📚 Cambridge Lower Secondary Mathematics Workbook 7 Answers | 剑桥初中数学练习册7答案详解与备考指南
Cambridge Lower Secondary Mathematics Workbook 7 is designed for learners aged 11 to 13 who are following the Cambridge Lower Secondary Mathematics curriculum. This article explains how to use the Workbook 7 answers effectively, highlights common errors, and provides worked solutions for typical Stage 7 topics such as integers, fractions, algebra, geometry, statistics and probability.
剑桥初中数学练习册7专为11至13岁、正在学习剑桥初中数学课程的学生设计。本文介绍如何有效使用练习册7的答案，指出常见错误，并针对七年级典型主题（如整数、分数、代数、几何、统计和概率）给出详细解题示范。
1. How to Use Workbook 7 Answers Effectively | 如何有效使用练习册7答案
Answers should be used as a learning tool rather than a shortcut. Attempt every question first, mark your work, then redo any question you got wrong before reading the worked solution.
答案应作为学习工具使用，而不是走捷径。先独立完成每道题，再对照答案批改；做错的题目应在阅读详细解答之前重新做一遍。
Keep a corrections notebook for mistakes. For each error, write down the question number, the correct method and a one-sentence reason for the mistake.
准备一本错题本。对于每一处错误，记下题号、正确方法，并用一句话写明错误原因。
When checking numerical answers, ask yourself whether the answer is reasonable. For example, adding two positive numbers must give a positive result larger than either number.
检查数值答案时，要问自己答案是否合理。例如，两个正数相加的结果一定是正数，并且比其中任何一个数都大。
Use the answer key after completing a whole exercise, not after every single question. This builds stamina and shows which topics need more review.
完成整节练习后再使用答案，而不要每做一题就看答案。这样可以训练答题耐力，并清楚地看出哪些主题需要复习。
Finally, always read the command word. If a question says 'Explain' or 'Show that', a bare number is never a full answer, even when the number is correct.
最后，一定要看清指令词。如果题目要求 "Explain"（解释）或 "Show that"（证明），那么即使数字算对了，只写一个数字也绝不是完整答案。
2. Integers and Place Value: Worked Answers | 整数与位值：典型题解析
A typical Workbook 7 question asks learners to add or subtract large numbers using correct place value. For example, calculate 4 582 + 3 769.
练习册7中的典型题目要求运用正确的位值进行大数加减。例如，计算 4 582 + 3 769。
5 + 7 + 1 = 13, so write 3 and carry 1. The thousands: 4 + 3 + 1 = 8. The answer is 8 351.
从个位开始：2 + 9 = 11，写 1 进 1。十位：8 + 6 + 1 = 15，写 5 进 1。百位：5 + 7 + 1 = 13，写 3 进 1。千位：4 + 3 + 1 = 8。答案是 8 351。
so the answer should be about 8 400. The exact answer 8 351 is close, so it is reasonable.
用估算检验：4 582 ≈ 4 600，3 769 ≈ 3 800，所以答案应在 8 400 左右。精确答案 8 351 与之接近，因此是合理的。
number line, adding a positive number moves right, and adding a negative number moves left.
对于 6 003 − 2 748 这样的减法，要小心地跨越零借位：6 003 − 2 748 = 3 255。再看负整数：−7 + 12 = 5，4 − 9 = −5。在数轴上，加正数向右移动，加负数向左移动。
nearest thousand, look at the hundreds digit 3, which is less than 5, so 47 382 ≈ 47 000.
位值同样用于四舍五入。把 47 382 精确到千位时，看百位数字 3，小于 5，所以 47 382 ≈ 47 000。
3. Fractions, Decimals and Percentages | 分数、小数与百分数
or 12: 3/4 = 9/12 and 5/6 = 10/12, so the sum is 19/12, which is 1 7/12 as a mixed number.
七年级学生必须能在分数、小数与百分数之间自如转换。计算 3/4 + 5/6 时，先找最小公分母 12：3/4 = 9/12，5/6 = 10/12，所以和为 19/12，化成带分数是 1 7/12。
To divide, invert the second fraction and multiply: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
分数相乘时，分子乘分子、分母乘分母：2/3 × 9/10 = 18/30 = 3/5。分数相除时，把第二个分数取倒数再相乘：3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
lace value: 0.375 = 375/1000 = 3/8. Percentages convert to decimals by dividing by 100: 45% = 0.45 = 45/100 = 9/20.
小数可按位值化为分数：0.375 = 375/1000 = 3/8。百分数除以 100 即得小数：45% = 0.45 = 45/100 = 9/20。
For a percentage-of question, convert the percentage to a decimal multiplier. So 15% of 260 = 0.15 × 260 = 39. To increase a value by 12%, multiply by 1.12; to decrease by 12%, multiply by 0.88.
遇到"求百分之几"的题目，先把百分数化成小数乘数。所以 260 的 15% = 0.15 × 260 = 39。把某数增加 12%，就乘 1.12；减少 12%，就乘 0.88。
Always simplify a final fraction and cancel before multiplying where possible. A marks are often awarded for a correctly simplified answer, and unsimplified fractions can lose that mark.
最后的分数一定要约分，并在可能的情况下先约分再相乘。答案写成分数时通常要求已约分，未约分可能失分。
4. Order of Operations, Powers and Roots | 运算顺序、乘方与开方
Workbook 7 answers frequently test the order of operations: brackets, indices, division and multiplication, then addition and subtraction.
练习册7的答案经常考查运算顺序：先括号，再乘方，然后除法和乘法，最后加法和减法。
ultiply and divide: 3 × 16 = 48 and 10 ÷ 2 = 5. Finally add and subtract: 5 + 48 − 5 = 48.
例题：5 + 3 × 4² − 10 ÷ 2。先算乘方：4² = 16。再算乘除：3 × 16 = 48，10 ÷ 2 = 5。最后算加减：5 + 48 − 5 = 48。
13 because 13 × 13 = 169. Cube numbers such as 2³ = 8 and 5³ = 125 also appear regularly.
平方数与平方根至少要背到 15² = 225。例如 √169 = 13，因为 13 × 13 = 169。立方数如 2³ = 8、5³ = 125 也经常出现。
Note that 5² means 5 × 5, not 5 × 2; this is one of the most common slips in Stage 7 work.
综合题可能写作：5² + 3³ = 25 + 27 = 52。注意 5² 表示 5 × 5，而不是 5 × 2；这是七年级作业中最常见的错误之一。
ted, evaluate the innermost bracket first: 2 × (3 + (4 − 1)²) = 2 × (3 + 9) = 2 × 12 = 24.
遇到嵌套括号时，先算最里面的括号：2 × (3 + (4 − 1)²) = 2 × (3 + 9) = 2 × 12 = 24。
5. Ratio, Proportion and Rate | 比、比例与速率
parts: 3 + 4 = 7. One part is 84 ÷ 7 = 12, so the shares are 3 × 12 = 36 and 4 × 12 = 48.
比例题要求按给定比分配总量。把 84 按 3 : 4 分配时，先求份数之和：3 + 4 = 7。一份是 84 ÷ 7 = 12，所以两部分为 3 × 12 = 36 和 4 × 12 = 48。
inal amount: 36 + 48 = 84. This single check catches most arithmetic errors in ratio work.
一定要检验各部分之和是否等于原数：36 + 48 = 84。这个检验能抓住比例题中绝大多数计算错误。
Direct proportion problems use a unit method. If 5 pens cost $3.50, then 1 pen costs $0.70, so 8 pens cost 8 × 0.70 = $5.60.
正比例题用"单位量法"。若 5 支笔售价 3.50 美元，则 1 支笔 0.70 美元，8 支笔就是 8 × 0.70 = 5.60 美元。
d of 150 ÷ 2.5 = 60 km/h. Rearranged, distance = speed × time and time = distance ÷ speed.
速率题联系路程、速度与时间。汽车 2.5 小时行驶 150 千米，平均速度为 150 ÷ 2.5 = 60 千米/时。变形可得：路程 = 速度 × 时间，时间 = 路程 ÷ 速度。
a map with scale 1 : 25 000, 4 cm represents 4 × 25 000 = 100 000 cm = 1 km on the ground.
地图比例尺也是比。比例尺为 1 : 25 000 的地图上，4 厘米代表实际 4 × 25 000 = 100 000 厘米 = 1 千米。
6. Algebra: Expressions, Formulae and Equations | 代数：表达式、公式与方程
fy 4a + 3b − 2a + 5b by grouping: 4a − 2a = 2a and 3b + 5b = 8b, so the answer is 2a + 8b.
合并同类项是七年级代数的基础。化简 4a + 3b − 2a + 5b 时先分组：4a − 2a = 2a，3b + 5b = 8b，所以答案是 2a + 8b。
If the bracket is subtracted, every sign inside changes: 4 − (x + 3) = 4 − x − 3 = 1 − x.
去括号要用分配律：3(2x + 5) = 6x + 15。若括号前是减号，括号内每一项都要变号：4 − (x + 3) = 4 − x − 3 = 1 − x。
sides. For 5x − 7 = 23, add 7 to both sides to get 5x = 30, then divide by 5 to get x = 6.
解一元一次方程，就是对两边做同样的运算。对于 5x − 7 = 23，两边同时加 7 得 5x = 30，再除以 5 得 x = 6。
= 2x + 11, so x = 5. Substitute back to check: 3(5 + 2) = 21 and 2 × 5 + 11 = 21. Correct.
出现括号时先展开：3(x + 2) = 2x + 11 变为 3x + 6 = 2x + 11，所以 x = 5。代回检验：3(5 + 2) = 21，2 × 5 + 11 = 21，正确。
ula is also common. If P = 2(l + w), then for l = 9 and w = 4, P = 2(9 + 4) = 2 × 13 = 26.
代入公式求值同样常见。若 P = 2(l + w)，当 l = 9、w = 4 时，P = 2(9 + 4) = 2 × 13 = 26。
7. Sequences, Functions and Graphs | 数列、函数与图像
A sequence question usually asks for the next term or the nth term rule. For 5, 9, 13, 17 the term-to-term rule is 'add 4', so the next term is 21.
数列题通常要求写出下一项或第 n 项的规律。对于 5, 9, 13, 17，相邻项的变化规律是"加 4"，所以下一项是 21。
e sequence: 4 × 1 = 4 but the first term is 5, so add 1. The nth term is therefore 4n + 1.
求第 n 项时，先看公差为 4，所以规律形如 4n。把 4n 与数列比较：4 × 1 = 4，而首项是 5，因此要加 1。所以第 n 项是 4n + 1。
h term rule with n = 1, n = 2 and n = 20. If n = 20 gives 4 × 20 + 1 = 81, the rule holds.
务必用 n = 1、n = 2 和 n = 20 检验第 n 项公式。若 n = 20 时得 4 × 20 + 1 = 81，说明规律成立。
, 3. Plotting these points gives a straight-line graph with gradient 3 and y-intercept −2.
函数机与坐标系是连在一起的。对于 y = 3x − 2，当 x = 0, 1, 2, 3 时，对应的 y = −2, 1, 4, 7。描出这些点可得一条直线，斜率为 3，y 轴截距为 −2。
s also appear. The midpoint of (2, 5) and (8, 11) is ((2 + 8) ÷ 2, (5 + 11) ÷ 2) = (5, 8).
中点与坐标题也会出现。(2, 5) 与 (8, 11) 的中点是 ((2 + 8) ÷ 2, (5 + 11) ÷ 2) = (5, 8)。
8. Geometry: Angles, Shapes and Constructions | 几何：角、图形与作图
Angle facts must be quoted, not just used. Angles on a straight line add up to 180°, angles at a point add up to 360°, and angles in a triangle add up to 180°.
使用角的性质时要写出理由，而不只是用结论。同一直线上的角之和为 180°，围绕一点的角之和为 360°，三角形内角和为 180°。
. The three angles are 84°, 56° and 40°, which sum to 180°. The check confirms the answer.
例题：三角形三个角为 3x、2x 和 40°。则 3x + 2x + 40 = 180，所以 5x = 140，x = 28。三个角分别是 84°、56°、40°，和为 180°。检验通过。
For parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add up to 180°. Naming the reason earns the method mark even if the arithmetic slips.
平行线中，内错角相等，同位角相等，同旁内角之和为 180°。写出理由名称，即使计算有误也能拿到方法分。
hexagon, (6 − 2) × 180° = 720°. The exterior angles of any polygon always add up to 360°.
多边形内角和公式为 (n − 2) × 180°，其中 n 是边数。六边形的内角和为 (6 − 2) × 180° = 720°。任何多边形的外角和恒为 360°。
a pair of compasses for arcs. In circle work, the radius is half the diameter, so d = 2r.
作图题要求规范使用工具：用直尺和铅笔画线段，用量角器画角，用圆规画弧。圆的相关知识中，半径是直径的一半，即 d = 2r。
9. Length, Area, Volume and Units | 长度、面积、体积与单位
ide. Rectangle: A = l × w = 8 × 5 = 40 cm². Triangle: A = ½ × b × h = ½ × 12 × 7 = 42 cm².
使用面积公式时，必须用垂直高度，而不是斜边。长方形：A = l × w = 8 × 5 = 40 平方厘米。三角形：A = ½ × b × h = ½ × 12 × 7 = 42 平方厘米。
60 cm². Trapezium: A = ½(a + b)h = ½(6 + 10) × 4 = 32 cm². Always write the unit squared.
平行四边形：A = b × h = 10 × 6 = 60 平方厘米。梯形：A = ½(a + b)h = ½(6 + 10) × 4 = 32 平方厘米。答案必须带平方单位。
3 × 5 = 60 cm³. Surface area is SA = 2(lw + lh + wh) = 2(12 + 20 + 15) = 2 × 47 = 94 cm².
长方体的体积 V = l × w × h = 4 × 3 × 5 = 60 立方厘米。表面积 SA = 2(lw + lh + wh) = 2(12 + 20 + 15) = 2 × 47 = 94 平方厘米。
m = 100 cm, 1 km = 1000 m, 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³ and 1 litre = 1000 cm³.
单位换算常常导致失分。要记住：1 厘米 = 10 毫米，1 米 = 100 厘米，1 千米 = 1000 米，1 平方米 = 10 000 平方厘米，1 立方米 = 1 000 000 立方厘米，1 升 = 1000 立方厘米。
| Quantity | Formula | Worked value |
| Rectangle area | l × w | 8 × 5 = 40 cm² |
| Triangle area | ½ × b × h | ½ × 12 × 7 = 42 cm² |
| Trapezium area | ½(a + b)h | ½(6 + 10) × 4 = 32 cm² |
| Cuboid volume | l × w × h | 4 × 3 × 5 = 60 cm³ |
10. Statistics: Averages and Charts | 统计：平均数与图表
The three averages
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