📚 IGCSE Edexcel Maths: Calculation Practice | IGCSE Edexcel 数学：计算题专项训练
Welcome to this focused revision guide on calculation skills for IGCSE Edexcel Mathematics. Strong number work and accurate computation are the foundations of success in both the calculator and non‑calculator papers. This article breaks down key topics, from BIDMAS to standard form, and provides plenty of worked examples to build your confidence.
欢迎阅读这份针对 IGCSE Edexcel 数学的计算题专项训练指南。扎实的数字运算能力和精准的计算是计算器和非计算器试卷取得成功的基础。本文逐一解析从运算顺序到标准形式的关键主题，并提供大量例题来帮助你建立信心。
1. Number Operations and BIDMAS | 数字运算与运算顺序
The order of operations is remembered by the acronym BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction. Always apply this sequence – division and multiplication share the same priority and are worked left to right, as are addition and subtraction.
运算顺序由首字母缩写 BIDMAS 记忆：括号、指数、除法、乘法、加法、减法。始终按照这个顺序计算——除法和乘法优先级相同，从左到右进行；加法和减法也是如此。
ivision and multiplication from left: 2 × 25 = 50, then 50 ÷ 5 = 10. Finally, 10 − 10 = 0.
例如，计算 10 − 2 × (4 + 1)² ÷ 5。先算括号内：4 + 1 = 5。接着指数：5² = 25。然后从左至右乘除：2 × 25 = 50, 接着 50 ÷ 5 = 10。最后 10 − 10 = 0。
5. Practise recognising where brackets are implied, especially in fractions and powers involving negative numbers.
常见的错误是先加后乘：3 + 4 × 5 等于 3 + 20 = 23，而不是 7 × 5 = 35。要练习识别隐含的括号，尤其是在涉及负数的分数和幂运算中。
When using a calculator, enter the expression exactly as written. Use the fraction key and bracket keys to avoid order mistakes. For (−3)², key in ( −3 ) x² to get 9, not −3², which gives −9.
使用计算器时，完全按照书写输入表达式。使用分数键和括号键避免顺序错误。对于 (−3)²，应输入 ( −3 ) x² 得到 9，而不是 −3²，后者结果为 −9。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are different ways of representing the same proportion. You must be able to convert fluently between them. To change a fraction to a decimal, divide the numerator by the denominator. For a percentage, multiply the decimal by 100.
分数、小数和百分数是表示同一比例的不同形式。你必须熟练地在它们之间转换。要将分数化为小数，用分子除以分母；化为百分数，则将小数乘以100。
For example, 3/8 as a decimal: 3 ÷ 8 = 0.375. As a percentage: 0.375 × 100 = 37.5%. Recurring decimals, such as 1/3 = 0.333..., can be written with a dot notation.
例如，3/8 化为小数：3 ÷ 8 = 0.375。化为百分数：0.375 × 100 = 37.5%。循环小数如 1/3 = 0.333... 可以用点记法表示。
tors and denominators: 3/5 × 2/7 = 6/35. For division, flip the second fraction and multiply: 4/9 ÷ 2/5 = 4/9 × 5/2 = 20/18 = 10/9.
分数加减时需要公分母。如 2/3 + 1/4，用 12 作公分母：(8/12) + (3/12) = 11/12。乘法直接分子乘分子、分母乘分母：3/5 × 2/7 = 6/35。除法将第二个分数颠倒再相乘：4/9 ÷ 2/5 = 4/9 × 5/2 = 20/18 = 10/9。
Below is a quick conversion table for common benchmarks:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/10 | 0.1 | 10% |
| 1/8 | 0.125 | 12.5% |
| 1/5 | 0.2 | 20% |
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333... | 33.3% |
| 1/2 | 0.5 | 50% |
| 3/4 | 0.75 | 75% |
Memorising these common equivalents speeds up both mental and written calculations.
记住这些常见的等值关系可以加快心算和笔算的速度。
3. Estimation and Rounding | 估算与四舍五入
Estimation is a valuable skill for checking the reasonableness of answers. Round numbers to one significant figure before performing the mental approximation. In IGCSE, you are often asked to show estimation alongside an accurate calculation.
估算是检查答案合理性的宝贵技能。在心算近似值之前，先将数字四舍五入到一位有效数字。在 IGCSE 考试中，常要求你在精确计算的同时展示估算过程。
imate becomes (50 × 3) / 1 = 150. The exact answer is about 159.0, so 150 is a good check.
示例：估算 (48.7 × 3.2) / 0.98 的值。将 48.7 四舍五入为 50，3.2 为 3，0.98 为 1。估算式变为 (50 × 3) / 1 = 150。精确答案约为 159.0，因此 150 是很好的核查。
Rounding can be to a specified number of decimal places (d.p.) or significant figures (s.f.). Significant figures count from the first non‑zero digit. For instance, 0.004076 rounded to 3 s.f. is 0.00408 (the zero before 4 is not counted). To 2 decimal places, 3.456 becomes 3.46.
四舍五入可以指定小数位数 (d.p.) 或有效数字位数 (s.f.)。有效数字从第一个非零数字开始计数。例如，0.004076 四舍五入到 3 位有效数字为 0.00408（4 之前的零不计）。保留两位小数，3.456 变为 3.46。
Always read the question carefully: if it says "give your answer to 2 decimal places", do not write 3.4 but 3.40, showing the required precision.
务必仔细读题：如果题目要求 "答案保留两位小数"，不要写成 3.4，而应写为 3.40，以体现题目要求的精度。
4. Ratios and Proportions | 比和比例
A ratio compares the sizes of two or more quantities. Ratios can be simplified like fractions by dividing by common factors. For example, 12:18 is simplified to 2:3 after dividing by 6.
比用于比较两个或多个量的大小。比可以像分数一样通过除以公因数来化简。例如，12:18 除以 6 后化简为 2:3。
To share a quantity in a given ratio, first add the parts. Divide £60 in the ratio 3:2: the total parts = 3 + 2 = 5. One part is £60 ÷ 5 = £12. So the shares are 3 × £12 = £36 and 2 × £12 = £24.
按给定比例分配数量时，先求总份数。将 60 英镑按 3:2 分配：总份数 = 3 + 2 = 5。一份为 £60 ÷ 5 = £12。因此分配为 3 × £12 = £36 和 2 × £12 = £24。
5. Use this to find quantities: for 20 litres of paint, blue needed = 1/5 × 20 = 4 litres.
比例问题常与分数关联。若颜料混合中蓝色与黄色之比为 1:4，则蓝色占比为 1/(1+4) = 1/5。利用这一比例求量：调配 20 升颜料，需蓝色 = 1/5 × 20 = 4 升。
et up equations such as y = kx or y = k/x and solve for the constant k using given values.
正比例意味着一个量翻倍时，另一个量也翻倍。反比例是指一个量翻倍时，另一个量减半。建立方程如 y = kx 或 y = k/x，并利用已知值求出常数 k。
5. Standard Form Calculations | 标准形式计算
such as 3.0 × 10⁸ m/s for the speed of light, or 1.6 × 10⁻¹⁹ C for the elementary charge.
标准形式将数字写成 a × 10ⁿ，其中 1 ≤ a < 10, n 为整数。它用于表示极大或极小的数字，比如光速 3.0 × 10⁸ m/s，或基本电荷 1.6 × 10⁻¹⁹ C。
0⁻²) = (2.5 × 4) × 10⁴⁺⁽⁻²⁾ = 10 × 10² = 1.0 × 10³. Notice the answer is rewritten so that 'a' stays between 1 and 10.
乘法：将 a 值相乘，指数相加。例如：(2.5 × 10⁴) × (4 × 10⁻²) = (2.5 × 4) × 10⁴⁺⁽⁻²⁾ = 10 × 10² = 1.0 × 10³。注意答案需重写，使得 a 保持在 1 到 10 之间。
he same power of 10: rewrite (3 × 10⁵) + (5 × 10⁶) as (0.3 × 10⁶) + (5 × 10⁶) = 5.3 × 10⁶.
除法：a 值相除，指数相减。(8 × 10⁷) ÷ (2 × 10³) = (8 ÷ 2) × 10⁷⁻³ = 4 × 10⁴。加减法需要将指数化为相同：将 (3 × 10⁵) + (5 × 10⁶) 改写为 (0.3 × 10⁶) + (5 × 10⁶) = 5.3 × 10⁶。
Use the EXP or ×10ˣ 2.5 × 10⁴ press 2.5 EXP 4. Always check that your screen shows the whole number correctly.
使用计算器上的 EXP 或 ×10ˣ 键快速输入标准形式。输入 2.5 × 10⁴ 可按 2.5 EXP 4。始终检查屏幕显示的数字是否正确。
6. Using a Calculator Effectively | 高效使用计算器
The IGCSE Edexcel exam includes a calculator paper, so mastering your calculator is essential. Know how to use the fraction key, the square and square root keys, and the bracket keys to structure complex expressions.
IGCSE Edexcel 考试包含计算器试卷，因此熟练掌握计算器使用至关重要。了解如何使用分数键、平方与平方根键以及括号键来构造复杂表达式。
For instance, to compute √3.5² + 2.4² er the square root. Avoid typing 3.5 x² + 2.4 x² = √, which many calculators misinterpret.
例如，计算 √3.5² + 2.4²，输入 √ ( 3.5 x² + 2.4 x² )。这确保整个和都在根号内。避免输入 3.5 x² + 2.4 x² = √，许多计算器会曲解此操作。
When dealing with trigonometric functions, check that the mode is set to degrees (D or DEG) for angle measures. For sin 30°, you should get 0.5. If your answer is −0.988, the calculator is in radian mode.
处理三角函数时，确保模式设置为 degrees（度，显示 D 或 DEG）。sin 30° 的结果应为 0.5；若得到 −0.988，说明计算器处于弧度模式。
Use the memory functions (M+, MR, MC) to store intermediate results. This avoids copying errors. Also, learn to use the reciprocal key (1/x) and the percentage key if available, although manual percentage calculations are safer.
利用存储功能（M+, MR, MC）保存中间结果，可以避免抄写错误。另外，学习使用倒数键 (1/x) 和百分比键（若有），但手动进行百分比运算更稳妥。
Before the exam, practise with the exact model you will use. Familiarity with the key layout saves precious time.
考前用考试当天使用的计算器型号进行练习。熟悉按键布局可以节省宝贵的时间。
7. Substituting into Algebraic Expressions | 代数表达式代入
Substituting numbers into expressions is a core algebra skill. Replace each variable with the given value, using brackets around negative numbers to avoid sign errors.
将数值代入表达式是一项核心代数技能。将每个变量替换为给定数值，并在负数外加括号以避免符号错误。
efully: 2(3)² − 3(−2) + 4 = 2(9) + 6 + 4 = 18 + 6 + 4 = 28. Notice that −3(−2) becomes +6.
示例：若 a = 3 且 b = −2，求 2a² − 3b + 4 的值。仔细代入：2(3)² − 3(−2) + 4 = 2(9) + 6 + 4 = 18 + 6 + 4 = 28。注意 −3(−2) 变为 +6。
ackets if needed. For v = u + at, with u = 5, a = −2, t = 4: v = 5 + (−2)(4) = 5 − 8 = −3.
代入分数时，如有需要则将整个分子或分母置于括号内。对于 v = u + at，当 u = 5, a = −2, t = 4 时：v = 5 + (−2)(4) = 5 − 8 = −3。
Double‑ ² = 16. For −x² with x = 4, it is −(4)² = −16. The placement of the negative sign matters.
核查幂运算：x = −4 时 x² 表示 (−4)² = 16。而 x = 4 时 −x² 为 −(4)² = −16。负号的位置很关键。
Practise with expressions involving square roots: √3x + 1 with x = 5 gives √15 + 1 = √16 = 4.
练习涉及平方根的表达式：√3x + 1 中代入 x = 5，得 √15 + 1 = √16 = 4。
8. Solving Linear Equations | 解一元一次方程
Linear equations can be solved by performing the same operation on both sides. The goal is to isolate the unknown, usually x. Step‑by‑step, undo addition/subtraction first, then multiplication/division.
一元一次方程可以通过在等式两边进行相同运算来求解。目标是分离未知数，通常是 x。按部就班，先处理加减，再处理乘除。
Example 1: 3x − 5 = 10. Add 5: 3x = 15. Divide by 3: x = 5. Check: 3(5) − 5 = 10, correct.
示例 1：3x − 5 = 10。两边加 5：3x = 15。除以 3：x = 5。检验：3(5) − 5 = 10，正确。
he bracket: 2x + 6 = 4x − 6. Bring x terms to one side: 6 + 6 = 4x − 2x → 12 = 2x → x = 6.
示例 2：2(x + 3) = 4x − 6。展开括号：2x + 6 = 4x − 6。移项：6 + 6 = 4x − 2x → 12 = 2x → x = 6。
+ 6(2) = 6((x − 1)/2) → 2x + 12 = 3(x − 1). Expand: 2x + 12 = 3x − 3. Bring together: 12 + 3 = 3x − 2x → 15 = x. So x = 15.
含有分数的方程可通过每一项乘以最小公分母来消去分母。解 (x/3) + 2 = (x − 1)/2。每一项乘以 6：6(x/3) + 6(2) = 6((x − 1)/2) → 2x + 12 = 3(x − 1)。展开：2x + 12 = 3x − 3。移项：12 + 3 = 3x − 2x → 15 = x。因此 x = 15。
Always substitute your final answer back into the original equation to verify. This catches simple sign or arithmetic mistakes.
始终将最终答案代回原方程进行验证，这能捕获简单的符号或算术错误。
9. Compound Measures and Unit Conversions | 复合量与单位换算
rce/area). Use triangle formulas to rearrange: speed = d/t, density = m/V, pressure = F/A.
复合量将两个或多个基本量联系起来。最常见的是速度（距离/时间）、密度（质量/体积）和压强（力/面积）。使用三角形公式进行变形：速度 = d/t，密度 = m/V，压强 = F/A。
0 seconds. So 72 km/h = 72 × 1000 m / 3600 s = 20 m/s. A quick method is to divide by 3.6.
单位换算必须小心处理。将 km/h 转换为 m/s，记住 1 km = 1000 m，1
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