is the population after n years. If the world population is 8 × 10⁹ and grows at 1.1% per year (r = 0.011), then after 10 years:

其中 P₀ 是初始人口，r 是以小数表示的年增长率，n 是年数，Pₙ 是 n 年后的人口。如果世界人口为 8 × 10⁹ 且每年增长 1.1%（r = 0.011），则 10 年后：

P₍₁₀₎ = 8 × 10⁹ × (1.011)¹⁰ ≈ 8 × 10⁹ × 1.116 ≈ 8.93 × 10⁹

So the population would be approximately 8.93 billion after 10 years. Note that exponential growth is not linear — the increase in the second year is larger than the increase in the first year, because 1.1% of a larger base is a larger amount.

因此 10 年后人口约为 89.3 亿。注意指数增长不是线性的——第二年的增量大于第一年的增量，因为较大基数的 1.1% 是较大的量。

7. Linear vs Exponential Growth | 线性增长与指数增长

It is important to distinguish linear and exponential growth. Linear growth adds a constant amount each year, while exponential growth multiplies by a constant factor each year. If a population of 10 million grows linearly by 200,000 per year, after 10 years it is 10 + 0.2 × 10 = 12 million.

区分线性和指数增长非常重要。线性增长每年增加固定数量，而指数增长每年乘以固定因子。如果一个人口为 1000 万的国家每年线性增长 20 万，10 年后人口为 1000 + 20 × 10 = 1200 万。

If the same population grows exponentially at 2% per year, after 10 years it is 10 × (1.02)¹⁰ ≈ 12.19 million. The difference is small at first, but over 50 years the linear model gives 10 + 0.2 × 50 = 20 million, while the exponential model gives 10 × (1.02)⁵⁰ ≈ 26.9 million.

如果同一人口以每年 2% 指数增长，10 年后为 10 × (1.02)¹⁰ ≈ 1219 万。起初差异很小，但 50 年后，线性模型给出 10 + 0.2 × 50 = 2000 万，而指数模型给出 10 × (1.02)⁵⁰ ≈ 2690 万。

In the exam, you may be asked to complete a table of values for both models and then plot them on the same graph. The linear model produces a straight line; the exponential model produces a curve that becomes steeper over time.

在考试中，你可能被要求完成两种模型的数值表，然后在同一张图上绘制。线性模型产生直线；指数模型产生随时间变得越来越陡的曲线。

8. Drawing and Interpreting Population Graphs | 绘制与解读人口图表

Scatter graphs are used to explore the relationship between population and other variables such as a country’s GDP, life expectancy or land area. You plot one variable on each axis and look for correlation. The line of best fit can be used to estimate missing values.

散点图用于探索人口与 GDP、预期寿命或土地面积等其他变量之间的关系。你在每个轴上绘制一个变量并寻找相关性。最佳拟合线可用于估计缺失值。

When drawing a scatter graph, always choose a suitable scale so that the points fill at least half of the grid. Use a ruler to draw the line of best fit, ensuring that approximately half of the points lie above the line and half below. Interpolation means estimating within the range of the data; extrapolation means estimating beyond the range — extrapolation is less reliable.

绘制散点图时，务必选择合适的刻度，使点至少占据网格的一半。用直尺绘制最佳拟合线，确保大约一半的点在线以上、一半在线以下。插值是在数据范围内进行估计；外推是在范围之外进行估计——外推的可靠性较低。

Population line graphs show change over time. The horizontal axis is the year and the vertical axis is the population. The gradient of the graph at any point represents the rate of growth — a steeper gradient means faster growth.

人口折线图显示随时间的变化。横轴是年份，纵轴是人口。图上任意点的梯度代表增长率——梯度越陡，增长越快。

9. Using Formulae to Predict Population | 用公式预测人口

Real population data can be modelled with a formula, then used for prediction. Suppose the population of a town is 50,000 in 2020 and decreases by 2% each year. The decay multiplier is 0.98, so after n years the population is Pₙ = 50,000 × 0.98ⁿ. After 5 years, P₅ = 50,000 × 0.98⁵ ≈ 50,000 × 0.904 ≈ 45,200.

真实人口数据可以用公式建模，然后用预测。假设某镇 2020 年人口为 5 万，每年减少 2%。衰减乘数为 0.98，因此 n 年后人口为 Pₙ = 50,000 × 0.98ⁿ。5 年后，P₅ = 50,000 × 0.98⁵ ≈ 50,000 × 0.904 ≈ 45,200。

You may be given a formula with specific constants and asked to substitute a value. For example, a model for a city’s population is P = 2.5 × 10⁶ × (1.03)ᵗ, where t is the number of years after 2020. To find the population in 2025, substitute t = 5: P = 2.5 × 10⁶ × (1.03)⁵ ≈ 2.5 × 10⁶ × 1.159 ≈ 2.90 × 10⁶.

你可能会得到一个带有特定常数的公式，并被要求代入一个值。例如，某城市人口的模型为 P = 2.5 × 10⁶ × (1.03)ᵗ，其中 t 是 2020 年后的年数。要求 2025 年的人口，代入 t = 5：P = 2.5 × 10⁶ × (1.03)⁵ ≈ 2.5 × 10⁶ × 1.159 ≈ 2.90 × 10⁶。

When solving such problems, remember to use the order of operations: calculate the power first, then multiply. Also check whether your answer should be rounded to a sensible degree of accuracy — population figures are usually quoted to 2 or 3 significant figures.

解此类问题时，记住运算顺序：先计算幂，再相乘。还要检查答案是否应四舍五入到合理的精度——人口数字通常保留 2 或 3 位有效数字。

10. Limits of Mathematical Models | 数学模型的局限性

No mathematical model is perfect. Exponential growth models assume that the growth rate stays constant, but in reality birth rates, death rates, migration and government policies all change over time. A model that predicts 15 billion people by 2100 may be inaccurate because it ignores factors such as disease, war or advances in medicine.

没有数学模型是完美的。指数增长模型假设增长率保持不变，但现实中出生率、死亡率、移民和政府政策都会随时间变化。一个预测 2100 年人口为 150 亿的模型可能不准确，因为它忽略了疾病、战争或医学进步等因素。

Models are also simplified versions of reality. A linear model may fit a short period of data well, but fail over a longer period. When you use a model in an exam, always state the assumption you have made, such as “assuming the growth rate remains constant”.

模型也是对现实的简化。线性模型可能很好地拟合短期数据，但在较长时间内失效。在考试中使用模型时，始终说明你所作的假设，例如”假设增长率保持不变”。

In exam questions, you may be asked to comment on the reliability of a prediction. A good answer should mention that the model only uses the given data, that outside factors are not considered, and that the prediction becomes less reliable the further into the future you go.

在考试题中，你可能被要求对预测的可靠性作出评论。好的答案应提到模型仅使用给定数据、未考虑外部因素，并且预测的时间越远，可靠性越低。

11. Estimation and Significant Figures | 估算与有效数字

Large population numbers are often estimated. To estimate the total population of three countries with populations of 131 million, 87 million and 54 million, round each to 1 significant figure first: 130 + 90 + 50 = 270 million. This gives a quick mental approximation of 2.7 × 10⁸.

大人口数字通常被估算。要估算三个人口分别为 1.31 亿、8700 万和 5400 万的国家的总人口，先将每个四舍五入到 1 位有效数字：130 + 90 + 50 = 2.7 亿。这给出了 2.7 × 10⁸ 的快速心算近似值。

When multiplying or dividing in standard form, you can estimate by rounding each number to 1 significant figure and then adjusting the power of ten. For example, (8.3 × 10⁹) ÷ (2.1 × 10³) ≈ (8 × 10⁹) ÷ (2 × 10³) = 4 × 10⁶.

标准形式的乘除中，可以通过将每个数字四舍五入到 1 位有效数字然后调整十的幂来进行估算。例如，(8.3 × 10⁹) ÷ (2.1 × 10³) ≈ (8 × 10⁹) ÷ (2 × 10³) = 4 × 10⁶。

Estimation is a quick check: use it to verify that your final answer is reasonable. If your calculated population comes out smaller than the original, you may have subtracted instead of multiplying by the growth multiplier.

估算是一种快速检查：用它来验证最终答案是否合理。如果你计算的人口比原来的还小，你可能做了减法而不是乘以增长乘数。

12. Exam-Style Practice | 考试风格练习

Here is a typical four-mark question. A country had a population of 60 million in 2020. It grows at 1.5% per year. (a) Write the growth multiplier. (b) Calculate the population in 2025 to the nearest million. (c) The land area is 500,000 km². Find the population density in 2025.

以下是一道典型的四分题。某国 2020 年人口为 6000 万。它以每年 1.5% 增长。(a) 写出增长乘数。(b) 计算 2025 年的人口，精确到百万。(c) 土地面积为 50 万平方公里。求 2025 年的人口密度。

(a) 1.015 (b) P = 60,000,000 × 1.015⁵ ≈ 60,000,000 × 1.0773 ≈ 64,640,000 ≈ 65 million (c) 64,640,000 ÷ 500,000 ≈ 129 people/km²

To score full marks, show each step clearly and write the final answer with its unit. If a question asks for the answer in standard form, convert at the end: 65 million = 6.5 × 10⁷.

要得满分，请清晰写出每一步，并写出带单位的最终答案。如果题目要求以标准形式给出答案，请在最后转换：6500 万 = 6.5 × 10⁷。

Practise interpreting percentile and quartile questions too, because population age data in IGCSE exams is often given as a grouped frequency table. The lower quartile is the value below which 25% of the data lies, the upper quartile below which 75% lies, and the interquartile range is their difference.

也要练习解读百分位数和四分位数的问题，因为 IGCSE 考试中的人口年龄数据通常以分组频数表给出。下四分位数是 25% 数据所在位置的值，上四分位数是 75% 数据所在位置的值，四分位距是它们的差。

Mastering population problems gives you confidence across many areas of the Edexcel IGCSE Mathematics syllabus. Review each topic above, attempt past paper questions on population data, and remember: always check units, use standard form for large numbers, and state your assumptions when using models.

掌握人口问题会帮助你在 Edexcel IGCSE 数学大纲的许多领域建立信心。复习上面每个主题，尝试有关人口数据的历年真题，并记住：始终检查单位、对大数使用标准形式、使用模型时说明你的假设。

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The World’s Population | 世界人口