📚 KS3 Maths: Calculation Practice Drill | KS3 数学：计算题专项训练
Welcome to the ultimate KS3 calculation practice drill, designed to build your confidence and fluency with numbers. In this article we cover the essential arithmetic skills tested throughout Key Stage 3: order of operations, integers, fractions, decimals, percentages and ratio, together with mental maths techniques that make written work faster and safer. Every section pairs a clear rule with a worked example, and the article closes with a timed practice set so you can check your fluency straight away. Work through the sections in order, or dip into the ones that cost you marks most often. By the end you will be fully prepared for classroom tests, end-of-topic assessments and real-world problem-solving. Let's get started with the fundamentals!
欢迎来到 KS3 计算题专项训练。本文旨在帮你建立对数字的信心与熟练度，系统覆盖 Key Stage 3 阶段的核心算术技能：运算顺序、整数、分数、小数、百分数和比例，并配以能让你算得更快、更稳的心算技巧。每个小节都把规则和例题配对呈现，文章最后还附有一套限时练习，方便你立刻检验自己的熟练度。你可以按顺序阅读，也可以直接跳到最容易丢分的小节。读完并做完练习后，你将能从容应对课堂测验、单元测评以及实际生活中的数学问题。让我们从基础开始吧！
1. What This Drill Covers | 本次训练涵盖什么
KS3 calculation is not a single topic but a set of interlocking number skills. Fluency in one area supports the others: secure place value helps with decimals, confident fraction work makes percentage questions quick, and a reliable order of operations protects every multi-step calculation you will ever do. The table below maps the skills in this article to what they actually let you do in an exam or in everyday life.
KS3 的计算并不是单一知识点，而是一整套彼此咬合的数感技能。一个板块的熟练会带动其他板块：牢固的数位概念让小数计算更稳，分数运算熟练后百分数题可以秒解，而可靠的运算顺序则守护着你做的每一道多步计算题。下表把本文涉及的能力与它们在考试和日常生活中的实际用途对应起来。
| Skill | 技能 | Typical Use | 典型用途 |
|---|---|
| Order of operations (BIDMAS) | 运算顺序 | Multi-step calculations, formula substitution | 多步计算、代入公式 |
| Integers | 整数 | Temperature, money, coordinates | 温度、金钱、坐标 |
| Fractions | 分数 | Recipes, sharing, probability | 食谱、分配、概率 |
| Decimals | 小数 | Measurement, money, science data | 测量、金钱、科学数据 |
| Percentages | 百分数 | Discounts, interest, statistics | 折扣、利息、统计 |
| Ratio | 比例 | Scaling, mixing, maps, exchange rates | 缩放、配比、地图、汇率 |
Notice that every row depends on non-calculator confidence as much as on calculator skill. KS3 papers frequently contain a non-calculator section, so mental strategies are examined content, not an optional extra.
请注意，表中每一行都同时依赖计算器技能和不使用计算器的信心。KS3 试卷通常设有不可使用计算器的部分，因此心算策略是被考查的内容，而不是可有可无的附加项。
2. Order of Operations (BIDMAS) | 运算顺序 (BIDMAS)
BIDMAS tells us the correct sequence when several operations appear together: Brackets, Indices, Division and Multiplication (worked from left to right), then Addition and Subtraction (also from left to right). Applying this rule avoids the most common calculation errors and guarantees that everyone reading your working gets the same answer.
BIDMAS 告诉我们多项运算同时出现时的正确顺序：括号（Brackets）、指数（Indices）、除法和乘法（Division and Multiplication，从左到右计算）、最后是加法和减法（Addition and Subtraction，同样从左到右）。应用这一规则可以避免最常见的计算错误，并确保任何人看你的步骤都会得到同一个答案。
8 + 2 × 3 = 8 + 6 = 14
incorrect. Indices must be dealt with before any multiplication, and brackets override everything else.
乘法优先于加法，所以要先乘后加。如果错误地先做加法，会得到 (8 + 2) × 3 = 30，这是不对的。指数必须先于任何乘法处理，而括号的优先级高于一切。
(3 + 4) × (5 − 2)² = 7 × 3² = 7 × 9 = 63
Work from the innermost bracket outward. Here we calculated inside the brackets first, then the index, and finally multiplied. When brackets are nested, the inner pair always goes first.
由最内层括号向外计算。这里先算括号内，再算指数，最后做乘法。当括号层层嵌套时，永远先算最里面那一对。
20 ÷ 10 = 2. The same applies to addition and subtraction: 10 − 4 + 3 = 9, not 10 − 7 = 3.
请记住，除法与乘法优先级相同，所以只需从左到右依次计算。例如 20 ÷ 5 × 2 = 4 × 2 = 8，而不是 20 ÷ 10 = 2。加法和减法同理：10 − 4 + 3 = 9，而不是 10 − 7 = 3。
A reliable habit is to underline each step you are about to perform before you do it, and to rewrite the whole expression after each stage. This keeps the line of working honest and makes checking much faster.
一个可靠的习惯是：在动手前先把你即将计算的那一步划出来，并在每算完一步后重写整个表达式。这样能让解题步骤保持清晰，也大大加快检查速度。
3. Integers and Negative Numbers | 整数与负数
Integers are whole numbers, positive, negative or zero. A number line is the safest mental model: moving right adds, moving left subtracts, and subtracting a negative number means moving right. Contexts such as temperature, bank balances, sea level and coordinates make these ideas concrete.
整数是正整数、负整数和零的统称。数轴是最稳妥的心智模型：向右移动是加，向左移动是减，而减去一个负数等于向右移动。温度、银行余额、海拔高度和坐标这些情境能让这些概念变得具体可感。
- −7 + 3 = −4 (start at −7, move 3 right) | 从 −7 向右移动 3 格
- −7 − 3 = −10 (start at −7, move 3 left) | 从 −7 向左移动 3 格
- 5 − (−4) = 5 + 4 = 9 (two minuses make a plus) | 两个负号相遇变为加号
ns differ the answer is negative. So (−3) × (−4) = +12, (−3) × 4 = −12 and (−24) ÷ 6 = −4.
乘除法的符号规律遵循同样的逻辑：两数同号，结果为正；两数异号，结果为负。因此 (−3) × (−4) = +12，(−3) × 4 = −12，(−24) ÷ 6 = −4。
−6 × (−2) + (−3) = 12 − 3 = 9
Here BIDMAS and integer rules combine: the multiplication is resolved first, then the addition of a negative. Always simplify sign combinations one at a time rather than trying to do everything in your head.
这道题把运算顺序与整数规律结合起来：先算乘法，再做与负数的加法。请一步一步地化简符号组合，不要试图在脑中一次完成全部运算。
4. Mental Maths Techniques | 心算技巧
Mental strategies convert awkward numbers into friendly ones. The aim is not to impress anyone but to reduce the chance of a slip and to give you a quick estimate against which a written answer can be checked.
心算策略的作用是把别扭的数字变成好算的数字。目的不是炫技，而是降低出错概率，并为你提供一个可以快速核对笔算答案的估值。
- Round and adjust: 47 + 38 = 47 + 40 − 2 = 85. | 先凑整再调整。
- Multiply by 5: multiply by 10, then halve: 68 × 5 = 680 ÷ 2 = 340. | 乘 5 等于乘 10 再除以 2。
- Multiply by 9: multiply by 10, then subtract the number: 47 × 9 = 470 − 47 = 423. | 乘 9 等于乘 10 再减去原数。
- Multiply by 25: multiply by 100, then divide by 4: 25 × 36 = 3600 ÷ 4 = 900. | 乘 25 等于乘 100 再除以 4。
- Use doubles: 63 + 65 = 2 × 64 = 128. | 利用双倍：靠近的两个数取其平均值翻倍。
- Factorise mentally: 25 × 16 = 25 × 4 × 4 = 100 × 4 = 400. | 分解因数让计算变得更简单。
10 and double: 340 ÷ 5 = 34 × 2 = 68. To divide by 4, halve twice: 156 ÷ 4 = 78 ÷ 2 = 39.
除法可以反过来使用同样的技巧。除以 5 时，先除以 10 再翻倍：340 ÷ 5 = 34 × 2 = 68。除以 4 时，连续除以 2 两次：156 ÷ 4 = 78 ÷ 2 = 39。
A final habit worth building is the sanity check: before writing an answer, decide roughly what it should be. If a subtraction is meant to be near 90 and your answer is 900, you have almost certainly lost or gained a zero.
最后一个值得养成的习惯是合理性检查：写下答案前，先大致判断结果应该是多少。如果一道减法题的结果应在 90 左右而你得出了 900，那几乎可以肯定你多写或少写了一个零。
5. Fractions: The Four Operations | 分数：四则运算
A fraction represents a division, so 3/4 means 3 ÷ 4. Simplifying and converting between mixed numbers and improper fractions are the two preparatory skills that make everything else easier.
分数本质上是一种除法，因此 3/4 就是 3 ÷ 4。约分，以及在带分数与假分数之间转换，是让后续一切都变简单的两项预备技能。
tor by their highest common factor: 18/24 = 3/4. To convert, 2 3/5 = (2 × 5 + 3)/5 = 13/5.
约分时，用分子和分母的最大公因数去除：18/24 = 3/4。带分数化为假分数时，2 3/5 = (2 × 5 + 3)/5 = 13/5。
Adding and subtracting requires a common denominator. Find the lowest common multiple of the denominators, rewrite each fraction, then combine the numerators only.
加法和减法需要通分。先找出分母的最小公倍数，把每个分数改写成同分母形式，然后只把分子相加或相减。
2/3 + 1/4 = 8/12 + 3/12 = 11/12
Multiplying is the easiest operation: multiply numerators together and denominators together, then simplify. Dividing means multiplying by the reciprocal of the second fraction.
乘法是最简单的一种：分子乘分子、分母乘分母，然后约分。除法则等价于乘以第二个分数的倒数。
2/3 × 4/5 = 8/15 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
ultiplying by the numerator first (45 × 3 = 135) also works, but dividing first keeps the numbers smaller.
考试中常见的题型是求某个量的几分之几：先除以分母，再乘以分子。3/5 of 45 = 45 ÷ 5 × 3 = 27。其实先乘分子（45 × 3 = 135）也可以，但先除能让数字更小、更好算。
6. Decimals: Place Value and Calculations | 小数：数位与计算
Decimal calculations are decided by place value, not by the number of digits. The column headings to the right of the decimal point are tenths, hundredths, thousandths and so on, each ten times smaller than the one before.
小数计算的关键在于数位，而不在于数字的个数。小数点右边的数位依次是十分位、百分位、千分位，每一位都比前一位小十倍。
he decimal points aligned, then fill any gaps with zeros: 4.7 − 0.85 = 4.70 − 0.85 = 3.85.
做加减法时，把数字竖式排列、对齐小数点，并用零补齐空位：4.7 − 0.85 = 4.70 − 0.85 = 3.85。
When multiplying, ignore the decimal points, multiply as whole numbers, then count the total number of decimal places in the question and put that many back into the answer.
做乘法时，先忽略小数点，按整数相乘，然后数一数题目中两个数一共有几位小数，就在结果中放回同样多的小数位。
0.6 × 0.4 = 0.24 1.3 × 0.2 = 0.26
= 48 ÷ 6 = 8. This preserves the value of the calculation because the ratio is unchanged.
除以小数时，把被除数和除数同时乘以相同的 10 的幂，使除数变成整数：4.8 ÷ 0.6 = 48 ÷ 6 = 8。由于比值不变，这样处理不会改变计算结果。
e ideas, and KS3 questions constantly move between them. Learn the anchors: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/5 = 0.2 = 20%, 1/8 = 0.125 = 12.5%, and 3/8 = 0.375 = 37.5%.
小数、分数和百分数其实是表达同一概念的三套语言，KS3 的题目会不断在它们之间切换。请记牢这些锚点：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，1/5 = 0.2 = 20%，1/8 = 0.125 = 12.5%，3/8 = 0.375 = 37.5%。
7. Percentages and Percentage Change | 百分数与百分数变化
Percent means "per hundred", so 35% ng to a decimal first turns almost every percentage question into a single multiplication.
percent 的意思是"每一百"，所以 35% = 35/100 = 0.35。先把百分数化成小数，几乎每一道百分数题都会变成一次简单的乘法。
15% of 240 = 0.15 × 240 = 36
The same answer can be reached by splitting the percentage: 10% of 240 = 24 and 5% of 240 = 12, so 15% = 24 + 12 = 36. This mental method is invaluable in non-calculator papers.
同样的结果也可以通过拆分百分数得到：240 的 10% 是 24，5% 是 12，所以 15% = 24 + 12 = 36。这种心算方法在不允许使用计算器的试卷中极其宝贵。
Percentage change is calculated using the formula below, which is worth memorising because it appears in many later topics.
百分数变化使用下面的公式计算，由于它会在后续许多知识点中反复出现，非常值得记住。
percentage change = (change ÷ original) × 100%
For increases and decreases, the fastest route is a single multiplier. A 15% increase multiplies by 1.15; a 15% decrease multiplies by 0.85. So a £80 item reduced by 15% costs 80 × 0.85 = £68.
处理增减时，最快的办法是使用单一的乘数。增加 15% 就是乘以 1.15，减少 15% 就是乘以 0.85。因此一件 80 英镑的商品降价 15% 后价格为 80 × 0.85 = 68 英镑。
never to add the percentage back on - instead divide by the multiplier. If a price becomes £96 after a 20% increase, the original was 96 ÷ 1.2 = £80.
逆向百分数题给出的是变化后的结果，要求你求原值。原则是：千万不要把百分数直接加回去，而应该除以乘数。如果某价格在上涨 20% 后变为 96 英镑，那么原价是 96 ÷ 1.2 = 80 英镑。
8. Ratio and Proportion | 比例
xactly like simplifying fractions - divide every part by the same number: 24 : 36 = 2 : 3.
比例用来比较同类的量。化简方法与约分完全相同——把每一项都除以同一个数：24 : 36 = 2 : 3。
Sharing in a given ratio is the classic ratio question, and it always follows the same three steps: add the parts, divide the total by the number of parts, then multiply by each part.
按给定比例分配是比例题的经典题型，永远遵循同样的三步：先把比各项相加得到总份数，用总量除以总份数求出每份的值，再分别乘以各部分。
£150 in the ratio 2 : 3 → 2 + 3 = 5 parts, £150 ÷ 5 = £30 per part, so £60 : £90
Ratios can also be written in the form 1 : n, which makes comparisons easier. The ratio 5 : 8 becomes 1 : 1.6, meaning the second quantity is 1.6 times the first. Unit conversion questions often use this form.
比例也可以写成 1 : n 的形式，便于比较。5 : 8 化为 1 : 1.6，说明第二个量是第一个量的 1.6 倍。单位换算类题目经常使用这种形式。
Direct proportion means quantities increase together at a constant rate. The unitary method handles these questions: if 4 pens cost £2.40, then 1 pen costs £0.60, and 10 pens cost £6.00. Always find the value of one unit first.
正比例意味着两个量以恒定速率同时增大。单位法可以解决这类问题：如果 4 支笔 2.40 英镑，那么 1 支笔 0.60 英镑，10 支笔就是 6.00 英镑。永远先求出"一个单位"的值。
9. Rounding, Estimation and Checking | 取整、估算与检验
Rounding is a precision decision, so the instruction matters as much as the number. To round to a given number of decimal places, look at the digit immediately after the cut-off point: 5 or more rounds up, 4 or less rounds down.
取整是一个关于精度的决定，因此题目要求与数字本身同样重要。要保留指定位数的小数，就看被截断位置后的第一位数字：5 及以上进位，4 及以下舍去。
- 3.647 to 1 d.p. = 3.6 | 3.647 保留 1 位小数得 3.6
- 3.647 to 2 d.p. = 3.65 | 3.647 保留 2 位小数得 3.65
- 24 800 to 2 s.f. = 25 000 | 24 800 保留 2 位有效数字得 25 000
- 0.004 72 to 2 s.f. = 0.004 7 | 0.004 72 保留 2 位有效数字得 0.004 7
Significant figures start from the first non-zero digit, which is why leading zeros never count. This is the single most common rounding mistake in KS3 papers.
有效数字从第一个非零数字开始计算，这正是前导零从不计入的原因。这也是 KS3 试卷中最常见的取整错误。
The exact answer is 80.36, so the estimate is close enough to catch a decimal place error.
估算则反向使用同样的思路：把每个数取到 1 位有效数字，然后心算。要估算 19.6 × 4.1，就用
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