📚 Year 9 Essential Maths 9H: Core Skills and Higher Applications | Year 9 数学 9H 核心技能与高阶应用
This revision guide covers the core Year 9 Higher topics from Essential Maths 9H, building strong foundations for GCSE. Work through each section, practise the worked examples and check your understanding with past-paper style questions.
本复习指南涵盖 Essential Maths 9H 的 Year 9 高阶核心主题，为 GCSE 打下坚实基础。逐节学习，练习示例，并用真题风格的问题检验理解。
1. Integers, Powers and Roots | 整数、幂与根
In Year 9 Higher, you must be confident with negative numbers and the rules of indices. When multiplying powers with the same base, add the indices; when raising a power to another power, multiply the indices; a negative index means a reciprocal.
在 Year 9 高阶课程中，你必须熟练掌握负数运算和指数法则。同底数幂相乘时指数相加；幂的乘方时指数相乘；负指数表示取倒数。
aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ a⁻ⁿ = 1/aⁿ
Square roots and cube roots can be estimated between integer bounds. For example, √50 lies between 7 and 8 because 7² = 49 and 8² = 64. You should also know that √a × √b = √ab and be able to simplify surds such as √50 = 5√2.
平方根和立方根可以在整数界限之间估算。例如 √50 介于 7 和 8 之间，因为 7² = 49 而 8² = 64。你还需要知道 √a × √b = √ab，并能化简根式，如 √50 = 5√2。
√a × √b = √ab √50 = 5√2
and n is an integer. This is essential for very large and very small numbers in science and maths.
使用科学计数法时，一个数写作 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。这对表示非常大和非常小的数在科学和数学中至关重要。
A × 10ⁿ 1 ≤ A < 10
2. Fractions, Decimals and Percentages | 分数、小数与百分比
multiplication, multiply the numerators and denominators directly; for division, multiply by the reciprocal.
分数加减需要通分。例如 2/3 + 1/4 = 8/12 + 3/12 = 11/12。分数乘法直接分子乘分子、分母乘分母；分数除法乘以倒数。
2/3 + 1/4 = 8/12 + 3/12 = 11/12
Converting between fractions, decimals and percentages is a core skill. A percentage increase or decrease uses a multiplier: increase by r% means multiply by (1 + r/100), and decrease by r% means multiply by (1 − r/100).
分数、小数和百分比之间的转换是核心技能。百分比增减使用乘数：增加 r% 表示乘以 (1 + r/100)，减少 r% 表示乘以 (1 − r/100)。
增加 r% → × (1 + r/100) 减少 r% → × (1 − r/100)
d discounts, are calculated by applying the multiplier repeatedly. For example, increasing £200 by 5% for 3 years gives 200 × 1.05³.
复合百分比变化，如利息或重复折扣，通过反复应用乘数来计算。例如，200 英镑以 5% 增长 3 年得到 200 × 1.05³。
£200 × 1.05³
3. Ratio and Proportion | 比与比例
mmon factor. If the ratio of boys to girls is 3:5, the total number of parts is 3 + 5 = 8.
比率比较整体的各个部分，可以通过将所有部分除以公因数来化简。如果男生与女生的比是 3:5，则总份数为 3 + 5 = 8。
3:5 → 总份数 = 3 + 5 = 8
To divide a quantity in a given ratio, find the value of one part first. For example, to share £240 in the ratio 3:5, one part is £
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