📚 7S Answers & Prelims | 7S 数学预考答案解析指南
This article walks through the essential answers and working methods for the 7S preliminary mathematics paper. It is designed to help students check their solutions, understand common errors, and build confidence before the final assessment.
本文梳理 7S 数学预考试卷的核心答案与解题方法，旨在帮助学生核对答案、理解常见错误，并在最终评估前建立信心。
1. Understanding the Prelims Format | 了解预考格式
The 7S prelims paper usually contains two sections: a non-calculator section testing mental arithmetic and written methods, and a calculator section covering problem solving and real-world contexts. Each question is marked for both method and final answer.
7S 预考试卷通常包含两个部分：不允许使用计算器的部分，考查心算和笔算方法；允许使用计算器的部分，考查问题解决和实际情境。每小题既看方法也看最终答案。
Typical question types include short-answer numerical questions, multiple-choice items, and structured word problems that require clear working. Marks are often awarded for substitution, rearrangement, and simplification, not only for the final value.
常见题型包括数值简答题、选择题，以及要求写出清晰步骤的应用题。得分点通常给在代入、移项和化简过程，而不仅仅是最后答案。
- Section A: Non-calculator arithmetic and algebra | A 部分：不允许使用计算器的算术与代数
- Section B: Calculator-based geometry, statistics, and problem solving | B 部分：允许使用计算器的几何、统计与问题解决
- Marking focuses on method, accuracy, and units | 评分关注方法、准确性和单位
2. Arithmetic and Number Operations | 算术与数运算
Number questions in 7S prelims focus on integers, order of operations (BODMAS), prime factors, squares, cubes, and roots. A typical answer requires showing each step: divide, multiply, add, and subtract in the correct order.
7S 预考中数与运算题重点考查整数、运算顺序（BODMAS）、质因数、平方数、立方数和根式。典型答案需要展示每一步：按照正确顺序进行除、乘、加、减。
. Writing down the intermediate step 3 + 8 = 11 helps avoid errors and earns method marks.
例如，计算 3 + 4 × 2 时，乘法先于加法，因此正确结果是 11，而不是 14。写出中间步骤 3 + 8 = 11 有助于避免错误并获得方法分。
60 = 2² × 3 × 5
Prime factorisation questions often ask for a number written as a product of primes. Divide repeatedly by the smallest prime factor until only prime numbers remain, then express the answer using index notation.
质因数分解题常要求将一个数写成质数乘积的形式。用最小质因数反复相除，直到只剩质数，然后用指数记法写出答案。
3. Algebraic Expressions and Equations | 代数表达式与方程
Algebra questions ask students to simplify expressions such as 3x + 5x or expand brackets like 2(x + 4). When solving equations, always perform the same operation on both sides and check by substitution.
代数题要求学生化简如 3x + 5x 的表达式，或展开如 2(x + 4) 的括号。解方程时，必须在等号两边同时进行相同运算，并通过代入检查答案。
uting x = 4 back into the left-hand side gives 2(4) + 3 = 11, which confirms the solution.
对于方程 2x + 3 = 11，两边同时减去 3 得到 2x = 8，再除以 2 得到 x = 4。将 x = 4 代回左边得 2(4) + 3 = 11，验证了解的正确性。
2x + 3 = 11 → 2x = 8 → x = 4
2x + 8, and 3(2x - 5) = 6x - 15. Watch the sign when the outside coefficient is negative.
展开括号时，用括号外的项乘以括号内的每一项。例如，2(x + 4) = 2x + 8，3(2x - 5) = 6x - 15。当外面系数为负数时，要特别注意符号。
4. Fractions, Decimals and Percentages | 分数、小数与百分比
Conversions between fractions, decimals and percentages are frequently examined. To compare values, convert them to the same form; to find a percentage of a quantity, multiply by the decimal equivalent.
分数、小数和百分数之间的转换是常考内容。比较数值时，应化成同一种形式；求一个数量的百分之几，用对应小数去乘。
1/4 = 0.25 = 25%
To find 15% of 240, first convert 15% to the decimal 0.15, then multiply 240 × 0.15 = 36. Alternatively, calculate 10% (24) and 5% (12), then add to get 36. Both methods are accepted if working is shown.
求 240 的 15%，先把 15% 化成小数 0.15，再计算 240 × 0.15 = 36。也可以先算 10%（24）和 5%（12），再相加得 36。只要写出过程，两种方法均可接受。
For percentage change, use the formula: change ÷ original × 100%. A price increase from $50 to $65 gives a change of $15, so the percentage increase is 15/50 × 100% = 30%.
求百分数变化时，使用公式：变化量 ÷ 原值 × 100%。价格由 50 美元涨到 65 美元，变化量为 15 美元，因此涨幅为 15/50 × 100% = 30%。
5. Geometry and Angle Properties | 几何与角度性质
In 7S geometry, students need to identify angle facts: angles on a straight line add to 180°, angles around a point add to 360°, and vertically opposite angles are equal. Diagrams should be labelled clearly.
在 7S 几何中，学生需要掌握角度性质：平角为 180°，周角为 360°，对顶角相等。图形应清晰标注。
a + b = 180° (angles on a straight line)
°, the angle vertically opposite is also 75°, and the adjacent angle is 180° - 75° = 105°.
两条直线相交时，对顶角相等。若一个角为 75°，其对顶角也为 75°，相邻角为 180° - 75° = 105°。
°, the third angle is 180° - 110° = 70°. Always state the angle fact used in your working.
题目还可能涉及三角形内角和为 180°。若三角形中两个角分别为 50° 和 60°，第三个角为 180° - 110° = 70°。解题时务必写出所用的角度性质。
6. Ratio, Proportion and Rates | 比、比例与速率
Ratio questions often require dividing a quantity in a given ratio or using a scale factor. Set up proportional relationships carefully and keep units consistent when working with rates such as km/h or m/s.
比和比例题通常要求按给定比例分配数量或使用比例因子。建立比例关系时需仔细，处理速率（如 km/h 或 m/s）时要保持单位一致。
3 + 2 = 5. One part is 120 ÷ 5 = 24 apples, so the shares are 3 × 24 = 72 and 2 × 24 = 48.
按 3:2 的比例分配 120 个苹果，先将份数相加：3 + 2 = 5。一份为 120 ÷ 5 = 24 个苹果，因此分配结果为 3 × 24 = 72 和 2 × 24 = 48。
y time: 180 ÷ 2.5 = 72 km/h. Always include the correct compound unit in the final answer.
对于速率问题，若一辆车在 2 小时 30 分钟内行驶 180 km，先将时间化为 2.5 小时，再用距离除以时间：180 ÷ 2.5 = 72 km/h。最终答案务必带上正确的复合单位。
7. Statistics and Data Handling | 统计与数据处理
Prelim papers include finding the mean, median, mode, and range from a list or frequency table. Always arrange data in order to find the median, and show the sum before dividing for the mean.
预考试卷包括从列表或频数表中求平均数、中位数、众数和极差。求中位数时务必将数据排序；求平均数时先求和再除以个数。
Mean = (sum of all values) ÷ (number of values)
s the middle value, 6. The mode is 8 because it occurs most often. The range is 9 - 2 = 7.
对于数据集 4, 8, 6, 9, 5, 8, 2，先排序：2, 4, 5, 6, 8, 8, 9。中位数为中间值 6。众数为 8，因为它出现次数最多。极差为 9 - 2 = 7。
If the data is in a frequency table, multiply each value by its frequency before summing.
平均数为 (2 + 4 + 5 + 6 + 8 + 8 + 9) ÷ 7 = 42 ÷ 7 = 6。如果数据在频数表中，先求出每个值乘以频数的积，再求和。
8. Common Pitfalls in 7S Papers | 7S 试卷常见失分点
Many marks are lost through missing units, incorrect sign use, or forgetting to answer the question in context. When a question asks for money, write two decimal places; when it asks for a fraction, simplify fully.
许多失分来自漏写单位、符号错误或忘记按题意作答。题目若问金额，要保留两位小数；若要求分数，要化为最简。
x because like terms are added without changing the power. Another frequent mistake is mishandling negative signs when expanding brackets.
代数中一个常见错误是把 2x + 3x 写成 5x²，但正确结果是 5x，因为合并同类项时指数不变。另一个常见错误是展开括号时处理负号不当。
- Always write units: 72 km/h, not just 72 | 务必写出单位：72 km/h，而不是只写 72
- Simplify fractions: 6/8 = 3/4 | 分数要化简：6/8 = 3/4
- Check sign rules: -2 × -3 = 6, but -2 × 3 = -6 | 检查符号法则：-2 × -3 = 6，而 -2 × 3 = -6
- Read the question: if it asks for the original price, do not give the discounted price | 仔细审题：若问原价，就不要答折后价
9. Worked Examples from Prelim Papers | 预考试题例题解析
Consider this typical 7S question: 'A book costs $12.80 after a 20% discount. Find the original price.' Since 80% corresponds to $12.80, divide by 80 and multiply by 100 to obtain $16.00.
考虑一个典型 7S 题：“一本书打八折后售价为 12.80 美元，求原价。”因为 80% 对应 12.80，用 12.80 ÷ 80 × 100 得到 16.00 美元。
80% of original price = 12.80 → original price = 12.80 ÷ 80 × 100 = 16.00
Another example: 'Solve 5(x - 2) = 20.' 30, and divide by 5 to get x = 6. Check by substituting: 5(6 - 2) = 20, which is correct.
另一个例子：“解方程 5(x - 2) = 20。”先展开括号得 5x - 10 = 20，然后两边加 10 得 5x = 30，再除以 5 得 x = 6。代入检验：5(6 - 2) = 20，结果正确。
For geometry, if two parallel lines are cut by a transversal and one alternate angle is 65°, the corresponding alternate angle is also 65°. Clearly state the angle property: alternate angles are equal when lines are parallel.
在几何中，若两条平行线被一条截线所截，其中一个内错角为 65°，则对应的内错角也为 65°。要明确写出角度性质：两直线平行时内错角相等。
10. Exam Techniques and Time Management | 考试技巧与时间管理
In the prelims, allocate time according to marks: about one minute per mark. Read the question twice, underline key values, and leave blank lines for any skipped question so you can return later.
预考中按分值分配时间：大约每分钟 1 分。题目读两遍，划出关键数据，跳过的题留出空行以便稍后回头完成。
Start with the questions you find easiest to build confidence, but do not spend too long on a single problem. If you are stuck for more than two minutes, move on and mark the question with a star in the margin.
先做最有把握的题目以建立信心，但不要在单一题目上花费过长时间。若卡住超过两分钟，就暂时跳过，并在页边标星以便回头。
- Show all working even if the answer is wrong | 即使答案有误，也要写出全部过程
- Use the mark allocation as a guide to how many steps are needed | 用分值提示需要写多少个步骤
- Leave five minutes at the end to check units and signs | 最后留五分钟检查单位和符号
- Do not erase working: cross out neatly if you change your method | 不要擦掉草稿过程：若改变方法，整齐地划掉即可
11. Final Review Checklist | 最终复习清单
Before the exam, check that you can: use BODMAS correctly, solve linear equations, convert between fraction-decimal-percentage, identify angle facts, and calculate mean/median/mode. Practise at least two full 7S prelim papers under timed conditions.
考前请确认自己能够：正确使用运算顺序、解一元一次方程、在分数-小数-百分数之间转换、识别角度性质，并会计算平均数/中位数/众数。至少计时完成两套完整的 7S 预考试卷。
- Can I factorise a number into its prime factors? | 我能否将一个数分解为质因数？
- Can I expand and simplify 2(3x + 4) - x? | 我能否展开并化简 2(3x + 4) - x？
- Can I find the mean of a frequency table? | 我能否从频数表中求平均数？
- Can I solve a word problem involving percentage change? | 我能否解决涉及百分数变化的应用题？
- Can I state at least three angle facts from memory? | 我能否凭记忆写出至少三条角度性质？
Create a one-page summary sheet with formulas, angle facts, and common conversions. Review this sheet the night before the prelim to reinforce your working memory.
制作一页包含公式、角度性质和常见换算的总结表。考前晚上复习这张表，强化工作记忆。
12. Conclusion | 总结
7S prelims answers are not just about the final number; they test clear working, correct notation, and the ability to reason logically. Review every mistake, rewrite the correct solution, and use this guide to target your revision.
7S 预考答案不只在于最后数字，更考查清晰的步骤、正确的符号和逻辑推理能力。复习每一个错误，重写正确解法，并用本指南进行针对性复习。
Consistent practice with full prelim papers builds the speed and accuracy needed for the final assessment. Focus on understanding why each step is taken, not just memorising procedures.
持续练习完整的预考试卷可以提升最终评估所需的速度和准确度。重点理解每一步为何这样做，而不只是死记步骤。
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