📚 Essential Maths Book 9S Answers: Key Points Explained | Essential Maths Book 9S 答案知识点精讲
Welcome to a focused guide through the key topics and typical answers found in Essential Maths Book 9S. Whether you are checking your work or revising for tests, these explanations break down the reasoning behind the solutions, helping you to build solid problem-solving skills across algebra, geometry, statistics and more.
欢迎来到 Essential Maths Book 9S 的重点知识点及典型答案解析指南。无论你是在核对作业还是为考试复习，以下内容都会详解答案背后的推理过程，帮助你在代数、几何、统计等方面建立扎实的解题能力。
1. Solving Linear Equations | 解一元一次方程
A typical question asks you to solve 3x + 7 = 22. The aim is to isolate x by performing inverse operations. First subtract 7 from both sides to get 3x = 15, then divide by 3, giving x = 5. Always check by substituting back: 3(5) + 7 = 15 + 7 = 22.
典型题目要求解方程 3x + 7 = 22。目标是运用逆运算将 x 单独留在一边。首先两边减 7 得到 3x = 15，再除以 3，得 x = 5。务必回代检验：3(5) + 7 = 15 + 7 = 22。
When equations involve brackets, such as 2(x – 4) = 10, expand first to produce 2x – 8 = 10. Then add 8 to both sides: 2x = 18, so x = 9.
如果方程含有括号，例如 2(x – 4) = 10，要先展开得到 2x – 8 = 10，然后两边加 8：2x = 18，所以 x = 9。
For equations with variables on both sides, like 5y + 2 = 3y + 10, collect like terms. Subtract 3y from both sides to get 2y + 2 = 10, then subtract 2: 2y = 8, so y = 4.
对于变量在等号两边的方程，例如 5y + 2 = 3y + 10，要把同类项合并。两边减 3y 得 2y + 2 = 10，再减 2：2y = 8，所以 y = 4。
2. Direct Proportion | 正比例
Direct proportion means that two quantities increase or decrease at the same rate. If 5 pens cost £3.50, the cost per pen is £3.50 ÷ 5 = £0.70. Therefore 8 pens cost 8 × £0.70 = £5.60. This can also be set up as a ratio: cost/quantity is constant.
正比例意味着两个量以相同速率增减。如果 5 支笔花费 £3.50，每支笔价格为 £3.50 ÷ 5 = £0.70。那么 8 支笔花费 8 × £0.70 = £5.60。也可以用比例式表达：总价/数量是常数。
The unitary method is the key approach in Book 9S answers: find the value of one item first, then scale up. Another method involves setting up equivalent fractions: 5/3.50 = 8/x, cross-multiplying gives 5x = 28, so x = 5.60.
Book 9S 答案中核心方法是单位法：先求出一个单位的量，再放大。另一种方法是建立等价分数：5/3.50 = 8/x，交叉相乘得 5x = 28，所以 x = 5.60。
3. Percentage Increase and Decrease | 百分数增减
To increase £240 by 15%, first find 10% (£24) and 5% (£12), then add: 15% = £36, so new amount is £240 + £36 = £276. Alternatively use the multiplier 1.15: 240 × 1.15 = 276.
将 £240 增加 15%，可以先算 10% (£24) 和 5% (£12)，相加得 15% = £36，新金额为 £240 + £36 = £276。也可使用乘数 1.15：240 × 1.15 = 276。
For percentage decrease, say reducing 80 kg by 35%, find 10% (8 kg) → 30% = 24 kg, 5% = 4 kg, total 28 kg. New mass is 80 – 28 = 52 kg. The multiplier method uses 0.65 directly: 80 × 0.65 = 52.
对于百分数减少，例如将 80 kg 减少 35%，先求 10% (8 kg) → 30% = 24 kg，5% = 4 kg，合计 28 kg。新质量为 80 – 28 = 52 kg。乘数法直接用 0.65：80 × 0.65 = 52。
4. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². For a triangle with legs 6 cm and 8 cm, the hypotenuse c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
在直角三角形中，斜边的平方等于两直角边的平方和：a² + b² = c²。对于直角边为 6 cm 和 8 cm 的三角形，斜边 c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm。
When finding a shorter side, rearrange the formula. For example, if hypotenuse = 13 m and one leg = 5 m, then a² = c² – b² = 169 – 25 = 144, so a = √144 = 12 m. Book 9S answers always show full rearrangement steps.
若要求直角边，需重写公式。例如斜边长 13 m，一直角边为 5 m，则 a² = c² – b² = 169 – 25 = 144，所以 a = √144 = 12 m。Book 9S 答案总是展示完整的移项步骤。
5. Area of Composite Shapes | 组合图形面积
To find the area of an L-shaped figure, split it into two rectangles. Suppose the shape consists of a 10 cm by 4 cm rectangle and a 6 cm by 5 cm rectangle. Calculate areas: 10 × 4 = 40 cm² and 6 × 5 = 30 cm², total area = 70 cm².
求 L 形图形的面积，可以拆分成两个矩形。假设图形由一个 10 cm x 4 cm 和一个 6 cm x 5 cm 的矩形组成。面积分别为 10 × 4 = 40 cm² 和 6 × 5 = 30 cm²，总面积 70 cm²。
For shapes with holes, like a rectangle with a square cut out, calculate the area of the full rectangle and subtract the area of the hole. If full rectangle is 12 cm by 8 cm (area 96 cm²) and the square hole is 3 cm by 3 cm (9 cm²), the shaded area is 96 – 9 = 87 cm².
对于有镂空的图形，比如一个长方形挖去一个正方形，先算完整长方形面积再减去孔洞面积。若完整长方形为 12 cm x 8 cm（96 cm²），正方形孔洞 3 cm x 3 cm（9 cm²），则阴影面积为 96 – 9 = 87 cm²。
6. Mean, Median and Range | 平均数、中位数与极差
The mean of a data set is found by summing all values and dividing by the number of values. For the numbers 4, 7, 8, 10, 12, the sum is 41, so the mean is 41 ÷ 5 = 8.2. The median is the middle value when ordered: here 8. If there is an even number of values, the median is the mean of the two middle numbers.
数据集的平均数由所有数值之和除以数据个数求得。对于数字 4、7、8、10、12，和为 41，平均数为 41 ÷ 5 = 8.2。中位数是将数据排序后中间的那个数，这里是 8。如果有偶数个数据，中位数是中间两个数的平均数。
The range is the difference between the maximum and minimum; here 12 – 4 = 8. Book 9S answers often emphasise using ordered lists to ensure correct median and range.
极差是最大值与最小值的差，这里 12 – 4 = 8。Book 9S 答案常强调先把数据按顺序列表，以确保中位数和极差正确。
7. Probability of Single Events | 单事件概率
Probability = number of favourable outcomes ÷ total number of possible outcomes. A fair six-sided dice has total outcomes 6. The probability of rolling a prime number (2, 3, 5) is 3/6 = 1/2. Answers are often simplified to simplest fractions, decimals or percentages.
概率 = 有利结果数 ÷ 所有可能结果总数。一个公平的六面骰子总结果数为 6。掷出质数（2、3、5）的概率为 3/6 = 1/2。答案通常化简为最简分数、小数或百分数。
If a bag contains 4 red, 3 blue and 2 green counters, the probability of drawing a red is 4/(4+3+2) = 4/9. The complementary probability of not drawing a red is 1 – 4/9 = 5/9.
如果一个袋子里有 4 个红球、3 个蓝球和 2 个绿球，抽到红球的概率为 4/(4+3+2) = 4/9。未抽到红球的互补概率为 1 – 4/9 = 5/9。
8. Straight‑Line Graphs | 一次函数图像
The equation of a straight line is often written as y = mx + c, where m is the gradient and c is the y‑intercept. For y = 2x + 3, the line crosses the y‑axis at (0,3) and has a slope of 2, meaning for every 1 unit right, go up 2 units.
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是截距。对于 y = 2x + 3，直线与 y 轴交于 (0,3)，斜率为 2，表示每向右移动 1 个单位，向上移动 2 个单位。
To plot the graph, choose x‑values (e.g. -2, -1, 0, 1, 2), calculate the corresponding y‑values, plot the points and join with a straight line. Book 9S answers always check for consistent gradients from plotted points.
画函数图像时，选取 x 值（如 -2, -1, 0, 1, 2），计算对应的 y 值，描点后用直尺连成直线。Book 9S 答案总会检查各点间的斜率是否一致。
9. Powers and Roots | 幂与方根
An exponent tells how many times to multiply a base by itself. For 3⁴, multiply 3 × 3 × 3 × 3 = 81. Any number to the power 0 equals 1, except 0 itself. Negative exponents represent reciprocals: 2⁻³ = 1/(2³) = 1/8.
指数表示底数自乘的次数。3⁴ 就是将 3 自乘 4 次：3 × 3 × 3 × 3 = 81。任何非零数的 0 次方都是 1。负指数代表倒数：2⁻³ = 1/(2³) = 1/8。
The square root symbol √ asks for what number multiplied by itself gives the radicand. √81 = 9 because 9² = 81. Cube root ³√64 = 4 because 4³ = 64. When simplifying expressions like √(x²), the answer is |x| for even roots in standard arithmetic.
平方根符号 √ 表示求哪个数自乘等于被开方数。√81 = 9，因为 9² = 81。立方根 ³√64 = 4，因为 4³ = 64。化简像 √(x²) 这样的表达式时，在普通算术中偶次根的结果是 |x|。
10. Transformations: Translation and Reflection | 变换：平移与反射
A translation moves a shape without rotating or resizing it. It is described by a column vector, e.g. (3, -2) means move 3 units right and 2 units down. Every point of the shape shifts by the same vector.
平移是将图形移动而不旋转或改变大小。用列向量描述，如 (3, -2) 表示向右移动 3 个单位、向下移动 2 个单位。图形上的每一个点都按相同向量移动。
A reflection flips a shape over a mirror line. For reflection in the x‑axis, (x, y) becomes (x, -y). For the y‑axis, (x, y) becomes (-x, y). When reflecting in y = x, coordinates swap. Book 9S answers require precise coordinate mapping.
反射是将图形沿对称轴翻转。关于 x 轴反射，(x, y) 变为 (x, -y)；关于 y 轴反射，(x, y) 变为 (-x, y)；关于 y = x 反射，坐标互换。Book 9S 答案要求精确的坐标对应。
11. Ratio and Proportion Problems | 比和比例问题
If a recipe uses flour and sugar in the ratio 3:2 and you have 450 g of flour, the amount of sugar can be found by scaling. The ratio 3:2 means for every 3 parts flour, 2 parts sugar. 450 g corresponds to 3 parts, so 1 part = 450 ÷ 3 = 150 g, thus sugar = 2 × 150 = 300 g.
如果一个食谱中面粉与糖的比例为 3:2，你有 450 g 面粉，可以通过比例缩放求糖的用量。比例 3:2 意味着 3 份面粉对应 2 份糖。450 g 对应 3 份，1 份 = 450 ÷ 3 = 150 g，因此糖 = 2 × 150 = 300 g。
When dividing £120 in the ratio 3:5, total parts = 8; each part = £120 ÷ 8 = £15. The first share is 3 × £15 = £45, the second 5 × £15 = £75. Always check that the shares add to the original amount.
将 £120 按 3:5 分配时，总份数 = 8；每份 = £120 ÷ 8 = £15。第一份为 3 × £15 = £45，第二份为 5 × £15 = £75。记得检查分配后的总和是否等于原数。
12. Angles in Polygons | 多边形内角
For an n-sided polygon, the sum of interior angles = (n – 2) × 180°. A hexagon (n=6) has interior angle sum = (6 – 2) × 180 = 720°. In a regular polygon, each interior angle = sum ÷ n, so for a regular hexagon each interior angle is 720 ÷ 6 = 120°.
n 边形的内角和 = (n – 2) × 180°。六边形（n=6）内角和 = (6 – 2) × 180 = 720°。在正多边形中，每个内角 = 内角和 ÷ n，所以正六边形每个内角是 720 ÷ 6 = 120°。
The exterior angles of any polygon always sum to 360°. In a regular polygon, each exterior angle = 360° ÷ n. These rules are frequently used in Book 9S to find missing angles in diagrams.
任何多边形的外角和始终为 360°。在正多边形中，每个外角 = 360° ÷ n。在 Book 9S 中，这些规则被频繁用于求解图形中的未知角。
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