📚 Essential Maths Book 9 Answers | 核心数学第九册知识点精讲
This article walks you through selected answers and key explanations from Essential Maths Book 9, a popular KS3 resource. Each section tackles a core topic, showing step-by-step solutions and the reasoning behind them. Whether you are checking your work or revising for a test, these model answers will help you understand the methods and avoid common mistakes. The coverage includes number, algebra, geometry, ratio, statistics, and more, all aligned to the KS3 curriculum.
本文带你梳理《核心数学第九册》中的精选答案与关键解释，这是一套广泛使用的 KS3 学习资料。每个小节聚焦一个核心主题，逐步展示解题过程与背后的思路。无论你是在核对作业还是备考复习，这些标准答案都能帮助你掌握方法、避开常见错误。内容涵盖数字、代数、几何、比率、统计等，全部贴合 KS3 课程大纲。
1. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, we first need a common denominator. For example, to work out 3/4 + 1/6, we find the least common multiple of 4 and 6, which is 12. Then we convert each fraction: 3/4 = 9/12 and 1/6 = 2/12. Adding gives 11/12. The answer is fully simplified because 11 and 12 share no common factors other than 1.
进行分数的加减法时，首先需要通分。例如计算 3/4 + 1/6，我们先找出 4 和 6 的最小公倍数 12。将每个分数转换：3/4 = 9/12，1/6 = 2/12。相加得到 11/12。因为 11 和 12 除了 1 之外没有其他公因数，答案已经是最简形式。
When subtracting, the same rule applies. For 5/8 − 1/3, the LCM of 8 and 3 is 24. So 5/8 = 15/24 and 1/3 = 8/24. Subtract to get 7/24. Again, 7 and 24 are coprime, so this is the final answer. Always check if simplification is possible — if numerator and denominator share a factor, divide them by it.
减法遵循同样规则。计算 5/8 − 1/3，8 和 3 的最小公倍数是 24。因此 5/8 = 15/24，1/3 = 8/24。相减得 7/24。7 和 24 互质，所以这是最终答案。每次都检查是否可以约分——如果分子和分母有公因数，就同时除以它。
2. Multiplying and Dividing Fractions | 分数的乘法与除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. For 2/3 × 5/7, we get 10/21. No cross‑cancellation is needed here, but if we had 3/8 × 4/9, we could cancel common factors before multiplying: divide 3 and 9 by 3, giving 1 and 3; divide 4 and 8 by 4, giving 1 and 2. Then multiply: 1/2 × 1/3 = 1/6. This saves simplification later.
分数乘法很简单：分子相乘，分母相乘。对于 2/3 × 5/7，得到 10/21。这里无需约分，但如果是 3/8 × 4/9，我们可以在相乘前先约去公因数：3 和 9 除以 3 得 1 和 3；4 和 8 除以 4 得 1 和 2。然后相乘：1/2 × 1/3 = 1/6。这样避免了后续的约分。
To divide fractions, flip the second fraction (find its reciprocal) and then multiply. For 3/5 ÷ 2/3, we change it to 3/5 × 3/2 = 9/10. If mixed numbers appear, convert them to improper fractions first. For example, 1 1/2 ÷ 3/4 becomes 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2.
分数除法要翻转第二个分数（取它的倒数），然后相乘。计算 3/5 ÷ 2/3，变为 3/5 × 3/2 = 9/10。如果出现带分数，先化为假分数。例如 1 1/2 ÷ 3/4 变为 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2。
3. Simplifying Algebraic Expressions | 代数式的化简
To simplify expressions, collect like terms. In 3a + 5b − 2a + 4b, combine a‑terms: 3a − 2a = a, and b‑terms: 5b + 4b = 9b. So the simplified expression is a + 9b. Note that you cannot combine a and b because they are different letters.
化简代数式时，要合并同类项。在 3a + 5b − 2a + 4b 中，合并 a 项：3a − 2a = a，合并 b 项：5b + 4b = 9b。化简结果为 a + 9b。注意 a 和 b 是不同字母，不能直接合并。
When multiplying terms, write numbers first, then letters in alphabetical order, and use indices. For 2x × 3x, multiply coefficients: 2 × 3 = 6, and variables: x × x = x², giving 6x². For mixed variables like 4a × 5b × 2a, we get 4 × 5 × 2 = 40, and a × a = a², plus b, so the result is 40a²b.
代数项相乘时，系数相乘，字母按字母表顺序书写并使用指数。例如 2x × 3x，系数相乘：2 × 3 = 6，变量相乘 x × x = x²，得 6x²。对于混合变量如 4a × 5b × 2a，4 × 5 × 2 = 40，a × a = a²，再乘以 b，结果为 40a²b。
4. Solving Linear Equations | 解一元一次方程
The goal is to isolate the unknown on one side of the equation. Take the equation 4x + 3 = 19. Subtract 3 from both sides: 4x = 16. Then divide both sides by 4: x = 4. Always check your answer by substituting it back: 4(4) + 3 = 16 + 3 = 19, correct.
目标是让未知数单独出现在方程的一边。以方程 4x + 3 = 19 为例，两边同时减去 3：4x = 16。然后两边同时除以 4：x = 4。一定要把答案代回原式检验：4(4) + 3 = 16 + 3 = 19，正确。
When variables appear on both sides, remove the smaller variable term first. For 5x − 2 = 3x + 8, subtract 3x from both sides: 2x − 2 = 8. Add 2 to both sides: 2x = 10, so x = 5. To solve equations with brackets, expand first: 2(x + 3) = 10 becomes 2x + 6 = 10, then 2x = 4, x = 2.
当未知数出现在方程两边时，先消去较小的未知数项。对于 5x − 2 = 3x + 8，两边同时减去 3x：2x − 2 = 8。两边加 2：2x = 10，所以 x = 5。解带括号的方程要先展开：2(x + 3) = 10 化为 2x + 6 = 10，然后 2x = 4，x = 2。
5. Working with Percentages | 百分数的运算
To find a percentage of an amount, convert the percentage to a decimal and multiply. For 15% of 240, we calculate 0.15 × 240 = 36. Alternatively, find 10% (24) and 5% (half of 10%, so 12), then add: 24 + 12 = 36. This mental method is useful in non‑calculator contexts.
求一个数的百分之几，将百分数转换为小数再相乘。求 240 的 15%，计算 0.15 × 240 = 36。另一种方法是先求 10%（24），再求 5%（10% 的一半，即 12），然后相加：24 + 12 = 36。这种心算方法在非计算器题目中非常实用。
To increase or decrease an amount by a percentage, use a multiplier. For a 20% increase, multiply by 1.20; for a 15% decrease, multiply by 0.85. For example, increasing 350 by 20% gives 350 × 1.2 = 420. Decreasing 480 by 15% gives 480 × 0.85 = 408. Reverse percentage problems require dividing by the multiplier: if a price of £76 includes a 20% profit, the cost was 76 ÷ 1.2 = £63.33 (to two decimal places).
按百分比增加或减少一个数时，使用乘数。增加 20% 就乘以 1.20；减少 15% 就乘以 0.85。例如 350 增加 20% 得到 350 × 1.2 = 420。480 减少 15% 得到 480 × 0.85 = 408。逆向百分数问题则需要除以乘数：如果标价 76 英镑中包含 20% 的利润，成本就是 76 ÷ 1.2 = 63.33 英镑（保留两位小数）。
6. Ratio and Proportion | 比与比例
Ratios compare quantities of the same kind. To simplify a ratio like 18:24, divide both sides by their highest common factor, 6, to get 3:4. When sharing an amount in a given ratio, first add the parts. If the ratio is 3:5, there are 8 parts in total. To share £64 in this ratio, each part is £64 ÷ 8 = £8. Then 3 parts = £24 and 5 parts = £40.
比用来比较同一类量的大小。化简比例如 18:24，两边同除以它们的最大公因数 6，得到 3:4。按给定比例分配一个数时，先求总份数。若比为 3:5，总份数是 8。将 64 英镑按此比例分配，每份为 64 ÷ 8 = 8 英镑。那么 3 份是 24 英镑，5 份是 40 英镑。
Proportion problems often involve scaling up or down. If 5 pens cost £4, then 15 pens cost 3 times as much, so £12. This is direct proportion. When two quantities are inversely proportional, doubling one halves the other. Use the unitary method (finding the value of one unit) to solve most proportion questions accurately.
比例问题通常涉及放大或缩小。如果 5 支笔花费 4 英镑，那么 15 支笔的价钱就是 3 倍，即 12 英镑。这是正比例关系。当两个量成反比时，一个量加倍，另一个量减半。使用统一单位法（先求一个单位的对应值）可以准确解决大部分比例问题。
7. Area and Perimeter of 2D Shapes | 平面图形的面积与周长
Perimeter is the distance around the outside of a shape. For a rectangle of length 8 cm and width 5 cm, perimeter = 2(8 + 5) = 26 cm. For compound shapes, break them into rectangles, find missing side lengths, then add all the outer sides. Always include units in your final answer.
周长是图形外部边界的总长度。一个长 8 厘米、宽 5 厘米的长方形，周长 = 2(8 + 5) = 26 厘米。对于复合图形，将其拆分成长方形，找出隐藏边长，然后把所有外围边长相加。最终答案一定要写明单位。
Area of a rectangle is length × width, so 8 × 5 = 40 cm². The area of a triangle is ½ × base × height. For a triangle with base 10 cm and height 6 cm, area = ½ × 10 × 6 = 30 cm². Remember the height must be perpendicular to the base. For a parallelogram, area = base × perpendicular height, and for a trapezium, area = ½(a + b) × h, where a and b are the parallel sides.
长方形面积 = 长 × 宽，因此 8 × 5 = 40 平方厘米。三角形面积 = ½ × 底 × 高。若一个三角形底 10 厘米，高 6 厘米，面积 = ½ × 10 × 6 = 30 平方厘米。注意高必须与底边垂直。平行四边形的面积 = 底 × 垂直高度，梯形面积 = ½(a + b) × h，其中 a 和 b 是两条平行边。
8. Averages and Range | 平均数与极差
The mean is calculated by adding all values and dividing by the number of values. For the data set 4, 8, 6, 10, 12, the sum is 40 and there are 5 numbers, so mean = 40 ÷ 5 = 8. The median is the middle value when data are ordered: 4, 6, 8, 10, 12 gives median 8. For an even number of values, take the mean of the two middle numbers.
算术平均数等于所有数值之和除以数据个数。数据集 4, 8, 6, 10, 12 的总和为 40，有 5 个数，因此平均数 = 40 ÷ 5 = 8。中位数是将数据排序后位于中间的值：4, 6, 8, 10, 12，中位数是 8。如果数据个数为偶数，则取中间两个数的平均数。
The mode is the most frequent value. In 3, 5, 5, 7, 9, 5, 12, the mode is 5. The range is the difference between the largest and smallest values: 12 − 3 = 9. The range measures spread, while averages measure central tendency. Choose the most appropriate average to describe a data set: if there are outliers, the median is often better than the mean.
众数是出现频率最高的数值。在 3, 5, 5, 7, 9, 5, 12 中，众数是 5。极差是最大值与最小值之差：12 − 3 = 9。极差衡量数据的分散程度，而平均数衡量集中趋势。选择最合适的平均数来描述数据集：如果存在极端值，中位数往往比平均数更合适。
9. Coordinates and Graphs | 坐标与图像
Points on a coordinate grid are written as (x, y). The x‑coordinate tells you how far to move horizontally from the origin (0,0), and the y‑coordinate how far vertically. To plot a straight line like y = 2x + 1, build a table of values. For x = 0, y = 1; x = 1, y = 3; x = 2, y = 5. Join the points with a straight line extending in both directions.
坐标网格上的点用 (x, y) 表示。x 坐标表示从原点 (0,0) 水平移动的距离，y 坐标表示垂直移动的距离。要画出 y = 2x + 1 这样的直线，先列出数值表。当 x = 0 时，y = 1；x = 1，y = 3；x = 2，y = 5。将这些点用直线连接并向两端延长。
To find the midpoint of two points, average the x‑coordinates and the y‑coordinates separately. Midpoint of (2, 5) and (8, 11) is ((2+8)÷2, (5+11)÷2) = (5, 8). Parallel lines have the same gradient, and perpendicular lines have gradients that multiply to −1. The gradient of a line between (x₁,y₁) and (x₂,y₂) is (y₂ − y₁)/(x₂ − x₁).
要找两点之间的中点，分别对 x 坐标和 y 坐标取平均值。(2, 5) 和 (8, 11) 的中点是 ((2+8)÷2, (5+11)÷2) = (5, 8)。平行线具有相同的斜率，垂直线的斜率乘积为 −1。过 (x₁,y₁) 和 (x₂,y₂) 两点的直线斜率为 (y₂ − y₁)/(x₂ − x₁)。
10. Angles and Lines | 角与线
Angles on a straight line add up to 180°. If one angle is 68°, the adjacent angle is 180° − 68° = 112°. Angles around a point sum to 360°. Vertically opposite angles are equal. When two parallel lines are cut by a transversal, corresponding angles are equal, alternate angles are equal, and co‑interior angles sum to 180°.
直线上的角之和为 180°。如果一个角是 68°，相邻角就是 180° − 68° = 112°。围绕一个点的周角之和为 360°。对顶角相等。当两条平行线被一条横截线所截时，同位角相等，内错角相等，同旁内角之和为 180°。
The interior angles of a triangle total 180°. In a quadrilateral, they total 360°. You can find missing angles by setting up simple equations. For example, in a triangle with angles x, 2x and 60°, we have x + 2x + 60 = 180 → 3x = 120 → x = 40°. Then the angles are 40°, 80° and 60°.
三角形的内角和为 180°。四边形的内角和为 360°。通过建立简单方程可以求出未知角。例如，一个三角形的三个角分别为 x、2x 和 60°，那么 x + 2x + 60 = 180 → 3x = 120 → x = 40°。因此三个角分别是 40°、80° 和 60°。
11. Probability Basics | 概率基础
Probability is a number between 0 and 1 that describes how likely an event is. Probability = (number of favourable outcomes) ÷ (total number of possible outcomes). When rolling a fair six‑sided die, the probability of rolling a 4 is 1/6. Probability of an even number is 3/6 = 1/2.
概率是一个 0 到 1 之间的数，用来描述某个事件发生的可能性大小。概率 = 有利结果数 ÷ 所有可能结果的总数。掷一枚均匀的六面骰子，掷出 4 的概率是 1/6。掷出偶数的概率是 3/6 = 1/2。
Events are mutually exclusive if they cannot happen at the same time. The probability of either event occurring is the sum of their individual probabilities. For example, probability of rolling a 2 or a 5 on a die is 1/6 + 1/6 = 2/6 = 1/3. If an event is certain, its probability is 1; if impossible, it is 0.
如果两个事件不可能同时发生，则称它们互斥。任一事件发生的概率等于各自概率之和。例如，掷骰子得到 2 或 5 的概率是 1/6 + 1/6 = 2/6 = 1/3。如果事件必然发生，概率为 1；如果事件不可能发生，概率为 0。
12. Units and Conversions | 单位与换算
Common metric conversions: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. For mass, 1 kg = 1000 g, 1 tonne = 1000 kg. For capacity, 1 L = 1000 mL, 1 L = 1000 cm³. To convert from a larger unit to a smaller one, multiply; from smaller to larger, divide. For example, 3.2 km to metres is 3.2 × 1000 = 3200 m.
常见公制换算：1 千米 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米。质量方面，1 千克 = 1000 克，1 吨 = 1000 千克。容量方面，1 升 = 1000 毫升，1 升 = 1000 立方厘米。大单位化小单位用乘法；小单位化大单位用除法。例如 3.2 千米化为米是 3.2 × 1000 = 3200 米。
Imperial units still appear in context: 1 inch ≈ 2.54 cm, 1 foot = 12 inches, 1 mile ≈ 1.6 km. For approximate conversions, remember 5 miles ≈ 8 km. Always check the required precision in word problems and state final answers with appropriate units.
英制单位在题目背景中仍会出现：1 英寸 ≈ 2.54 厘米，1 英尺 = 12 英寸，1 英里 ≈ 1.6 千米。进行估算时，记住 5 英里 ≈ 8 千米。在应用题中注意题目要求的精度，并用正确的单位给出最终答案。
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