📚 250: Mastering Growth and Decay in IGCSE Maths | 250：掌握IGCSE数学中的增长与衰减
The number 250 is more than just a three-digit figure; in the context of IGCSE Edexcel Mathematics, it can represent an initial investment, a starting population, or a principal amount used to explore percentage increase and decrease. This article uses 250 as a core example to demonstrate compound interest, depreciation, exponential growth and decay — all essential topics for your exam.
数字250不仅仅是一个三位数；在IGCSE Edexcel数学中，它可代表初始投资、起始人口或本金，用于探索百分比增加和减少。本文以250为核心示例，演示复利、折旧、指数增长与衰减——这些都是考试的关键主题。
1. What Does 250 Represent? | 数字250代表什么？
In many IGCSE problems, 250 might be the initial amount of money deposited in a bank, the original value of a car, or the starting number of bacteria in a culture. By using this concrete number, we can clearly see how quantities grow or shrink over time.
在许多IGCSE题目中，250可能是存入银行的初始金额、汽车的原值或培养液中细菌的起始数量。通过使用这个具体数字，我们可以清楚地看到数量如何随时间增长或缩小。
The choice of 250 allows simple mental calculation while still illustrating the power of exponential change. It also appears frequently in exam questions where numbers are designed to avoid overly messy arithmetic.
选择250便于简单的心算，同时仍能说明指数变化的威力。它也经常出现在考试题中，题目中的数字设计就是为了避免过于繁琐的计算。
2. Simple Interest with 250 | 250的单利计算
Before diving into compound interest, it is useful to revise simple interest. Simple interest means the interest earned each year is based only on the original principal.
在深入复利之前，先复习单利是很有用的。单利意味着每年获得的利息仅基于原始本金。
For a principal of £250 invested at an annual simple interest rate of 4% for 3 years, the interest is:
I = (P × R × T) / 100 = (250 × 4 × 3) / 100 = £30
对于本金250英镑，按年利率4%投资3年，利息为：
I = (P × R × T) / 100 = (250 × 4 × 3) / 100 = 30英镑
The total amount after 3 years is therefore £250 + £30 = £ In IGCSE, you are often asked to find either the total amount or just the interest earned.
因此3年后的总金额为250 + 30 = 280英镑。在IGCSE中，你经常被要求计算总金额或仅计算所获利息。
3. Compound Interest Formula | 复利公式
Unlike simple interest, compound interest earns "interest on interest". Each year, interest is added to the principal, and the following year's interest is calculated on the new, larger amount.
与单利不同，复利是“利滚利”。每年利息被加到本金上，下一年的利息则基于新的、更大的金额计算。
The compound interest formula is:
A = P(1 + r/100)ⁿ
where A is the final amount, P is the principal, r is the annual interest rate, and n is the number of years.复利计算公式为：
A = P(1 + r/100)ⁿ
其中A为最终金额，P为本金，r为年利率，n为年数。ounding happens more frequently than once a year, we adjust the formula: A = P(1 + r/100m)mn, where m is the number of compounding periods per year.
当复利频率高于每年一次时，我们调整公式：A = P(1 + r/100m)mn，其中m是每年的复利期数。
4. Annual Compounding with 250 | 250的年度复利计算
Let us take £250 invested at an annual compound interest rate of 5% for 3 years. Using the formula:
A = 250(1 + 5/100)³ = 250(1.05)³
我们以250英镑为例，按年复利5%投资3年。使用公式：
A = 250(1 + 5/100)³ = 250(1.05)³
7625, so A = 250 × 1.157625 = 289.40625. Rounded to the nearest penny, the final amount is £289.41.
逐步计算：1.05³ = 1.05 × 1.05 × 1.05 = 1.157625，因此 A = 250 × 1.157625 = 289.40625。四舍五入到最接近的便士，最终金额为289.41英镑。
Here is the year-by-year growth in a table:
以下是逐年增长的表格：
| Year (年) | Amount at start (年初金额) | Interest (利息) | Amount at end (年末金额) |
|---|---|---|---|
| 1 | £250.00 | £12.50 | £262.50 |
| 2 | £262.50 | £13.13 | £275.63 |
| 3 | £275.63 | £13.78 | £289.41 |
Compare this to the simple interest total of £280 — compound interest gives an extra £9.41 over 3 years.
与单利总额280英镑相比——复利在3年内多出了9.41英镑。
5. Monthly Compounding with 250 | 250的月复利
Sometimes interest is compounded monthly. For £250 at an annual rate of 4.8% compounded monthly for 1 year, we use m = 12.
有时利息按月复利。对于250英镑，年利率4.8%，按月复利1年，我们使用m = 12。
A = 250 (1 + 4.8/(100×12))12×1 = 250 (1 + 0.004)¹² = 250 (1.004)¹²
Calculating 1.004¹² gives approximately 1.04907, so A ≈ 250 × 1.04907 = £262.27. The effective annual rate is slightly higher than 4.8% due to more frequent compounding.
计算1.004¹²，得到约1.04907，因此 A ≈ 250 × 1.04907 = 262.27英镑。由于复利频率更高，实际年利率略高于4.8%。
This demonstrates a key principle: the more frequently interest is compounded, the greater the final amount, assuming the same nominal annual rate.
这说明了一个关键原理：在名义年利率相同的情况下，复利频率越高，最终金额越大。
6. Depreciation: Reducing Balance Method | 折旧：余额递减法
Depreciation works like compound interest in reverse — the value of an asset decreases by a fixed percentage each year. The formula is:
V = P(1 – r/100)ⁿ
where V is the value after n years, P is the original value, and r is the annual rate of depreciation.折旧类似于反向复利——资产价值每年以固定百分比减少。公式为：
V = P(1 – r/100)ⁿ
其中V是n年后的价值，P是原始价值，r是年折旧率。Suppose a laptop originally worth £250 depreciates at 20% per year. After 3 years, its value is:
V = 250(1 – 20/100)³ = 250(0.8)³ = 250 × 0.512 = £128
假设一台原价250英镑的笔记本电脑每年折旧20%。3年后，它的价值为：
V = 250(1 – 20/100)³ = 250(0.8)³ = 250 × 0.512 = 128英镑
Notice that the multiplier (1 – r/100) is always less than 1, so repeated multiplication causes the value to shrink exponentially.
注意乘数 (1 – r/100) 始终小于1，因此反复相乘会导致价值呈指数级缩小。
7. Exponential Growth with 250 Bacteria | 250个细菌的指数增长
Exponential growth is not only for money. Imagine a colony of 250 bacteria that doubles every hour. The number of bacteria after t hours follows:
N = 250 × 2ᵗ
指数增长不仅适用于金钱。假设一个250个细菌的菌落每小时翻倍。t小时后的细菌数量遵循：
N = 250 × 2ᵗ
0 bacteria. This rapid growth is typical of many natural and financial processes covered in IGCSE.
5小时后，N = 250 × 2⁵ = 250 × 32 = 8000个细菌。这种快速增长是IGCSE中许多自然和金融过程的典型特征。
In the Edexcel IGCSE syllabus, you may encounter questions that ask you to form an equation from a word problem and then solve for time using logarithms or trial and improvement.
在Edexcel IGCSE大纲中，你可能会遇到要求从文字题建立方程，然后使用对数或试错法求解时间的问题。
8. IGCSE Exam-style Question | IGCSE考试题型
Question: Sarah invests £250 in a savings account that pays compound interest at a rate of 3.5% per annum. How much will she have after 5 years? Give your answer to the nearest penny.
问题：莎拉将250英镑投资于一个年复利为3.5%的储蓄账户。5年后她将有多少钱？答案精确到便士。
Solution: P = 250, r = 3.5, n = 5.
A = 250(1 + 3.5/100)⁵ = 250(1.035)⁵
Compute 1.035⁵ ≈ 1.18768, so A ≈ 250 × 1.18768 = 296.92. Sarah will have £296.92.解答：P = 250，r = 3.5，n = 5。
A = 250(1 + 3.5/100)⁵ = 250(1.035)⁵
计算1.035⁵ ≈ 1.18768，因此 A ≈ 250 × 1.18768 = 296.92。莎拉将拥有296.92英镑。This type of straightforward application question is common in Edexcel IGCSE Paper 1 and Paper 2. Always show the substitution step before calculating.
这类直接应用题在Edexcel IGCSE的卷一卷二中很常见。务必在计算前展示代入步骤。
9. Half-Life and Decay with 250 | 半衰期与以250为初始值的衰减
Radioactive decay is another example of exponential decay. If a 250-gram sample of a radioactive substance has a half-life of 3 years, the mass remaining after t years can be modelled by:
M = 250 × (1/2)t/3
放射性衰变是指数衰减的另一个例子。如果一个250克的放射性物质样本半衰期为3年，则t年后的剩余质量可表示为：
M = 250 × (1/2)t/3
31.25 grams. This ties directly to the Edexcel topic of exponential functions and graphs.
9年后，M = 250 × (1/2)³ = 250 × 1/8 = 31.25克。这与Edexcel指数函数与图像的主题直接相关。
10. Key Tips for Edexcel IGCSE Maths | Edexcel IGCSE数学关键提示
When tackling growth and decay problems, keep these points in mind:
在处理增长与衰减问题时，请记住以下几点：
Always identify whether the change is an increase (multiplier > 1) or a decrease (multiplier between 0 and 1).
始终判断变化是增加（乘数>1）还是减少（乘数在0到1之间）。
Write down the multiplier clearly: for a 4% increase use 1.04, for a 12% decrease use 0.88.
清楚地写下乘数：增加4%使用1.04，减少12%使用0.88。
compounding periods, not necessarily years. For quarterly compounding over 2 years, n = 8.
在复利中，n代表复利期数，而不一定是年数。对于2年期的季度复利，n=8。
Use brackets carefully on your calculator, especially when dealing with powers and fractional rates.
在计算器上小心使用括号，尤其是在处理幂次和分数利率时。
Check whether the question asks for the total amount or just the interest earned.
检查题目要求的是总金额还是仅利息部分。
11. Avoiding Common Mistakes | 避免常见错误
One typical mistake when using 250 as a principal is forgetting to convert the percentage correctly. For example, writing (1 + 3.5) instead of (1 + 3.5/100) will give a wildly incorrect answer.
使用250作为本金时的一个典型错误是忘记正确转换百分比。例如，写成(1 + 3.5)而不是(1 + 3.5/100)会得到完全错误的答案。
Another pitfall is mixing up the exponent. For depreciation over 4 years, the power is 4, not 5. Always count the number of time periods carefully.
另一个陷阱是混淆指数。对于4年的折旧，指数是4，而不是5。请始终仔细计算时间周期数。
= P(1 ± r/100)ⁿ, remember that n and r must be in the same time units. If interest is 1.5% per month and you invest for 2 years, n = 24, not 2.
此外，使用公式 A = P(1 ± r/100)ⁿ 时，记住n和r必须使用相同的时间单位。如果月利率为1.5%，投资2年，则n=24，而非2。
12. Summary: The Power of 250 | 总结：250的力量
Starting with 250, we have seen how a modest number can grow significantly through compound interest or shrink through depreciation. Whether it is money in a bank, bacteria in a lab, or a decaying isotope, the core mathematical structure is the same.
从250出发，我们看到了一个不起眼的数字如何通过复利显著增长，或通过折旧不断缩小。无论是银行里的钱、实验室里的细菌，还是衰变的同位素，其核心数学结构是相同的。
Mastering these concepts gives you a strong foundation for a large portion of the Edexcel IGCSE syllabus, including percentage change, indices, exponential functions, and real-world problem solving. Keep practising with different starting values — and remember, 250 is always a great number to test your understanding.
掌握这些概念，将为你在Edexcel IGCSE大纲中很大一部分内容打下坚实基础，包括百分比变化、指数、指数函数和现实问题求解。继续用不同的起始值练习——并记住，250永远是检验你理解的好数字。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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