📚 Mastering 9C Maths Homework: Answers Title Page | 掌握 9C 数学作业本答案标题页
Many students open a 9C homework book and go straight to the answers title page, hoping for a quick check. However, copying answers without understanding the methods will not help in exams. This article explains the key topics that usually appear on a Year 9C maths homework title page, with worked examples and common pitfalls. Use it as a revision guide rather than a shortcut.
许多学生打开 9C 作业本后会直接翻到答案标题页，希望快速核对。但是，只抄答案而不理解方法对考试毫无帮助。本文将解析 Year 9C 数学作业本标题页上通常出现的核心主题，配以例题和常见错误。请把它当作复习指南，而不是捷径。
1. Number Operations and Order of Operations | 数与运算及运算顺序
Addition and Subtraction. In 20 − 6 ÷ 2, division comes before subtraction, so 6 ÷ 2 = 3, then 20 − 3 = 17. If the question is (20 − 6) ÷ 2, the bracket changes everything: 20 − 6 = 14, then 14 ÷ 2 = 7. Always look for grouping symbols before calculating. A common error is to work strictly left to right and get 7 for the first expression; that is wrong because there are no brackets.
在 9C 答案标题页上，数与运算题常常考查运算顺序，即 BIDMAS：括号、指数、除法与乘法、加法与减法。在 20 − 6 ÷ 2 中，除法优先于减法，所以 6 ÷ 2 = 3，然后 20 − 3 = 17。如果是 (20 − 6) ÷ 2，括号改变一切：20 − 6 = 14，然后 14 ÷ 2 = 7。计算前一定要先找分组符号。常见错误是严格从左到右计算，把第一个式子算成 7；这是错的，因为没有括号。
20 − 6 ÷ 2 = 20 − 3 = 17
(20 − 6) ÷ 2 = 14 ÷ 2 = 7
t of 81 is 9, since 9 × 9 = 81. A frequent mistake is to ignore the sign or to treat −3² as (−3)². Write out the meaning of the power before calculating.
负数、乘方和根号也会出现在这一部分。例如，(−3)² = 9，但 −3² = −9，因为指数只作用于 3。81 的平方根是 9，因为 9 × 9 = 81。常见错误是忽略符号，或者把 −3² 当成 (−3)²。计算前先把乘方的含义写清楚。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
and percentages are interchangeable forms of the same value. For example, 3/4 = 0.75 = 75%. To find 15% r reverse percentage problems, do not simply subtract or add the percentage. If a price is £96 after a 20% increase, the original price is 96 ÷ 1.20 = £80, not 96 × 0.80.
分数、小数和百分数是同一个值的不同表达形式。例如，3/4 = 0.75 = 75%。求 240 的 15%，先把百分数化成小数：0.15 × 240 = 36。对于反推百分数问题，不要简单地加减百分数。如果某商品涨价 20% 后是 96 英镑，原价是 96 ÷ 1.20 = 80 英镑，而不是 96 × 0.80。
When adding or subtracting fractions, find a common denominator first. When multiplying fractions, multiply numerators and denominators directly. When dividing by a fraction, multiply by its reciprocal. A common error is to cancel across addition signs, such as treating (2 + 3)/6 as 2/6 + 3/6 incorrectly without simplifying properly.
分数加减时，先找公分母。分数相乘时，分子乘分子、分母乘分母。除以一个分数，等于乘它的倒数。常见错误是在加法中直接约分，例如把 (2 + 3)/6 错误地拆开约分而不按步骤化简。
3. Ratio, Proportion and Rates | 比、比例与速率
an unknown. To simplify a ratio, divide all parts by their highest common factor. To share £180 in the ratio 2 : 3 : 5, add the parts: 2 + 3 + 5 = 10. One part is £180 ÷ 10 = £18, so the shares are £36, £54 and £90. Always check that the shares add back to the original total.
比例题常常要求化简、按比例分配或求未知量。化简比时，把所有部分同时除以最大公因数。把 180 英镑按 2 : 3 : 5 分配，先加总份数：2 + 3 + 5 = 10。一份是 180 ÷ 10 = 18 英镑，所以三份分别是 36、54 和 90 英镑。最后要检查各份之和是否等于原总量。
y = mass ÷ volume. A common error is mixing units, such as using minutes with km/h. Convert units before substituting into the formula.
速率问题涉及速度、密度或单价。速度 = 路程 ÷ 时间，所以一辆车 2 小时行驶 150 千米，平均速度是 75 千米/时。密度 = 质量 ÷ 体积。常见错误是单位混用，例如把分钟和千米/时混在一起。代入公式前先统一单位。
4. Algebraic Expressions and Substitution | 代数式与代入求值
s the distributive law: 3(2x − 5) = 6x − 15. When substituting, replace the variable with its value and use brackets to avoid sign errors. Substitute x = −2 into x² + 3x: (−2)² + 3(−2) = 4 − 6 = −2.
9C 标题页上的代数题通常从合并同类项开始。例如，5x + 3y − 2x + 4y 化简为 3x + 7y。展开括号使用分配律：3(2x − 5) = 6x − 15。代入求值时，用数值替换字母，并加括号避免符号错误。把 x = −2 代入 x² + 3x： (−2)² + 3(−2) = 4 − 6 = −2。
ⁿ. For a power of a power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ. A common mistake is to multiply the indices when adding terms, but index laws apply to multiplication and division, not addition.
指数法则也很重要。同底数相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。同底数相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。常见错误是在加法中乱用指数法则；指数法则适用于乘除，不适用于加减。
5. Linear Equations and Rearranging Formulae | 一元一次方程与公式变形
o get 2x = 16, so x = 8. Always substitute your answer back into the original equation to check it.
解一元一次方程时，要对等式两边做相同的运算。解 5x − 7 = 3x + 9，两边同时减 3x 得 2x − 7 = 9，再加 7 得 2x = 16，所以 x = 8。最后把答案代回原方程检验。
common error is to move terms without changing their signs, or to divide only one side. Write each step clearly so the inverse operations are visible.
公式变形同样遵循等式平衡原则。要让 x 成为 y = 4x − 3 的主项，两边加 3：y + 3 = 4x，再除以 4：x = (y + 3)/4。常见错误是移项不变号，或者只对一边做除法。把每一步写清楚，让逆运算看得见。
6. Coordinates, Graphs and Straight Lines | 坐标、图像与直线
3) is (13 − 5)/(6 − 2) = 8/4 = 2. The midpoint of the same two points is ((2 + 6)/2, (5 + 13)/2) = (4, 9). Plotting points accurately helps you see whether the line rises or falls.
直线图像通常写成 y = mx + c，其中 m 是斜率，c 是 y 轴截距。两点 (2, 5) 和 (6, 13) 之间的斜率是 (13 − 5)/(6 − 2) = 8/4 = 2。同样两点的中点是 ((2 + 6)/2, (5 + 13)/2) = (4, 9)。准确描点有助于判断直线上升还是下降。
Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to −1, so if one gradient is 2, the perpendicular gradient is −1/2. A common error is to confuse the gradient with the intercept, or to read the axes in the wrong order. Always label axes and check the scale.
平行线斜率相同。垂直直线的斜率乘积为 −1，所以如果一条线斜率是 2，垂直线的斜率就是 −1/2。常见错误是把斜率和截距混淆，或者看错坐标轴顺序。要标注坐标轴并检查刻度。
7. Sequences and nth Term | 数列与第 n 项
is 3, so the nth term is 3n + 1. Check by substituting n = 1: 3 × 1 + 1 = 4. The term-to-term rule is "add 3".
数列题要求续写规律、找相邻项规则或求第 n 项。在数列 4, 7, 10, 13 中，公差是 3，所以第 n 项是 3n + 1。代入 n = 1 检验：3 × 1 + 1 = 4。相邻项规则是 "加 3"。
Quadratic sequences have a constant second difference, and their nth term contains n². For example, 2, 5, 10, 17 has first differences 3, 5, 7 and second difference 2, so the nth term is n² + 1. A common mistake is to assume every sequence is linear. Always check whether the first difference changes.
二次数列的二阶差分为常数，其第 n 项含有 n²。例如，2, 5, 10, 17 的一阶差是 3, 5, 7，二阶差是 2，所以第 n 项是 n² + 1。常见错误是假定所有数列都是线性的。要检查一阶差是否变化。
8. Angles, Polygons and Bearings | 角、多边形与方位角
Angle facts are fundamental. Angles on a straight line add to 180°, angles around a point add to 360°, and angles in a triangle add to 180°. With parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°. Mark parallel lines and transversals clearly before writing reasons.
角度基本事实是基础。直线上的角相加为 180°，围绕一点的角相加为 360°，三角形内角和为 180°。在平行线中，内错角相等，同位角相等，同旁内角相加为 180°。写理由之前，先标清平行线和截线。
f sides. The exterior angle of a regular polygon is 360°/n. Bearings are measured clockwise from north and are written with three figures, such as 075°. A common error is measuring anticlockwise or using two figures.
对于多边形，内角和为 (n − 2) × 180°，其中 n 是边数。正多边形的外角是 360°/n。方位角从正北顺时针测量，写成三位数，例如 075°。常见错误是逆时针测量，或者只写两位数。
9. Area, Perimeter, Circles and Pythagoras | 面积、周长、圆与勾股定理
mference of a circle is 2πr or πd, and the area is πr². Make sure you use the radius, not the diameter, in πr². If the diameter is 10 cm, the radius is 5 cm and the area is 25π cm².
面积和周长在 9C 答案页上经常出现。长方形面积是长 × 宽。三角形面积是 ½ × 底 × 高。圆的周长是 2πr 或 πd，面积是 πr²。用 πr² 时一定要用半径，不要用直径。如果直径是 10 厘米，半径就是 5 厘米，面积是 25π 平方厘米。
Pythagoras' add the squares but forget to take the square root, or to label the hypotenuse incorrectly. The hypotenuse is always opposite the right angle.
勾股定理指出，在直角三角形中，a² + b² = c²，其中 c 是斜边。如果 a = 6，b = 8，那么 c² = 36 + 64 = 100，所以 c = 10。常见错误是加完平方后忘记开方，或者标错斜边。斜边总是直角所对的边。
10. Statistics, Probability and Averages | 统计、概率与平均数
Statistics questions test mean, median, mode and range. The mean is the sum of values divided by the number of values. The median is the middle value after ordering the data. The mode is the most frequent value, and the range is the highest value minus the lowest
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