📚 Mastering Essential Maths Book 7C: Question Types & Solutions | KS3 数学：Essential Maths Book 7C 答案题型解析
The Essential Maths Book 7C is a core resource for KS3 students, building fluency in key topics such as number, algebra, geometry, and statistics. This guide analyses the most common question types and provides step-by-step solutions, helping students understand not just the final answer but the reasoning behind it.
Essential Maths Book 7C 是 KS3 阶段的核心教材，帮助学生在数、代数、几何和统计等主题上建立扎实的运算能力。本指南分析最常见的题型并给出逐步解答，帮助学生不仅掌握最终答案，更理解背后的推理过程。
1. Number and Place Value | 数与位值
Place value questions often ask for the value of a specific digit in large numbers or decimals. For example, in the number 45 623.789, the digit 6 represents 600 (six hundred), while the digit 8 represents 0.08 (eight hundredths).
位值问题常要求找出大数或小数中某一位的具体数值。例如，在数字 45 623.789 中，数字 6 表示 600（六百），而数字 8 表示 0.08（百分之八）。
A typical question: Write the value of the digit 3 in the number 4.376. The answer is 0.3 or 3/10 (three tenths). To round 7892 to the nearest hundred, look at the tens digit (9); since it is 5 or above, round up to 7900.
常见题型：写出数字 4.376 中数字 3 的数值。答案是 0.3 或 3/10（十分之三）。将 7892 四舍五入到最近的百位数时，看十位数字（9），由于大于等于 5，向上舍入为 7900。
| Digit | Place Value |
|---|---|
| 7 (in 7892) | 7000 |
| 8 | 800 |
| 9 | 90 |
| 2 | 2 |
Common mistake: confusing '0.3' with '0.03'. Always count the decimal places after the point.
常见错误：混淆“0.3”与“0.03”。务必数清小数点后的位数。
2. Addition and Subtraction in Context | 情景中的加减法
Multi-step word problems test calculation accuracy. For instance: 'A cinema has 324 seats. 156 adults and 89 children enter. How many seats are left?' First, 156 + 89 = 245, then 324 - 245 = 79. Answer: 79 seats remain.
多步应用题考查计算准确性。例如：“一家电影院有 324 个座位，入场了 156 位成人和 89 位儿童。还剩多少空座？”先算 156 + 89 = 245，再算 324 - 245 = 79。答案：还剩 79 个座位。
the positive, so -5 + 9 = 4. Using a number line helps visualise the jump from -5 up to 4.
涉及负数的题目经常出现：计算 -5 - (-9)。减去一个负数等于加上它的相反数，所以 -5 + 9 = 4。借助数轴可以直观看到从 -5 跳到 4 的过程。
-5 - (-9) = -5 + 9 = 4
Always check borrowing in column subtraction, particularly when zeros are involved, e.g. 2003 - 867.
列竖式减法时务必检查借位，尤其是出现连续的零时，比如 2003 - 867。
3. Multiplication, Division, and BODMAS | 乘除法与运算顺序
9. Then indices: 8² = 64. Next multiplication: 9 × 3 = 27. Finally addition: 27 + 64 = 91.
运算顺序在 7C 练习中至关重要。典型题目：计算 (15 - 6) × 3 + 8²。按照 BODMAS/BIDMAS 顺序，先算括号：15 - 6 = 9。再算指数：8² = 64。接着乘法：9 × 3 = 27。最后加法：27 + 64 = 91。
d 47 × 8 = 376. Adding them gives 1786. Answers must be checked for place value alignment.
两位数乘法如 47 × 38 要注意书写格式。将 38 拆成 30 和 8：47 × 30 = 1410，47 × 8 = 376。相加得 1786。答案必须核对数位对齐。
4 = 21.25 or 21¼. Students should be confident converting remainders into fractional form.
除法题常将余数表示为分数或小数：85 ÷ 4 = 21.25 或 21¼。学生应熟练掌握将余数转化为分数形式。
4. Fundamentals of Fractions | 分数基础
Equivalent fractions and simplifying are tested. Write 18/24 in simplest form. The highest common factor of 18 and 24 is 6, so divide both by 6 to get 3/4.
等效分数与约分是考点。将 18/24 化为最简形式。18 和 24 的最大公因数是 6，分子分母同除以 6 得到 3/4。
18/24 ÷ 6/6 = 3/4
Comparing fractions requires a common denominator. Which is larger, 3/5 or 5/8? LCM of 5 and 8 is 40. 3/5 = 24/40, 5/8 = 25/40, so 5/8 is larger.
比较分数大小时需通分。3/5 和 5/8 哪个更大？5 和 8 的最小公倍数是 40。3/5 = 24/40，5/8 = 25/40，因此 5/8 更大。
Finding a fraction of an amount: Work out 2/5 of 35. Divide by the denominator: 35 ÷ 5 = 7, then multiply by the numerator: 7 × 2 = 14.
求一个数的几分之几：计算 35 的 2/5。先除以分母：35 ÷ 5 = 7，再乘以分子：7 × 2 = 14。
Mixed numbers and improper fractions conversions appear regularly. 23/4 = 11/4 because 2 × 4 + 3 = 11.
带分数与假分数互化经常出现。23/4 = 11/4，因为 2 × 4 + 3 = 11。
5. Decimals and Percentages | 小数与百分数
Interconversion between fractions, decimals and percentages is a core skill. 0.125 equals 125/1000, which simplifies to 1/8, and as a percentage it is 12.5%.
分数、小数、百分数的互化是一项核心技能。0.125 等于 125/1000，约分为 1/8，换成百分数为 12.5%。
original numbers have one decimal place each, so the answer has two decimal places: 0.12.
小数乘法：0.6 × 0.2。先忽略小数点：6 × 2 = 12。原数各有一位小数，共两位小数，答案为 0.12。
0.6 × 0.2 = 0.12
Percentage of amounts: Find 15% of £200. 10% = £20, 5% = £10, so 15% = £30. Alternatively, multiply 200 by 0.15.
求百分比数值：计算 £200 的 15%。10% = £20，5% = £10，因此 15% = £30。也可用 200 × 0.15 计算。
A table often helps students remember conversions:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/10 | 0.1 | 10% |
6. Algebra: Expressions and Simple Equations | 代数：表达式与简易方程
a + 2b - a + 3b. Group a terms: 3a - a = 2a. Group b terms: 2b + 3b = 5b. Answer: 2a + 5b.
合并同类项化简表达式是常见题型。化简 3a + 2b - a + 3b。合并 a 项：3a - a = 2a。合并 b 项：2b + 3b = 5b。答案：2a + 5b。
2x = 8. Divide both sides by 2: x = 4. Always verify by substituting back: 2(4) + 3 = 11.
解一步或两步方程：解 2x + 3 = 11。两边同时减 3：2x = 8。两边除以 2：x = 4。务必代入验证：2(4) + 3 = 11。
2x + 3 = 11 → x = 4
a = 3 and b = -2, find the value of 4a - b. Substituting gives 4(3) - (-2) = 12 + 2 = 14.
代入求值题：已知 a = 3，b = -2，求 4a - b 的值。代入得 4(3) - (-2) = 12 + 2 = 14。
7. Geometry: Angles and Lines | 几何：角与线
°. If one angle is 67°, the other is 180° - 67° = 113°. Around a point, angles total 360°.
测试直线与周角的角度性质。直线上所有角的和为 180°。若一个角是 67°，另一个角为 180° - 67° = 113°。绕一点一周的角总和为 360°。
Vertically opposite angles are equal. In a diagram with two intersecting lines, the angle opposite 45° is also 45°. This is frequently used alongside complementary and supplementary relationships.
对顶角相等。在两线相交的图中，45° 的对顶角也是 45°。这一性质常与互余、互补关系结合考查。
sosceles triangles have two equal base angles; identifying them helps find missing angles.
三角形内角和为 180°。已知两角分别为 54° 和 73°，则第三角 = 180° - (54° + 73°) = 53°。等腰三角形两底角相等，识别这一点可求出未知角。
8. Measurement: Perimeter and Area | 测量：周长与面积
26 cm. For composite shapes, ensure all outer sides are added without missing hidden lengths.
矩形与组合图形的周长：长 8 cm、宽 5 cm 的矩形周长为 2(8+5) = 26 cm。对于组合图形，应确保所有外边长都相加，不要遗漏隐藏边。
ase × height) ÷ 2. A triangle with base 10 m and height 6 m has area (10 × 6) ÷ 2 = 30 m².
矩形面积 = 长 × 宽。上述矩形面积为 8 × 5 = 40 cm²。三角形面积 = (底 × 高) ÷ 2。底 10 m、高 6 m 的三角形面积 = (10 × 6) ÷ 2 = 30 m²。
Area of triangle = (b × h) ÷ 2
Answers must include correct units (cm, cm², m, etc.) and distinguish between perimeter (linear) and area (square). Converting units, such as from mm² to cm², often causes errors.
答案必须注明正确单位（cm、cm²、m 等），并区分周长（长度单位）与面积（平方单位）。单位换算如 mm² 转为 cm² 常导致出错。
9. Statistics: Mean and Charts | 统计：平均数与图表
(most frequent). The median, after ordering 5,7,8,8,12, is 8, and the range is 12 - 5 = 7.
计算平均数：得分 5、8、12、7、8 总和为 40。共有 5 个数，因此平均数为 40 ÷ 5 = 8。众数为 8（出现最频繁）。中位数按顺序 5、7、8、8、12 处于中间的是 8，全距为 12 - 5 = 7。
Interpreting bar charts and pictograms: A bar chart showing favourite fruits may ask, 'How many more chose apples than pears?' frequency of pears from apples. In pictograms, check the key (e.g. one symbol = 4 pupils).
解读条形图和象形图：显示最喜欢的水果的条形图可能问“选择苹果的比选择梨子的多几人？”用苹果的频数减去梨子的频数。在象形图中，注意图例（例如一个符号代表 4 名学生）。
A common answer style: 'The mean is 8, which represents the typical score, but the range of 7 shows a fair spread in data.' This connects calculation with interpretation.
常见答案风格：“平均数为 8，代表典型的得分水平，但全距 7 显示数据分布较广。”这将计算与解释联系起来。
10. Ratio and Proportion | 比和比例
Simplifying ratios: Write 8:12 in simplest form. The highest common factor of 8 and 12 is 4, so divide both by 4 to get 2:3. Ratios must be kept in the same order as the problem states.
化简比：将 8:12 写成最简形式。8 和 12 的最大公因数为 4，同除以 4 得 2:3。比的顺序必须与题目给出的顺序一致。
Sharing in a given ratio: Share £60 in the ratio 2:3. Total parts = 2 + 3 = 5. One part = £60 ÷ 5 = £12. The first share is 2 × £12 = £24, the second share is 3 × £12 = £36.
按比例分配：将 £60 按 2:3 分配。总份数 = 2 + 3 = 5。一份 = £60 ÷ 5 = £12。第一份为 2 × £12 = £24，第二份为 3 × £12 = £36。
Share £60 in 2:3 → £24 and £36
Proportion problems: If 5 pencils cost 75p, what do 8 pencils cost? Find the unit cost: 75 ÷ 5 = 15p per pencil. Then 8 × 15p = 120p or £1.20.
比例问题：若 5 支铅笔 75p，8 支铅笔多少钱？先求单价：75 ÷ 5 = 15p 每支。然后 8 × 15p = 120p，即 £1.20。
11. Sequences and Patterns | 数列与规律
term = 4n + 1 (since 4×1 + 1 = 5). To find the 20th term, substitute n=20: 4(20) + 1 = 81.
求线性数列的第 n 项：5, 9, 13, 17,... 公差为 4。第 n 项 = 4n + 1（因为 4×1 + 1 = 5）。求第 20 项，代入 n=20：4(20) + 1 = 81。
Generating terms: 'Write the first three terms of the sequence with nth term 3n - 2.' For n=1: 3(1)-2=1; n=2: 4; n=3: 7. Answer: 1, 4, 7.
写出数列的前几项：“写出第 n 项为 3n - 2 的数列的前三项。” n=1 时：3(1)-2=1；n=2：4；n=3：7。答案：1, 4, 7。
Picture sequences, such as dot patterns, follow similar linear rules. Count dots and model with 2n+3 type expressions.
图案序列（如点阵）遵循类似的线性规律。
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