📚 Essential Maths Book 7S Answers: Year 7 Core Skills Review Guide | 七年级 Essential Maths Book 7S 答案解析与核心技能复习指南
This revision guide reviews typical answers from Essential Maths Book 7S and shows how to check each result using clear, repeatable methods. Use it after you have attempted the questions yourself, not before, so that mistakes become learning points. The focus is on method, common errors, and quick self-checking rather than simply copying results.
本复习指南复习 Essential Maths Book 7S 中的典型答案，并展示如何用清晰、可重复的方法核对每个结果。请先自己尝试题目，再使用本指南，这样错误才能成为学习要点。本指南侧重解题方法、常见错误和快速自查，而非简单抄写结果。
The guide follows the main topic order of the book: place value, the four operations, directed numbers, fractions, decimals, percentages, ratio, algebra, equations, graphs, geometry, measurement, statistics and probability. Each section gives a model answer check, a list of frequent mistakes, and a short self-test routine.
本指南按照教材的主要专题顺序展开：位值、四则运算、有向数、分数、小数、百分数、比、代数、方程、图像、几何、测量、统计与概率。每一节都给出答案核对范例、常见错误清单和简短自测流程。
1. How to Use the Answer Section | 如何使用答案解析
Always do your working in pencil and write the final answer in pen. Mark your own work using a different colour. If an answer is wrong, classify the error as reading (R), method (M), or calculation (C). This classification helps target revision. Keep a short error log with the question number, your answer, and the correct answer.
始终用铅笔写过程，用钢笔写最终答案。用不同颜色批改自己的作业。如果答案错误，将错误分为读题错误 R、方法错误 M 或计算错误 C。这种分类有助于有针对性复习。用简短的错题本记录题号、你的答案和正确答案。
Before you mark a question, ask yourself three things: did I copy the numbers correctly, did I use the correct order of operations, and is my answer sensible for the size of the numbers? This quick habit prevents small mistakes from becoming lost marks.
批改题目之前，先问自己三件事：数字抄对了吗，运算顺序正确吗，答案的大小是否合理。这个快速习惯可以防止小错误导致丢分。
| Self-check step | Question to ask |
|---|---|
| 1 | Did I copy the numbers and symbols correctly? |
| 2 | Did I follow the correct order of operations? |
| 3 | Is my answer a sensible size for the question? |
When an answer is correct, write one sentence explaining why the method works. This turns a correct answer into a revision note. When an answer is wrong, rewrite the full solution without looking at the answer key, then compare line by line.
答案正确时，用一句话解释该方法为什么成立。这样可以把正确答案变成复习笔记。答案错误时，不要看答案，先完整重写一遍解法，再逐行与答案对照。
2. Whole Numbers and Place Value | 整数与位值
Place value is the foundation for all written methods. When marking, compare the digits in each column and check that zeros have been used as place holders. For example, in 305 072, the digit 3 has value 300 000, the digit 5 has value 5 000, and the digit 2 has value 2. A missing zero often changes the value completely.
位值是所有书面方法的基础。批改时，逐列比较数字，并检查零是否被用作占位符。例如，在 305 072 中，数字 3 的值为 300 000，数字 5 的值为 5 000，数字 2 的值为 2。漏掉一个零常常会完全改变数值。
Common errors include writing 3 000 000 as 300 000, confusing -7 with 7 in a table of values, and rounding 4 985 to 4 900 instead of 5 000. To check rounding, underline the digit in the rounding place, look at the digit to its right, and apply the rule: 5 or more rounds up, 4 or less rounds down.
常见错误包括把 3 000 000 写成 300 000，在数值表中把 -7 与 7 混淆，以及把 4 985 四舍五入为 4 900 而不是 5 000。检查四舍五入时，划出需要保留的数位，看它右边的一位，然后应用规则：5 或以上进位，4 或以下舍去。
A useful self-check is to write the number in words and then back into figures. For 2 040 006, read "two million, forty thousand and six". If the words and figures do not match, the place value is wrong.
一个有用的自查方法是用文字写出该数，再写回数字。对于 2 040 006，读作 "two million, forty thousand and six"。如果文字与数字不一致，说明位值错了。
3. Order of Operations and Directed Numbers | 运算顺序与有向数
mistake is to calculate 2 + 3 × 4 as 20 instead of 14. The correct working is 2 + 12 = 14.
公认的运算顺序是：括号、乘方、从左到右的除法与乘法，然后是从左到右的加法与减法。常见错误是把 2 + 3 × 4 算成 20 而不是 14。正确过程是 2 + 12 = 14。
dividing two numbers with the same sign gives a positive result; two numbers with different signs give a negative result.
对于有向数，记住减去负数等于加上正数：a − (−b) = a + b。同号两数相乘或相除得正；异号两数相乘或相除得负。
(−3) × (−4) = +12
(−3) × (+4) = −12
−8 ÷ (−2) = +4
ning. In the first case the negative sign is inside the bracket and is squared; in the second case only the 2 is squared.
仔细检查乘方：(−2)² = 4，但 −2² = −4。括号改变了含义。第一种情况负号在括号内，一起平方；第二种情况只有 2 平方。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three ways of writing the same value. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify.
分数、小数和百分数是同一个值的三种写法。分数化小数，用分子除以分母。小数化百分数，乘以 100。百分数化分数，写成百分之几再化简。
3/4 = 0.75 = 75%
0.6 = 3/5 = 60%
45% = 0.45 = 9/20
um is 11/12. A common error is to add numerators and denominators separately, giving 3/7, which is incorrect.
做分数加减时，先找公分母。对于 2/3 + 1/4，最小公分母是 12，所以 2/3 = 8/12，1/4 = 3/12。和为 11/12。常见错误是把分子和分母分别相加，得到 3/7，这是错误的。
d/c. Always simplify the final answer and check whether it is larger or smaller than the original numbers.
分数相乘时，分子乘分子，分母乘分母。分数相除时，乘以倒数：a/b ÷ c/d = a/b × d/c。最后一定要化简答案，并检查它比原来的数是变大还是变小。
5. Ratio, Proportion and Scaling | 比、比例与缩放
A ratio compares quantities in a fixed order. The ratio 3 : 5 means three parts to five parts, with eight parts in total. To share an amount in a given ratio, add the parts, divide the total by the number of parts, then multiply by each share.
比按固定顺序比较数量。比 3 : 5 表示三份对五份，总共八份。按给定比分配一个量时，先加总份数，用总量除以总份数，再分别乘以各份数。
Example: share £ in the ratio 3 : 5. Total parts = 3 + 5 = 8. One part = 72 ÷ 8 = 9. The shares are 3 × 9 = £27 and 5 × 9 = £45. Check: 27 + 45 = 72.
例：把 72 英镑按 3 : 5 分配。总份数 = 3 + 5 = 8。一份 = 72 ÷ 8 = 9。两份分别为 3 × 9 = 27 英镑和 5 × 9 = 45 英镑。验证：27 + 45 = 72。
For scaling, use the unitary method. If 4 pens cost £1.20, one pen costs £0.30, so 7 pens cost £2.10. A common mistake is to divide by the wrong quantity or to forget to multiply by the required number.
缩放问题可以用归一法。如果 4 支笔 1.20 英镑，那么 1 支笔 0.30 英镑，所以 7 支笔 2.10 英镑。常见错误是用错被除数，或者忘记乘以所要求的数量。
Direct proportion means both quantities increase together. Inverse proportion means one increases as the other decreases. Check whether the answer should be larger or smaller than the original.
正比例表示两个量一起增大。反比例表示一个量增大时另一个量减小。检查答案应该比原来的数大还是小。
6. Algebraic Expressions and Simplifying | 代数表达式与化简
For example, 3a + 5a = 8a, but 3a + 5b cannot be simplified because a and b are different.
代数表达式用字母代表数。同类项有相同的字母组合，可以合并。例如，3a + 5a = 8a，但 3a + 5b 不能化简，因为 a 和 b 不同。
When expanding brackets, multiply each term inside by the term outside. For 3(2x + 5), the result is 6x + 15. A common error is to multiply only the first term and forget the second.
展开括号时，用括号外的项分别乘以括号内的每一项。对于 3(2x + 5)，结果是 6x + 15。常见错误是只乘第一项，忘记第二项。
3(2x + 5) = 6x + 15
5a − 2a + 4b = 3a + 4b
x × x = x²
n gives 6x + 15 = 6 × 4 + 15 = 24 + 15 = 39. Both match, so the simplification is correct.
用代入法检查表达式。如果 x = 4，那么 3(2x + 5) = 3(8 + 5) = 3 × 13 = 39。化简后的表达式给出 6x + 15 = 6 × 4 + 15 = 24 + 15 = 39。两者一致，所以化简正确。
7. Solving Linear Equations | 解一元一次方程
For 4x + 7 = 31, subtract 7 from both sides to get 4x = 24, then divide by 4 to get x = 6.
解方程时，两边同时进行相同的运算，保持等式平衡。目标是让未知字母单独在一边。对于 4x + 7 = 31，两边同时减 7 得 4x = 24，再除以 4 得 x = 6。
k: 4 × 6 + 7 = 24 + 7 = 31. This check takes only a few seconds and catches most mistakes.
把答案代回原方程检查：4 × 6 + 7 = 24 + 7 = 31。这个检查只需几秒钟，却能发现大多数错误。
x + 9, subtract 2x from both sides to get 3x − 3 = 9, then add 3 to get 3x = 12, so x = 4.
当未知数出现在两边时，把含字母的项集中到一边，数字集中到另一边。对于 5x − 3 = 2x + 9，两边减 2x 得 3x − 3 = 9，再加 3 得 3x = 12，所以 x = 4。
A common error is to forget to change the sign when moving a term across the equals sign. Remember that adding or subtracting the same term from both sides is the safe method.
常见错误是把项移过等号时忘记变号。记住，两边同时加或减同一项是最安全的方法。
8. Sequences, Functions and Graphs | 数列、函数与图像
A sequence follows a rule. For the sequence 5, 8, 11, 14, the term-to-term rule is "add 3" arts with 3n. If n = 1, 3n = 3, but the first term is 5, so add 2. The nth term is 3n + 2.
数列遵循某个规则。对于数列 5, 8, 11, 14，相邻项规则是 "加 3"。线性数列的第 n 项形式为 an + b。这里公差是 3，所以规则以 3n 开始。如果 n = 1，3n = 3，但第一项是 5，所以加 2。第 n 项是 3n + 2。
4: 3 × 4 + 2 = 14, which matches the fourth term. If it does not match, the rule is wrong.
验证时，代入 n = 4：3 × 4 + 2 = 14，与第四项一致。如果不一致，规则就是错的。
For coordinates, remember that the x-axis is horizontal and the y-axis is vertical. The point (3, −2) means 3 right and 2 down. A common error is to plot (3, −2) as (−2, 3). Always read the x-coordinate first.
对于坐标，记住 x 轴水平，y 轴竖直。点 (3, −2) 表示向右 3、向下 2。常见错误是把 (3, −2) 画成 (−2, 3)。一定要先读 x 坐标。
cept c. A positive gradient slopes upwards from left to right; a negative gradient slopes downwards.
对于直线图像，方程 y = mx + c 的斜率为 m，y 轴截距为 c。正斜率从左到右上升；负斜率从左到右下降。
9. Geometry: Angles, Perimeter and Area | 几何：角、周长与面积
Angle facts must be quoted accurately. Angles on a straight line add to 180°. Angles around a point add to 360°. Vertically opposite angles are equal. In a triangle, the interior angles add to 180°.
角的事实必须准确引用。直线上的角相加为 180°。围绕一点的角相加为 360°。对顶角相等。三角形内角和为 180°。
a + b + c = 180°
Perimeter of rectangle = 2(l + w)
Area of rectangle = l × w
Area of triangle = ½ × base × height
rallelogram, area = base × perpendicular height. For a trapezium, area = ½(a + b)h, where a and b are the parallel sides.
求三角形面积时，高必须垂直于底，而不是斜边。常见错误是把斜边当作高。平行四边形面积 = 底 × 垂直高。梯形面积 = ½(a + b)h，其中 a 和 b 是平行边。
Check units carefully. Perimeter is measured in cm, m or km. Area is measured in cm², m² or km². Mixing units is a common source of lost marks.
仔细检查单位。周长用 cm、m 或 km。面积用 cm²、m² 或 km²。混淆单位是常见的丢分原因。
10. Volume, Surface Area and Units | 体积、表面积与单位
face area is the total area of all faces. A cuboid has six faces: three pairs of equal rectangles.
长方体体积 = 长 × 宽 × 高。正方体体积 = 棱长³。表面积是所有面的总面积。长方体有六个面：三对相等的长方形。
V = l × w × h
Surface area of cuboid = 2(lw + lh + wh)
1 cm³ = 1 ml
1 m³ = 1 000 000 cm³
nverting area units, square the conversion factor: 1 m² = 10 000 cm². When converting volume units, cube the factor: 1 m³ = 1 000 000 cm³.
单位换算是高频考点。长度：1 km = 1 000 m，1 m = 100 cm，1 cm = 10 mm。质量：1 吨 = 1 000 kg，1 kg = 1 000 g。容量：1 升 = 1 000 ml。换算面积单位时，换算系数要平方：1 m² = 10 000 cm²。换算体积单位时，系数要立方：1 m³ = 1 000 000 cm³。
To check a volume answer, estimate the size. A box 10 cm by 10 cm by 10 cm has volume 1 000 cm³, which is 1 litre. If your answer is 10 000 cm³, the box would be much larger than expected.
检查体积答案时，估算大小。一个 10 cm × 10 cm × 10 cm 的盒子体积为 1 000 cm³，即 1 升。如果答案是 10 000 cm³，盒子会比预期大得多。
11. Statistics and Probability | 统计与概率
For statistics, the mean is the total divided by the number of values. The median is the middle value when the data are in order. The mode is the most frequent value. The range is the largest value minus the smallest value.
统计方面，平均数是总和除以数据个数。中位数是数据排序后的中间值。众数是出现次数最多的值。极差是最大值减最小值。
5 + 5 + 7 + 10) ÷ 5 = 30 ÷ 5 = 6. The median is 5. The mode is 5. The range is 10 − 3 = 7.
例：对于数据 3, 5, 5, 7, 10，平均数为 (3 + 5 + 5 + 7 + 10) ÷ 5 = 30 ÷ 5 = 6。中位数为 5。众数为 5。极差为 10 − 3 = 7。
For probability, the probability of an event is the number of favourable outcomes divided by the total number of equally likely outcomes. Probabilities range from 0 to 1. A probability of 0 means impossible; a probability of 1 means certain.
概率方面，事件概率等于有利结果数除以等可能结果总数。概率在 0 到 1 之间。概率 0 表示不可能；概率 1 表示必然。
P(event) = favourable outcomes ÷ total outcomes
0 ≤ P(A) ≤ 1
P(not A) = 1 − P(A)
A common error is to write a probability as a fraction that has not been simplified, or to use the wrong total. Always list the sample space first if you are unsure.
常见错误是概率分数没有化简，或者总数用错。如果不确定，先列出样本空间。
12. Exam Technique and Final Checklist | 考试技巧与最终自查清单
Read each question twice before starting. Underline key information such as units, the required form of the answer, and whether working must be shown. Show every step clearly; method marks are often available even when the final answer is wrong.
开始答题前把每道题读两遍。划出关键信息，如单位、答案要求的形式，以及是否需要展示过程。清楚写出每一步；即使最终答案错误，也常常有方法分。
Manage time by doing the questions you find easiest first, then return to harder ones. Leave two or three minutes at the end to check calculations, units and signs. Do not leave a question blank if you can make a reasonable start.
先做最容易的题，再回头做难题，以此管理时间。最后留两三分钟检查计算、单位和符号。只要能从合理的地方入手，就不要留空。
Final self-check list:
最终自查清单：
- Did I copy every number and symbol correctly? | 我抄对了每一个数字和符号吗？
- Did I use the correct order of operations? | 我用了正确的运算顺序吗？
- Are my units correct and consistent? | 我的单位正确且一致吗？
- Is my answer sensible in size? | 我的答案大小合理吗？
- Have I simplified fractions and expressions? | 我化简分数和表达式了吗？
- Have I checked by substitution where possible? | 在可能的地方，我用代入法检查了吗？
- Have I answered the question that was asked? | 我回答了所问的问题吗？
Use the answer section actively: mark, classify, correct and rewrite. Over time, the same errors will appear less often, and your confidence in Year 7 core skills will grow.
主动使用答案区：批改、分类、纠正并重写。随着时间推移，同样的错误会越来越少，你对七年级核心技能的信心也会增强。
Published by TutorHao | Year 7 Mathematics Revision Series | aleveler.com
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