Essential Maths 8H: Compressed Homework Answers & Key Concepts | Essential Maths 8H 作业答案压缩版：知识点精讲

📚 Essential Maths 8H: Compressed Homework Answers & Key Concepts | Essential Maths 8H 作业答案压缩版：知识点精讲

This article distills the core concepts behind typical Essential Maths 8H homework exercises. Whether you’re checking answers or revising for tests, these compressed notes cover number, algebra, geometry, statistics, and probability at the KS3 Higher tier. Each section pairs a quick reference solution approach with essential theory, helping you understand not just the ‘what’ but the ‘why’.

本文提炼了 Essential Maths 8H 典型家庭作业背后的核心概念。无论你在核对答案还是备考复习，这份压缩笔记涵盖了 KS3 高阶要求下的数、代数、几何、统计和概率知识。每个部分都将快速解题方法与基本理论配对，帮助你不仅知道“是什么”，更理解“为什么”。

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1. Fractions, Decimals and Percentages | 分数、小数与百分比

Converting between fractions, decimals and percentages is a key skill. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To find a percentage of an amount, write the percentage as a fraction or decimal and multiply.

分数、小数和百分比的互换是一项关键技能。分数转小数，用分子除以分母；小数转百分比，乘以100。求一个数的百分比，将百分比写成分数或小数再相乘。

Example: Convert 3/8 to a decimal and a percentage. 3 ÷ 8 = 0.375, and 0.375 × 100 = 37.5%.

例：将 3/8 转换为小数和百分比。3 ÷ 8 = 0.375，0.375 × 100 = 37.5%。

When adding or subtracting fractions, find the lowest common multiple of the denominators. For mixed numbers, change them to improper fractions first. Always simplify your final answer.

加减分数时，先找分母的最小公倍数。带分数务必先化为假分数。最后的答案一定要化简。

Work out 2/5 + 1/4. LCM of 5 and 4 is 20, so 2/5 = 8/20 and 1/4 = 5/20. Sum = 13/20. As a decimal this is 0.65 or 65%.

计算 2/5 + 1/4。5 和 4 的最小公倍数是 20，因此 2/5 = 8/20，1/4 = 5/20。和为 13/20，化成小数是 0.65，即 65%。

To calculate a percentage of a quantity without a calculator, find 10% first, then scale. For example, 15% of £80: 10% is £8, 5% is £4, so 15% = £12.

不用计算器求一个数量的百分比，可以先找出 10%，再推算。比如 £80 的 15%：10% 是 £8，5% 是 £4，所以 15% = £12。

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2. Indices and Standard Form | 指数与标准形式

The three basic index laws are: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Any non‑zero number raised to the power of zero equals 1.

三条基本指数法则是：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。任何非零数的零次方都等于 1。

A negative index gives the reciprocal: a⁻ⁿ = 1/aⁿ. For example, 4⁻² = 1/4² = 1/16.

负指数表示倒数：a⁻ⁿ = 1/aⁿ。例如 4⁻² = 1/4² = 1/16。

Fractional indices represent roots: a¹/² = √a, a¹/³ = ∛a. In general, a¹/ⁿ = ⁿ√a.

分数指数表示开方：a¹/² = √a，a¹/³ = ∛a。一般地，a¹/ⁿ = ⁿ√a。

Standard form writes a number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. Large number 6500 = 6.5 × 10³. Small number 0.0042 = 4.2 × 10⁻³.

标准形式将一个数写成 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。大数 6500 = 6.5 × 10³。小数 0.0042 = 4.2 × 10⁻³。

To multiply numbers in standard form, multiply the A values and add the exponents. (3 × 10⁴) × (2 × 10⁵) = (3×2) × 10⁴⁺⁵ = 6 × 10⁹.

标准形式的乘法：将 A 值相乘，指数相加。(3 × 10⁴) × (2 × 10⁵) = (3×2) × 10⁴⁺⁵ = 6 × 10⁹。

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3. Ratio and Proportion | 比与比例

A ratio compares parts of a whole. To simplify a ratio, divide all parts by their highest common factor. The ratio 12:18 simplifies to 2:3 by dividing by 6.

比用于比较整体中的各部分。化简比时，各项同除以最大公因数。12:18 除以 6，简化为 2:3。

When sharing a quantity in a given ratio, find the total number of shares first. Divide the quantity by the total shares, then multiply by the required parts.

按给定比例分配数量时，先求总份数。用总数量除以总份数，再乘以所求份数。

Example	Steps
Share £60 in the ratio 2:3	Total parts = 5. One part = £60 ÷ 5 = £12. First share = 2 × £12 = £24. Second = 3 × £12 = £36.

例：将 £60 按 2:3 分配。总份数 5，每份 £12。第一份 2×£12 = £24，第二份 3×£12 = £36。

Direct proportion means two quantities increase at the same rate. If 3 pens cost £2.70, one pen costs £0.90, so 10 pens cost £9.00.

正比例意味着两个量以相同速率增加。若 3 支笔 £2.70，则 1 支 £0.90，10 支 £9.00。

Use the unitary method: find the value of one unit first, then scale to the required amount.

使用单位法：先求一个单位的值，再按需放大。

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4. Algebraic Simplification | 代数化简

Collect like terms: terms that have the same variable part can be combined. 5x + 2y + 3x − y simplifies to 8x + y.

合并同类项：含有相同字母部分的项可以合并。5x + 2y + 3x − y 化简为 8x + y。

To expand a single bracket, multiply each term inside by the term outside. 3(2x − 4) = 6x − 12. Watch the signs carefully.

单括号展开时，括号外的项乘以括号内每一项。3(2x − 4) = 6x − 12。注意符号。

Expanding two brackets uses FOIL: (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.

展开双括号使用 FOIL 法则：(x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6。

Factorising is the reverse of expanding. Look for the highest common factor (HCF). 10x + 15 = 5(2x + 3). For an expression like x² + 5x + 6, find two numbers that multiply to 6 and add to 5: (x + 2)(x + 3).

因式分解是展开的逆运算。先找最高公因式（HCF）。10x + 15 = 5(2x + 3)。对形如 x² + 5x + 6 的式子，找两个数，其积为 6，和为 5，即 (x + 2)(x + 3)。

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5. Solving Linear Equations | 解一元一次方程

Use the balance method: whatever you do to one side, you must do to the other. Solve 2x + 5 = 13. Subtract 5: 2x = 8. Divide by 2: x = 4.

使用天平法：一边做的运算，另一边也必须做。解 2x + 5 = 13。两边减 5：2x = 8。两边除以 2：x = 4。

When brackets appear, expand first. 3(2x − 1) = 15 → 6x − 3 = 15 → 6x = 18 → x = 3.

当有括号时，先展开。3(2x − 1) = 15 → 6x − 3 = 15 → 6x = 18 → x = 3。

Equations with unknowns on both sides: collect x terms on one side. 5x + 2 = 3x + 10 → 5x − 3x = 10 − 2 → 2x = 8 → x = 4.

方程两边都有未知数时，将含 x 的项移到一边。5x + 2 = 3x + 10 → 5x − 3x = 10 − 2 → 2x = 8 → x = 4。

If the equation contains fractions, multiply every term by the lowest common denominator. x/3 + 1 = x/2 gives 2x + 6 = 3x → x = 6.

若方程含分数，每项乘以最小公分母。x/3 + 1 = x/2 变为 2x + 6 = 3x → x = 6。

Always verify your solution by substituting it back into the original equation.

务必将解代入原方程检验。

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6. Sequences and the nth Term | 数列与第 n 项

A linear sequence has a constant difference between consecutive terms. For 5, 9, 13, 17, …, the difference is +4.

线性数列相邻项之间差为常数。数列 5, 9, 13, 17, … 的公差为 +4。

The nth term of an arithmetic sequence is given by an+b, where a is the common difference and a + b is the first term. For the sequence above, nth term = 4n + 1.

等差数列的第 n 项公式为 an+b，其中 a 是公差，a+b 为首项。上述数列的第 n 项为 4n + 1。

To find the 50th term, substitute n = 50: 4×50 + 1 = 201.

求第 50 项，代入 n = 50：4×50 + 1 = 201。

If a sequence does not start at n=1, adjust accordingly. For a sequence of patterns, build a table of term number and value, then find the rule.

若数列不从 n=1 开始，要相应调整。对图形数列，先建立项数与值的表格，再寻找规则。

Quadratic sequences have a second common difference. The nth term is of the form an² + bn + c. In KS3, you may only be asked to recognise them.

二次数列的二阶差为常数，其第 n 项形式为 an² + bn + c。KS3 阶段通常只需识别。

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7. Angles in Polygons | 多边形内角

The sum of the interior angles of an n‑sided polygon is (n−2) × 180°. For a hexagon (n=6), sum = (6−2)×180° = 720°.

n 边形内角和为 (n−2)×180°。六边形 (n=6) 内角和 = (6−2)×180° = 720°。

In a regular polygon all interior angles are equal. One interior angle = sum ÷ n. A regular pentagon’s interior angle = 540° ÷ 5 = 108°.

正多边形所有内角相等，一个内角 = 内角和 ÷ n。正五边形内角 = 540° ÷ 5 = 108°。

The sum of exterior angles of any convex polygon is always 360°. The exterior angle of a regular polygon is 360° ÷ n.

任何凸多边形的外角和恒为 360°。正多边形的外角 = 360° ÷ n。

You can use interior + exterior = 180° to move between them. If an interior angle is 135°, the exterior is 45°, so the polygon has 360° ÷ 45° = 8 sides.

利用内角+外角 = 180° 可相互转换。若内角为 135°，外角为 45°，则边数 = 360° ÷ 45° = 8。

Always give a reason when finding missing angles, such as ‘angles on a straight line sum to 180°’ or ‘base angles in an isosceles triangle are equal’.

求未知角时务必写出理由，如“平角为180°”或“等腰三角形底角相等”。

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8. Circles: Circumference and Area | 圆的周长和面积

Circumference C = πd or 2πr, where r is the radius. Use π ≈ 3.14 or the π button on your calculator. Leave answers in terms of π unless told otherwise.

周长 C = πd 或 2πr，r 为半径。使用 π ≈ 3.14 或计算器上的 π 键。除非题目要求，答案可保留 π。

Area of a circle A = πr². Remember to square the radius first, then multiply by π.

圆面积 A = πr²。记住先

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