📚 Essential Maths Book 8 Support Answers Explained | Essential Maths Book 8 Support 答案与知识点详解
Welcome to our in‑depth walkthrough of the Essential Maths Book 8 Support answer key. This companion guide breaks down each major topic using the same clear step‑by‑step reasoning that underpins the Support series. Whether you are a student consolidating your understanding or a parent helping with homework, you will find detailed explanations for number, fractions, decimals, percentages, algebra, geometry, measures and statistics questions.
欢迎阅读 Essential Maths Book 8 Support 答案精讲。本指南按照 Support 系列的思路，将每个重要知识点的答案拆解为清晰的步骤。不论你是想巩固知识的学生，还是辅导孩子作业的家长，都可以在这里找到关于整数、分数、小数、百分比、代数、几何、测量与统计等题目的详细讲解。
1. Mastering Whole Number Arithmetic | 整数运算精通与答案解析
roblems such as 357 + 498 or 24 × 36, and then check your answer using inverse operations.
整数运算是所有后续数学学习的支柱。本节集中讲解大数的加减乘除以及运算顺序（BIDMAS）。Book 8 Support 中有许多习题要求计算如 357 + 498 或 24 × 36 这样的题目，并用逆运算进行验算。
The final sum is 6105. Reversing with subtraction confirms the result: 6105 − 2349 = 3756.
例题：计算 3756 + 2349。Support 答案提示我们将数位对齐，先算个位（6 + 9 = 15，写 5 进 1），然后十位、百位、千位，最终得到 6105。用减法逆运算 6105 − 2349 = 3756 即验证正确。
. The answer key stresses estimation first: 50 × 20 = 1000, so we know 1081 is reasonable.
乘法部分 Support 练习册常用网格法。计算 47 × 23：拆分为 (40 + 7) × (20 + 3) 填入网格，四个积分别得到 800、120、140 和 21，相加得 1081。答案强调先估算：50 × 20 = 1000，所以 1081 是合理的。
2. Fractions: Adding, Subtracting, Multiplying and Dividing | 分数：加减乘除答案精讲
Working with fractions confidently is a core KS3 target. Essential Maths Book 8 Support revisits equivalent fractions, mixed numbers and all four operations. Many answers show the need for a common denominator when adding or subtracting, and “keep‑change‑flip” for division.
熟练处理分数是 KS3 的重要目标。Essential Maths Book 8 Support 复习了等值分数、带分数以及四则运算。许多答案展示了加减时要先通分，除法中要“倒乘”的规则。
fractions: 3/2 + 8/3. Use common denominator 6 → 9/6 + 16/6 = 25/6, which simplifies to 4 ⅙ rt answer key always encourages converting back to a mixed number unless otherwise stated.
例题：1 ½ + 2 ⅔。先将带分数化为假分数：3/2 + 8/3。通分分母为 6 → 9/6 + 16/6 = 25/6，化简为 4 ⅙。Support 答案提示，除非另有要求，一般都要化回带分数。
iply across: (2 × 7) / (3 × 8) = 14/24 = 7/12 after simplifying by 2. When dividing, as in ⅘ ÷ ½, keep the first fraction, change ÷ to ×, and flip the second: ⅘ × 2 = 8/5 = 1 ⅗. The answer key underlines that flipping is only for division.
乘法运算如 ⅔ × ⅞，直接分子乘分子分母乘分母：(2 × 7) / (3 × 8) = 14/24，约去 2 得 7/12。除法如 ⅘ ÷ ½，保留第一个分数，除号变乘号，第二个分数倒置：⅘ × 2 = 8/5 = 1 ⅗。答案指出只有除法才需要倒置。
3. Decimals: Place Value and Operations | 小数：数位与运算答案
Decimals appear throughout Book 8 Support, from ordering sizes to adding, subtracting, multiplying and dividing by powers of ten. The answer booklet consistently lines up decimal points when adding or subtracting.
小数贯穿了 Book 8 Support，从比较大小到加减乘除以及乘除以 10 的幂。答案册在加减法时一直强调小数点对齐。
h digit moves two places left, giving 560. The Support answers frequently remind learners: “Don’t just add two zeros – understand that the digits shift columns.”
例题：14.07 + 9.3。小数点对齐：14.07 + 9.30 = 23.37。答案常补零占位以避免混淆。乘以 100 时，如 5.6 × 100，所有数位向左移动两位，得到 560。Support 答案经常提醒：“不要只加两个零——要理解数位在列中移动。”
Dividing a decimal by a whole number is often shown with short division. For 7.68 ÷ 4, divide 7 by 4 (1 remainder 3), bring down 6 to make 36, divide by 4 to get 9, bring down 8 to make 8, divide by 4 to get 2. The result is 1.92. The answer key places the decimal point directly above in the working.
小数除以整数常用短除法。例如 7.68 ÷ 4，7 除以 4 商 1 余 3，落下 6 组成 36 除以 4 得 9，再落下 8 除以 4 得 2，结果为 1.92。答案在运算过程中将小数点直接标在上方。
4. Percentages: Conversions and Calculations | 百分比：转换与计算答案
Essential Maths Book 8 Support builds percentage skills through conversion between fractions, decimals and percentages, and by finding percentages of quantities. The answer key often uses the “find 10% then scale” method.
Essential Maths Book 8 Support 通过分数、小数和百分比的相互转换以及求一个量的百分之几来建立百分比技能。答案常用“先找 10% 再缩放”的方法。
To find 35% of 340, the Support answer shows: 10% of 340 = 34; 5% is half of that, 17; so 35% plier: 0.35 × 340 = 119. Both methods are presented to reinforce mathematical flexibility.
求 340 的 35%，Support 答案演示：340 的 10% = 34；5% 是它的一半，即 17；所以 35% = 3 × 34 + 17 = 102 + 17 = 119。同时也可以用小数乘法：0.35 × 340 = 119。两种方法并行展示以培养灵活性。
Percentage increase and decrease questions also feature. For an increase of 15% on £80, the answer shows 15% of 80 = 12, so new price £92. Using a single multiplier 1.15 × 80 = 92 is highlighted as a time‑saver. The key stresses writing the correct multiplier for a decrease, e.g. a 30% discount uses 0.70.
也有百分比增减题。例如在 £80 的基础上增加 15%，答案先算 80 的 15% = 12，新价 £92。使用一次性的乘数 1.15 × 80 = 92 则更快捷。答案强调减少时要用正确的乘数，比如七折优惠用 0.70。
5. Ratio and Proportion Simplified | 比与比例简化答案
Ratio questions in Book 8 Support range from simplifying ratios to dividing a quantity into a given ratio and solving proportion problems. The answer key treats ratios just like fractions: divide both terms by their highest common factor.
Book 8 Support 中的比和比例题从化简比到按给定比例分配数量，再到解决比例问题。答案将比视为分数：用最大公因数同时除前后项。
2, so 24÷12 : 36÷12 = 2:3. The answer always ensures no common factor remains. For sharing £560 in the ratio 3:5, add parts = 8. One part = £560 ÷ 8 = £70. First share = 3 × £70 = £210, second = 5 × £70 = £350. A quick check: £210 + £350 = £560.
例题：化简 24:36。最大公因数为 12，故 24÷12 : 36÷12 = 2:3。答案确保再无公因数。按 3:5 分配 £560：总份数 = 8，每份 = £560 ÷ 8 = £70，第一份 3 × £70 = £210，第二份 5 × £70 = £350。验证：£210 + £350 = £560。
Direct proportion is illustrated with tables. If 5 pens cost £4, the cost of 8 pens is found by working out the unit price (1 pen costs £0.80) then multiplying by 8 (£6.40). The answer key sometimes uses the “× / ÷” scaling method: 5 to 8 is ×1.6, so £4 × 1.6 = £6.40.
正比例关系用表格展现。若 5 支笔 £4，8 支笔的价格先求单价（1 支 £0.80）再乘 8 得 £6.40。答案有时也用比例缩放法：5 到 8 是 ×1.6，所以 £4 × 1.6 = £6.40。
6. Algebra: Working with Expressions | 代数：表达式操作答案
The Support book gently introduces algebraic notation, collecting like terms and substitution. Answer pages show clear steps: identify like terms (same letter and power), group them, then combine.
Support 练习册温和地引入代数符号、合并同类项和代入求值。答案页展示清晰的步骤：识别同类项（相同字母及指数），分组，合并。
Simplify 3a + 5b – 2a + 4b. Group a‑terms: 3a – 2a = a. Group b‑ + 9b. The answer key often uses colours or underlining in the book, but here we stress re‑writing the expression with terms ordered to avoid mistakes.
化简 3a + 5b – 2a + 4b。a 项合并：3a – 2a = a。b 项合并：5b + 4b = 9b。最终答案 a + 9b。书中答案常用颜色或下划线，这里我们强调重写表达式并排列有序以避免错误。
Substitution: If x = 3 and y = –2, evaluate 2x² – xy. The Support answer works stepwise: x² = 9, so 2 × 9 = 18; –xy = –(3 × –2) = –(– ackets around negatives are strongly recommended by the answer key to prevent sign errors.
代入求值：已知 x = 3, y = –2，求 2x² – xy。Support 答案逐步计算：x² = 9，所以 2 × 9 = 18；–xy = –(3 × –2) = –(–6) = +6。最后 18 + 6 = 24。答案强烈建议将负数括起来，避免符号错误。
7. Solving Linear Equations Step by Step | 一步步解线性方程答案
Book 8 Support introduces equations with one unknown via the balance method. The answer key insists on performing the same operation on both sides and clearly showing each line of working.
Book 8 Support 用天平法引入一元一次方程。答案坚持在等式两边进行相同运算，并清晰展示每一步。
Solve 5x – 3 = 2x + 9. Step 1: subtract 2x from both sides → 3x – = 12. Step 3: divide by 3 → x = 4. The Support answer always includes a check: LHS = 5(4) – 3 = 17, RHS = 2(4) + 9 = 17, so solution is correct.
解方程 5x – 3 = 2x + 9。第一步：两边减 2x → 3x – 3 = 9。第二步：两边加 3 → 3x = 12。第三步：除以 3 → x = 4。Support 答案都会验算：左 = 5(4) – 3 = 17，右 = 2(4) + 9 = 17，解正确。
ations with brackets like 2(3y + 1) = 16 are tackled by expanding first. 6y + 2 = 16, then – The key reinforces that the bracket must be fully multiplied out before collecting terms.
带括号的方程如 2(3y + 1) = 16，先展开：6y + 2 = 16，两边减 2 → 6y = 14，除以 6 → y = 14/6 = 7/3，答案写为 2 ⅓。重点强调必须先展开括号再移项。
8. Geometry: Angles and Lines | 几何：角度与直线答案
Angle facts on a straight line, around a point, and in triangles are essential KS3 geometry topics. The Support answer booklet visually maps angles and uses simple equations to find missing values.
直线上的角、点周围的角以及三角形内角和是 KS3 几何的基本内容。Support 答案册用图示法和简单方程求未知角。
ical question: four angles are 120°, 80°, y and 90° → 120 + 80 + 90 + y = 360, so y = 70°.
若直线上的三个角为 40°、70° 和 x，则 40 + 70 + x = 180，得 x = 70°。答案强调直线上的邻角之和为 180°。点周围角的和为 360°。典型题：四个角分别为 120°、80°、y 和 90°，则 120 + 80 + 90 + y = 360，得 y = 70°。
180°. An isosceles triangle with base angles of 50° each has its third angle equal to 180 – us to look for markings showing equal sides. Vertically opposite angles are also highlighted; they appear frequently in “find the missing angle” exercises.
三角形内角和为 180°。底角均为 50° 的等腰三角形，第三个角为 180 – (50 + 50) = 80°。答案提醒留意标识相等边的符号。对顶角相等也是重点，在“求角度”习题中频繁出现。
9. Perimeter, Area and Volume: Measurements in Action | 周长、面积与体积：实际测量答案
Measurement questions in Book 8 Support involve perimeters of composite shapes, area of triangles and parallelograms, and volume of cuboids. The answer key always includes units and often a diagram reference.
Book 8 Support 的测量题涉及组合图形的周长、三角形和平行四边形的面积以及长方体的体积。答案总是带上单位，并常附有图示参考。
½ × 6 × 4 = 12 m². The Support answers check that the height is perpendicular to the base.
长方形周长，长 15 cm 宽 8 cm：2 × (15 + 8) = 46 cm。对于由矩形和三角形组成的组合图形，要加总所有外边长。三角形面积 = ½ × 底 × 高。例：底 6 m，高 4 m → 面积 = ½ × 6 × 4 = 12 m²。Support 答案会检查高是否垂直于底。
units). Often, capacity in litres is linked: 1 litre = 1000 cm³, so 120 cm³ = 0.12 litres.
长方体体积 = 长 × 宽 × 高。一个 4 cm × 3 cm × 10 cm 的盒子体积为 120 cm³。答案严格区分体积（立方单位）与表面积（平方单位）。常联系升的容量：1 升 = 1000 cm³，所以 120 cm³ = 0.12 升。
10. Statistics: Handling Data and Charts | 统计：数据处理与图表答案
Data handling questions in Essential Maths Book 8 Support cover reading bar charts, pictograms, and simple pie charts, as well as calculating the mean, median, mode and range. The answer key guides how to extract information accurately and avoid common misreads.
Essential Maths Book 8 Support 中的数据处理题涵盖读条形图、象形图和简单饼图，以及计算平均数、中位数、众数和极差。答案指导如何准确提取信息，避免常见误读。
5, 7, 9, 12. With an even count, median = mean of the middle two: (5+7)/2 = 6. Range = 12 – 5 = 7. Mean = (5+12+7+5+9+5)/6 = 43/6 ≈ 7.17 (answer key may round to 7.2).
一组数据：5, 12, 7, 5, 9, 5。众数是 5（出现最频）。中位数：先排序 5,5,5,7,9,12，偶数个时中位数为中间两数的平均数 (5+7)/2=6。极差=12–5=7。平均数=(5+12+7+5+9+5)/6=43/6≈7.17（答案可能保留至 7.2）。
Bar chart interpretation: if a bar for “Year 7” reaches 45 on the frequency axis, and “Year 8” reaches 60, the answer key simply reads the height. For pictograms where one symbol represents 10 pupils, half a symbol represents 5. Pie chart questions often rely on angles: a 90° sector represents ¼ of the total, so if total is 500, the sector represents 125.
条形图解读：“7 年级”柱高频率轴 45，“8 年级”60，答案直接读取高度。象形图中如果一个符号代表 10 名学生，半个符号就是 5 名。饼图题常用角度：90° 扇形代表总数的 ¼，若总数为 500，该扇形代表 125。
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