📚 Essential Maths 7S Homework Answers: Key Knowledge Points Explained | Essential Maths 7S 作业答案详解：知识点精讲
This article goes through the most important topics covered in Essential Maths 7S homework exercises, providing clear explanations for common answers and the reasoning behind each step. Whether you are checking your work or deepening your understanding, these notes will help you master the key skills at KS3 level.
本文梳理了 Essential Maths 7S 作业中涉及的核心知识点，为典型题目的答案提供清晰的解释和每一步的推理过程。不论你是在核对作业，还是想加深理解，这些精讲都能帮助你牢牢掌握 KS3 阶段的重点数学技能。
1. Place Value and Writing Numbers | 数位与写数
Understanding place value is the foundation of all number work. For example, in the number 4 702 156, the digit 7 is in the hundred-thousands place, so its value is 700 000. When writing numbers in words, separate groups of three digits with a space or a comma, and read each group as hundreds, tens and ones followed by the group name (thousand, million, etc.).
理解数位是所有数字运算的基础。以数字 4 702 156 为例，数字 7 在十万位，因此它的值是 700 000。用英文写数字时，用空格或逗号将数字每三位分成一组，然后依次读出每一组的百位、十位和个位，并加上该组的名称（千、百万等）。
Homework answers often ask you to write numbers in expanded form: 4 702 156 = 4 000 000 + 700 000 + 2 000 + 100 + 50 + 6. Being confident with this representation helps with later work on addition, subtraction and even standard form.
作业答案中经常要求你用展开式表示一个数：4 702 156 = 4 000 000 + 700 000 + 2 000 + 100 + 50 + 6。熟练掌握这种表示法，对后面的加法、减法甚至标准形式的运算都很有帮助。
2. Order of Operations (BIDMAS/BODMAS) | 运算顺序
The order of operations must be followed carefully: Brackets, Indices (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right). A common mistake is to always do multiplication before division regardless of order. Remember, they have equal priority, so work from left to right.
计算时必须严格遵守运算顺序：括号、指数（幂）、除法和乘法（从左到右）、加法和减法（从左到右）。一个常见错误是总以为乘法优先于除法，而不看先后顺序。要记住，乘法和除法优先级相同，应该从左向右计算。
For example: 12 – 3 × 4 ÷ 2 + 5. First, multiply and divide from left to right: 3 × 4 = 12, then 12 ÷ 2 = 6. The expression becomes 12 – 6 + 5. Now add and subtract from left to right: 12 – 6 = 6, then 6 + 5 = 11. The correct answer is 11.
例如：12 – 3 × 4 ÷ 2 + 5。首先从左向右处理乘法和除法：3 × 4 = 12，然后 12 ÷ 2 = 6。式子变成 12 – 6 + 5。再从左向右进行加减：12 – 6 = 6，然后 6 + 5 = 11。正确答案是 11。
3. Negative Numbers in Context | 负数的实际应用
Negative numbers appear in temperature, bank balances and elevation. Adding a negative is the same as subtracting its positive value, and subtracting a negative is the same as adding. A number line is the best tool for visualising these moves.
负数经常出现在温度、银行余额和海拔高度等情境中。加上一个负数等于减去它的正数，减去一个负数等于加上它的正数。数轴是帮助理解这些移动的最佳工具。
When answering homework questions such as “The temperature rises from –4 °C to 6 °C. Find the rise,” the correct approach is to subtract the starting temperature from the final temperature: 6 – (–4) = 6 + 4 = 10 °C. Do not simply count the spaces on a thermometer without considering the sign change.
在回答作业题，比如“气温从 –4 °C 上升到 6 °C，求上升了多少度”时，正确的方法是用最终温度减去起始温度：6 – (–4) = 6 + 4 = 10 °C。不要只是数温度计上的格数而忽略符号变化。
4. Factors, Multiples and Primes | 因数、倍数与质数
A factor is a whole number that divides exactly into another number. A multiple is the result of multiplying a number by an integer. A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13.
因数是指能整除另一个数的整数。倍数是指一个数乘以整数得到的结果。质数恰好只有两个因数：1 和它本身。最小的几个质数是 2、3、5、7、11、13。
Essential Maths 7S includes questions on highest common factor (HCF) and lowest common multiple (LCM). To find the HCF of 48 and 60, list all factors of each, then pick the largest common one. 48 has factors 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; 60 has 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The HCF is 12. For LCM, list multiples until you find the first common one: multiples of 8 are 8, 16, 24, 32, 40, 48; multiples of 12 are 12, 24, 36, 48. The LCM is 24.
Essential Maths 7S 中包含了求最大公因数（HCF）和最小公倍数（LCM）的题目。要求 48 和 60 的最大公因数，可以先列出每个数的所有因数，然后找出最大的公共因数。48 的因数有 1, 2, 3, 4, 6, 8, 12, 16, 24, 48；60 的因数有 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60。最大公因数是 12。求最小公倍数时，列出倍数直到找到第一个公共的倍数：8 的倍数有 8, 16, 24, 32, 40, 48；12 的倍数有 12, 24, 36, 48。最小公倍数是 24。
5. Fractions: Equivalent, Simplifying and Comparing | 分数：等值、化简与比较
Equivalent fractions are made by multiplying or dividing the numerator and denominator by the same non-zero number. To simplify a fraction to its lowest terms, divide both parts by their HCF. Answers in homework must usually be given in simplest form.
等值分数是通过将分子和分母同时乘以或除以同一个非零数得到的。要把分数化成最简形式，用分子和分母的最大公因数同时除它们。作业中的答案通常要求写成最简分数。
Comparing fractions with different denominators requires finding a common denominator. For example, to compare 3/5 and 5/8, find the LCM of 5 and 8, which is 40. 3/5 = 24/40, 5/8 = 25/40, so 5/8 is larger. Always convert both fractions to the same denominator before comparing.
比较不同分母的分数需要先找到公分母。例如，比较 3/5 和 5/8，先求 5 和 8 的最小公倍数，即 40。3/5 = 24/40，5/8 = 25/40，因此 5/8 更大。比较之前务必将两个分数转化为同分母。
6. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, they must have the same denominator. If they do not, find the LCM of the denominators, convert each fraction, then add or subtract the numerators. The denominator stays the same. Finally, simplify if possible.
分数的加减法必须先确保分母相同。如果分母不同，先求出分母的最小公倍数，把每个分数通分，然后将分子相加或相减。分母保持不变。最后，如果可以，要化成最简分数。
For mixed numbers, a reliable method is to convert them to improper fractions first. Example: 2 2/3 + 1 1/4 = 8/3 + 5/4. The LCM of 3 and 4 is 12. 8/3 = 32/12, 5/4 = 15/12. Sum = 47/12 = 3 11/12. Avoid the mistake of adding whole parts and fraction parts separately without checking common denominators.
对于带分数，一个可靠的方法是先把它们化为假分数。例如：2 2/3 + 1 1/4 = 8/3 + 5/4。3 和 4 的最小公倍数是 12。8/3 = 32/12，5/4 = 15/12。和为 47/12 = 3 11/12。要避免在没有检查公分母的情况下直接将整数部分和分数部分分别相加的错误。
7. Multiplying and Dividing Fractions | 分数的乘法与除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. Cancel common factors before multiplying to make the calculation easier.
分数乘法很简单：分子相乘作分子，分母相乘作分母。在相乘之前先约去公因数，可以让计算更简单。
Dividing by a fraction means multiplying by its reciprocal. For example, 4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10, which simplifies to 6/5 or 1 1/5. Never try to divide straight across without flipping the second fraction. This is one of the most heavily tested skills in Essential Maths 7S.
除以一个分数等于乘以它的倒数。例如，4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10，化简得 6/5 或 1 1/5。千万不要不把第二个分数倒过来就直接用分子除以分子、分母除以分母。这是 Essential Maths 7S 中考查最多的技能之一。
8. Decimals: Place Value and Rounding | 小数：数位与四舍五入
Decimals extend the place value system to the right: tenths, hundredths, thousandths. When rounding to a given number of decimal places, look at the next digit. If it is 5 or more, round up. Example: round 3.576 to 2 decimal places → look at the third decimal digit, which is 6, so round the 7 up to 8 → 3.58.
小数将数位系统向右扩展：十分位、百分位、千分位。在按要求保留几位小数时，要看下一位数字。如果是 5 或以上就进位。例如：把 3.576 保留两位小数 → 看第三位小数 6，所以将 7 进位变成 8 → 3.58。
Multiplying or dividing decimals by 10, 100, 1000 moves the digits left or right. Each zero shifts the decimal point one place. Homework answers should show understanding of this shift, not just the final number.
小数乘以或除以 10、100、1000 时，数字分别向左或向右移动。每有一个零，小数点就移动一位。作业答案中应该体现出对这种移位的理解，而不仅仅是写出最后的数字。
9. Percentages, Fractions and Decimals | 百分数、分数与小数的互化
A percentage is a number out of 100. To convert a percentage to a fraction, write it over 100 and simplify. 65% = 65/100 = 13/20. To convert a fraction to a percentage, find an equivalent fraction with a denominator of 100, or multiply by 100%.
百分数表示百分之多少。把百分数化成分数时，写成分母为 100 的分数，然后化简。65% = 65/100 = 13/20。把分数化成百分数时，可以求出分母为 100 的等值分数，或者乘以 100%。
Finding a percentage of an amount: 30% of 80 = 30/100 × 80 = 24. For non-calculator methods, find 10% first and build up. 10% of 80 is 8, so 30% = 3 × 8 = 24. Understanding these connections ensures correct answers in mixed-word problems.
求一个数的百分之几：80 的 30% = 30/100 × 80 = 24。对于非计算器方法，可以先求 10% 再推算。80 的 10% 是 8，因此 30% = 3 × 8 = 24。理解这些联系能确保在混合应用题中得出正确答案。
10. Ratio and Proportion | 比与比例
Ratios compare quantities. They can be simplified like fractions by dividing both sides by the same number. The ratio 8:12 simplifies to 2:3. When sharing an amount in a given ratio, find the total number of parts, then work out the value of one part.
比用来比较几个量。像分数一样，可以通过在两边同时除以同一个数来化简比。比 8:12 化简后是 2:3。当按给定比例分配一个数量时，先求出一共有多少份，然后求出每一份的值。
Example: share £56 in the ratio 3:5. Total parts = 3 + 5 = 8. One part = £56 ÷ 8 = £7. The shares are £7 × 3 = £21 and £7 × 5 = £35. Always check that the parts add up to the original total.
例如：将 56 英镑按 3:5 分配。总份数 = 3 + 5 = 8。一份为 £56 ÷ 8 = £7。两部分分别是 £7 × 3 = £21 和 £7 × 5 = £35。一定要检验两部分之和是否等于原来的总数量。
11. Basic Algebra: Simplifying Expressions | 基础代数：化简表达式
In algebra, letters represent unknown numbers. Only like terms can be added or subtracted: 3a + 5a = 8a, but 3a + 5b cannot be simplified further. When multiplying, combine numbers and letters: 2 × a = 2a, and 4a × 3b = 12ab.
在代数中，字母代表未知数。只有同类项才能相加或相减：3a + 5a = 8a，但 3a + 5b 不能再化简。相乘时，将数字相乘，字母写在一起：2 × a = 2a，4a × 3b = 12ab。
Substituting values into expressions is another common task. If a = 3 and b = 4, evaluate 2a + 3b – 5. Replace a and b: 2×3 + 3×4 – 5 = 6 + 12 – 5 = 13. Write out the substitution step clearly to avoid sign errors.
代入求值也是一项常见要求。若 a = 3，b = 4，计算 2a + 3b – 5。代入 a 和 b：2×3 + 3×4 – 5 = 6 + 12 – 5 = 13。要清晰地写出代入的步骤，以免符号出错。
12. Geometry: Angles and Properties of Shapes | 几何：角与图形性质
Angles on a straight line add up to 180°, and angles around a point add up to 360°. Vertically opposite angles are equal. In Essential Maths 7S, students often need to use these facts to find missing angles without a protractor.
直线上的角之和为 180°，绕一点一周的角之和为 360°。对顶角相等。在 Essential Maths 7S 中，学生经常需要运用这些性质，不用量角器而求出未知角。
Triangles have interior angles that sum to 180°. If a triangle has angles of 45° and 55°, the third angle is 180° – 45° – 55° = 80°. Answers should include the degree symbol and a short reason.
三角形内角和为 180°。如果一个三角形的两个角分别是 45° 和 55°，第三个角就是 180° – 45° – 55° = 80°。答案要包含度数符号和简短的理由。
Perimeter is the distance around the outside of a shape; area for rectangles is length × width. For a rectangle of 4 cm by 6 cm, perimeter = 2 × (4 + 6) = 20 cm, area = 4 × 6 = 24 cm². Always include units.
周长是图形外边一周的长度；长方形的面积 = 长 × 宽。一个长 6 cm、宽 4 cm 的长方形，周长 = 2 × (4 + 6) = 20 cm，面积 = 4 × 6 = 24 cm²。一定要写单位。
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