📚 Mastering BIDMAS: Order of Operations | 掌握 BIDMAS：运算顺序
, the answer is not always what you might expect. Mathematicians follow a strict set of rules called the order of operations, often remembered by the acronym BIDMAS or BODMAS. This article, based on the exercise from page 187 of your Cambridge Checkpoint Mathematics coursebook, will guide you through every step so you can solve multi-step calculations with confidence.
当你看到一个算式如 3 + 6 × 2 时，答案并不总是你想象的那样。数学家们遵循一套严格的规则，称为运算顺序，通常用缩略词 BIDMAS 或 BODMAS 来记忆。本文基于 Cambridge Checkpoint 数学教材第187页的练习，将引导你逐步掌握每一步，从而自信地求解多步计算。
1. What is BIDMAS? | 什么是 BIDMAS？
BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition and Subtraction. It tells you the order in which you must perform operations when an expression contains more than one of these. Always work from the top of the list to the bottom.
BIDMAS 代表括号（Brackets）、指数（Indices）、除法（Division）、乘法（Multiplication）、加法（Addition）和减法（Subtraction）。它告诉你当一个表达式中包含多种运算时，你必须按照怎样的顺序进行计算。始终从列表的顶部到底部依次处理。
2. Brackets First | 括号优先
Whatever is inside brackets must be calculated before anything else. If there are nested brackets, work from the innermost pair outwards. Remember that a bracket can also be a division line in a fraction, meaning you must simplify the numerator and denominator separately before dividing.
括号内的所有内容必须最先计算。如果有嵌套括号，则从最内层的括号开始向外计算。请记住，分数线也可以看作一种括号，这意味着你必须在做除法之前先分别化简分子和分母。
3. Indices (Powers) | 指数（幂）
After brackets, evaluate any powers or roots. For example, 4² means 4 × 4, and √9 means the positive number that squares to give 9. Indices come before division, multiplication, addition or subtraction.
处理完括号后，接着计算所有的幂或方根。例如，4² 表示 4 × 4，而 √9 表示平方后得到 9 的正数。指数运算优先于除法、乘法、加法或减法。
4. Division and Multiplication – Left to Right | 除法和乘法——从左到右
ese two operations remain, work from left to right. Do not always multiply before dividing – follow the direction of the calculation. For instance, 10 ÷ 2 × 5 gives 25, not 1.
除法和乘法具有相同的优先级。当只剩下这两种运算时，应按从左到右的顺序计算。不要总是先乘后除——必须遵循算式的方向。例如，10 ÷ 2 × 5 的结果是 25，而不是 1。
5. Addition and Subtraction – Left to Right | 加法和减法——从左到右
Like multiplication and division, addition and subtraction share the same priority. Once all other operations have been completed, perform addition and subtraction as they appear from left to right. This avoids common sign errors.
与乘除法类似，加法和减法也拥有相同的优先级。当所有其他运算完成后，按照从左到右的顺序进行加法和减法。这样可以避免常见的符号错误。
6. Worked Example 1: Simple Expression | 例题1：简单表达式
ccording to BIDMAS, multiplication comes before addition. So 6 × 2 = 12, then 3 + 12 = 15.
计算 3 + 6 × 2。没有括号和指数。根据 BIDMAS，乘法优先于加法。因此 6 × 2 = 12，然后 3 + 12 = 15。
3 + 6 × 2 = 3 + 12 = 15
7. Worked Example 2: Including Brackets | 例题2：包含括号
− 5 would give a different result (9), showing how brackets change the whole calculation.
化简 (2 + 4) × 3 − 5。首先处理括号：2 + 4 = 6。然后乘法：6 × 3 = 18。最后减法：18 − 5 = 13。如果没有括号，2 + 4 × 3 − 5 将得出不同的结果（9），这显示了括号如何改变整个计算。
(2 + 4) × 3 − 5 = 6 × 3 − 5 = 18 − 5 = 13
8. Worked Example 3: Including Indices | 例题3：包含指数
mistakenly multiplied 5 × 3 first, you would get 15² + 4 = 225 + 4 = 229, which is wrong.
计算 5 × 3² + 4。先算指数：3² = 9。然后乘法：5 × 9 = 45。最后加法：45 + 4 = 49。如果你错误地先计算 5 × 3，就会得到 15² + 4 = 225 + 4 = 229，这是错误的。
5 × 3² + 4 = 5 × 9 + 4 = 45 + 4 = 49
9. Common Mistakes to Avoid | 常见错误
r mistake is adding before subtracting when subtraction appears first; always move left to right.
一个常见错误是忽略乘除法从左到右的规则。例如，在 24 ÷ 6 × 2 中，许多学生会先计算 6 × 2 = 12，然后 24 ÷ 12 = 2。正确的从左到右顺序是 24 ÷ 6 = 4，再 4 × 2 = 8。另一个错误是当减法出现在前面时，却先做加法；一定要从左到右计算。
- Wrong: 24 ÷ 6 × 2 = 24 ÷ 12 = 2 / 错误：24 ÷ 6 × 2 = 24 ÷ 12 = 2
- Correct: 24 ÷ 6 × 2 = 4 × 2 = 8 / 正确：24 ÷ 6 × 2 = 4 × 2 = 8
10. Practice Questions (Page 187 Style) | 练习题（类似第187页）
Try these problems to build your skills. Apply BIDMAS carefully and show each step.
尝试以下题目来锻炼你的技能。仔细运用 BIDMAS，并写出每一步。
1) 7 + 3 × 4
2) (10 − 3) × 2 + 6
3) 20 − 2³ × 2
4) 36 ÷ (4 + 2) × 5
5) 5² + 8 ÷ 2 − 1
Solutions / 解答:
1) 7 + 12 = 19
2) 7 × 2 + 6 = 14 + 6 = 20
3) 20 − 8 × 2 = 20 − 16 = 4
4) 36 ÷ 6 × 5 = 6 × 5 = 30
5) 25 + 4 − 1 = 28
11. Using BIDMAS in Algebra | 在代数中使用 BIDMAS
. Then add: 6 + 16 = 22. The same rules prevent mistakes when simplifying algebraic terms.
当你将数字代入代数式时，BIDMAS 同样适用。例如，若 a = 3 且 b = 4，那么表达式 2a + b² 表示 2 × 3 + 4²。先算指数：4² = 16；再算乘法：2 × 3 = 6；然后相加：6 + 16 = 22。同样的规则可以防止化简代数项时出错。
12. Summary and Key Points | 总结与要点
Always follow BIDMAS: Brackets first, then Indices, then Division and Multiplication from left to right, and finally Addition and Subtraction from left to right. Write out your steps clearly to avoid confusion, and double-check any calculation that involves more than two operations. The exercises on page 187 are designed to reinforce these habits – practice them regularly, and the order of operations will become automatic.
请始终遵循 BIDMAS：先算括号，再算指数，接着从左到右进行乘除，最后从左到右进行加减。清晰写出每一步以避免混淆，并仔细检查任何涉及两种以上运算的计算。第187页的练习旨在巩固这些习惯——定期练习，运算顺序自然会变得自如。
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