Tag: KS3

📚 Cambridge KS3 Maths: Solving Linear Equations (p141 Q2) | 剑桥 KS3 数学：解一元一次方程（第141页第2题）

Welcome to this Cambridge KS3 Maths revision guide. We will work through a key algebra topic: solving linear equations in one variable. This includes one-step and two-step equations, equations with brackets, equations with fractions, and equations with unknowns on both sides. The methods shown here are directly relevant to Cambridge Checkpoint questions, including the exercise found on page 141 question 2.

欢迎阅读本篇剑桥 KS3 数学复习指南。我们将系统学习一个核心代数主题：解一元一次方程。内容包括一步和两步方程、含括号的方程、含分数的方程以及未知数在等式两边的方程。这里展示的方法与 Cambridge Checkpoint 试题直接相关，包括第141页第2题的练习。

1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation is an algebraic statement that says two expressions are equal, and the unknown variable appears only to the first power. For example, 3x + 5 = 20 and 4(x – 2) = 16 are linear equations. The word ‘linear’ comes from the fact that if you drew y = 3x + 5, you would get a straight line.

一元一次方程是一个代数等式，表示两个表达式相等，而且未知数只以一次方的形式出现。例如 3x + 5 = 20 和 4(x – 2) = 16 都是一元一次方程。单词 ‘linear’ 源于这样一个事实：如果画出 y = 3x + 5 的图像，你会得到一条直线。

3x + 5 = 20

x = 5

In this example, x = 5 is the solution because substituting 5 gives 3 × 5 + 5 = 20, which is true. Linear equations can have one solution, no solution, or infinitely many solutions, but at KS3 we mostly meet equations with exactly one solution.

在这个例子中，x = 5 是解，因为代入 5 得到 3 × 5 + 5 = 20，等式成立。线性方程可能有一个解、

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📚 Solving Linear Equations: KS3 Cambridge Maths | 解一元一次方程：KS3剑桥数学

Linear equations are the foundation of algebra at KS3. On page 140 of the Cambridge Checkpoint Maths series, you will meet equations such as x + 3 = 7, 4x = 20 and 2(x – 1) = x + 5. Mastering these skills allows you to solve problems in geometry, science and everyday life. This article explains the key methods step by step and gives you the confidence to tackle any Cambridge KS3 equation question.

线性方程是 KS3 代数的基础。在剑桥 Checkpoint 数学系列的第 140 页，你会遇到像 x + 3 = 7、4x = 20 和 2(x – 1) = x + 5 这样的方程。掌握这些技能能帮助你解决几何、科学和日常生活中的问题。本文逐步讲解核心方法，帮助你自信应对任何剑桥 KS3 方程题。

1. What Is a Linear Equation? | 什么是线性方程？

A linear equation is an equation in which the unknown, usually written as x, is raised only to the power of 1. This means there are no x² terms, no square roots of x, and no fractions where x appears in the denominator. For example, 3x + 5 = 14 is linear, but x² + 2 = 6 is not. The word ‘linear’ comes from the fact that the graph of y = 3x + 5 is a straight line.

线性方程是指未知数（通常写作 x）的指数只为 1 的方程。也就是说，方程中没有 x² 项，没有 x 的平方根，也没有 x 出现在分母中的分数。例如，3x + 5 = 14 是线性方程，但 x² + 2 = 6 不是。’线性’ 一词来源于 y = 3x + 5 的图像是一条直线。

In KS3, you solve linear equations by finding the value of x that makes the equation true. This value is called the solution or root. The equals sign acts like a balance: whatever you do to one side, you must do to the other side to keep the equation balanced.

在 KS3 阶段，你通过求出使方程成立的 x 值来解线性方程。这个值叫做方程的解或根。等号的作用就像一个天平：你对等式的一边做了什么，就必须对另一边做同样的操作，以保持方程平衡。

2. The Balance Method | 天平法

The balance method is the most reliable way to solve linear equations. Imagine each side of the equation is a pan on a balance scale. If you add, subtract, multiply or divide one side by a number, you must do exactly the same to the other side. This keeps the pans level and the equation true.

天平法是解线性方程最可靠的方法。把方程的两边想象成天平的两个托盘。如果你对一边进行加、减、乘或除以一个数，你必须对另一边做完全相同的操作。这样托盘才能保持水平，方程才成立。

For example, to solve x + 6 = 15, subtract 6 from both sides: x + 6 – 6 = 15 – 6, which simplifies to x = 9. You can check by substituting 9 back into the original equation: 9 + 6 = 15, so the solution is correct.

例如，解 x + 6 = 15，两边同时减去 6：x + 6 – 6 = 15 – 6，化简得 x = 9。你可以把 9 代回原方程检验：9 + 6 = 15，所以解是正确的。

Writing the operation on both sides on the same line, or directly below, helps you show method marks in Cambridge exams. Even when you can see the answer mentally, always show the balancing step.

把两边进行的操作写在同一行或正下方，有助于你在剑桥考试中拿到过程分。即使你能直接看出答案，也要写出平衡步骤。

3. Solving with Addition and Subtraction | 用加法与减法解方程

When a number is added to or subtracted from the unknown, undo that operation by using its inverse. If the equation says x – 4 = 10, add 4 to both sides: x – 4 + 4 = 10 + 4, giving x = 14. If the equation says x + 7 = 3, subtract 7 from both sides: x = 3 – 7 = -4.

当未知数被加上或减去一个数时，用逆运算来消去它。如果方程是 x – 4 = 10，两边都加 4：x – 4 + 4 = 10 + 4，得到 x = 14。如果方程是 x + 7 = 3，两边都减 7：x = 3 – 7 = -4。

Many KS3 students find negative solutions challenging. Remember that subtracting a larger number from a smaller one gives a negative result. Practise with examples such as x + 9 = 2 and x – 5 = -3 until you are comfortable moving between positive and negative values.

许多 KS3 学生对负数解感到困难。记住，用较小的数减去较大的数会得到负数。多练习像 x + 9 = 2 和 x – 5 = -3 这样的例子，直到你能熟练处理正负数之间的转换。

Always line up the equation vertically so that the added or subtracted number appears clearly on both sides. This reduces careless mistakes when the solution is negative.

始终把方程垂直排列，使两边加或减的数字清晰可见。这样在解为负数时能减少粗心错误。

4. Solving with Multiplication and Division | 用乘法与除法解方程

If the unknown is multiplied by a number, divide both sides by that coefficient. For example, 5x = 35 becomes 5x ÷ 5 = 35 ÷ 5, so x = 7. If x is divided by a number, multiply both sides by that denominator. For x/4 = 9, multiply both sides by 4: x = 36.

如果未知数乘以一个数，两边都除以这个系数。例如，5x = 35 变为 5x ÷ 5 = 35 ÷ 5，所以 x = 7。如果 x 除以一个数，两边都乘以这个分母。对于 x ÷ 4 = 9，两边都乘以 4：x = 36。

Be careful with negative coefficients: in -3x = 12, divide both sides by -3 to get x = -4. A common sign error is to write x = 4 because the negative sign is forgotten.

注意负系数：在 -3x = 12 中，两边都除以 -3，得到 x = -4。一个常见的符号错误是忘记负号而写成 x = 4。Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com

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📚 Ratio and Proportion: Cambridge KS3 Core Skills | 比例与比例推理：剑桥初中数学核心技法

Ratios and proportions appear throughout the Cambridge KS3 mathematics course, from sharing money and mixing ingredients to reading scale drawings and solving direct proportion problems. The skills in this article match the style of page 134 question 2, where you must choose the correct part of a ratio and apply it to a real situation.

比和比例贯穿剑桥初中数学课程的始终，从分配金钱、混合配料到读取比例尺图和解决正比例问题。本文所讲的技能与第134页第2题的题型一致，这类题目要求你正确找出比的各项，并将其应用到实际情境中。

1. Understanding Ratio | 理解比

A ratio compares two or more quantities in the same unit. For example, if a fruit bowl has 4 apples and 6 oranges, the ratio of apples to oranges is written as 4 : 6.

比用来比较两个或两个以上相同单位的量。例如，一个水果碗里有 4 个苹果和 6 个橙子，苹果与橙子的比就写作 4 : 6。

The order is always important: 4 : 6 is not the same as 6 : 4. The first number must match the first item named in the question.

顺序始终非常重要：4 : 6 不等于 6 : 4。第一个数字必须对应题目中最先提到的项目。

A ratio has no units after it is simplified. It only tells us the relative size of the parts, not the actual amounts.

比在化简后没有单位。它只表示各部分之间的相对大小，并不表示实际数量。

2. Simplifying Ratios | 化简比

To simplify a ratio, divide every part by the highest common factor, or HCF. For 4 : 6, the HCF is 2, so 4 ÷ 2 : 6 ÷ 2 = 2 : 3.

化简比时，要用最大公因数（HCF）去除比的每一项。例如 4 : 6 的最大公因数是 2，所以 4 ÷ 2 : 6 ÷ 2 = 2 : 3。

If a ratio contains decimals or fractions, multiply every part by the same power of 10 or the same denominator first. This makes all parts whole numbers.

如果比中含有小数或分数，要先给每一项乘上相同的 10 的幂或相同的分母，使所有部分都变成整数。

• Example: 0.5 : 1.5 = 5 : 15 = 1 : 3 | 示例：0.5 : 1.5 = 5 : 15 = 1 : 3
• Example: 1/2 : 1/4 = 2 : 1 | 示例：1/2 : 1/4 = 2 : 1

3. Writing Ratios in the Form 1 : n | 写成 1 : n 的形式

Cambridge questions often ask you to write a ratio in the form 1 : n. To do this, divide both sides of the ratio by the first number.

剑桥考试常要求将比写成 1 : n 的形式。方法是用第一个数去除比的两边。

For example, 3 : 12 becomes 1 : 4 because 3 ÷ 3 = 1 and 12 ÷ 3 = 4.

例如，3 : 12 可以写成 1 : 4，因为 3 ÷ 3 = 1，12 ÷ 3 = 4。

a : b → 1 : (b ÷ a)

Use this form when you need to compare one part to the other relative to a single unit. It is especially useful for scale drawings and rates.

当你需要以单位 1 为基准比较两个部分时，可以使用这种形式。它在比例尺图和速率问题中尤其有用。

4. Sharing in a Given Ratio | 按给定比例分配

To share a quantity in the ratio a : b, first add the parts to find the total number of parts. Then divide the quantity by that total to find the value of one part.

要按 a : b 分配一个量，先把各项相加求出总份数。然后用总量除以总份数，得到一份的大小。

Finally multiply the value of one part by each ratio term. This gives the actual amount for each share.

最后用一份的大小分别乘比的每一项，就能得到每一部分的实际数量。

One part = Total ÷ (a + b)

Example: Share £200 in the ratio 2 : 3.

示例：按 2 : 3 分配 200 英镑。

Total parts = 2 + 3 = 5, so one part = £200 ÷ 5 = £40. The shares are £40 × 2 = £80 and £40 × 3 = £120.

总份数 = 2 + 3 = 5，所以一份 = 200 ÷ 5 = 40 英镑。两部分分别是 40 × 2 = 80 英镑和 40 × 3 = 120 英镑。

Always check that the shares add back to the original total: £80 + £120 = £200.

一定要检验各部分相加是否等于原来的总量：80 + 120 = 200 英镑。

5. Three-Part Ratios | 三项比

When three quantities are compared, such as red : blue : green = 3 : 5 : 7, the total number of parts is 3 + 5 + 7 = 15.

当比较三个量时，例如红 : 蓝 : 绿 = 3 : 5 : 7，总份数为 3 + 5 + 7 = 15。

Three-part ratio problems work exactly the same way as two-part ratio problems. Find one part first, then multiply by each term.

三项比的解题方法与两项比完全相同。先求出一份的大小，再分别乘以每一项。

Example: 300 ml of juice is made from syrup, water and sparkling water in the ratio 1 : 2 : 3.

示例：300 毫升果汁由糖浆、水和气泡水按 1 : 2 : 3 的比例混合而成。

Total parts = 1 + 2 + 3 = 6. One part = 300 ÷ 6 = 50 ml. Amounts are 50 ml, 100 ml and 150 ml.

总份数 = 1 + 2 + 3 = 6。一份 = 300 ÷ 6 = 50 毫升。各量分别为 50 毫升、100 毫升和 150 毫升。

Check: 50 + 100 + 150 = 300 ml and the simplified ratio is 50 : 100 : 150 = 1 : 2 : 3.

检验：50 + 100 + 150 = 300 毫升，化简后为 50 : 100 : 150 = 1 : 2 : 3。

6. Direct Proportion | 正比例

Two quantities are in direct proportion if one increases at the same rate as the other. The ratio between the two quantities remains constant.

如果两个量以相同的速率增长，它们就成正比例。两个量之间的比保持不变。

For example, if 5 pens cost £3, then 10 pens cost £6. The cost per pen is £3 ÷ 5 = £0.60.

例如，5 支笔花费 3 英镑，那么 10 支笔花费 6 英镑。每支笔的价格是 3 ÷ 5 =

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📚 Mastering Percentages for Cambridge KS3 Maths | 掌握剑桥 KS3 数学百分数

A percentage is a fraction with denominator 100. The symbol % means ‘out of 100’, so 35% = 35/100 = 0.35. In KS3, percentages link fractions, decimals, ratio and real-life problems such as discounts, tax and interest.

百分数是一个分母为 100 的分数。符号 % 表示“每一百”，因此 35% = 35/100 = 0.35。在 KS3 阶段，百分数将分数、小数、比与实际生活问题（如折扣、税费和利息）联系起来。

1. What Is a Percentage? | 什么是百分数？

A percentage is a way of expressing a number as a part of 100. It allows different quantities to be compared on the same scale. For example, 25% means 25 out of every 100, which can also be written as the fraction 25/100 = 1/4 or the decimal 0.25.

百分数是一种把数字表示为每一百中的多少的方式。它使不同的数量能够在同一尺度上进行比较。例如，25% 表示每一百中有 25，也可以写成分数 25/100 = 1/4 或小数 0.25。

Every percentage can be converted into a fraction or decimal, and these conversions are the foundation for many KS3 calculations.

每个百分数都可以转换为分数或小数，这些转换是许多 KS3 计算的基础。

2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数互化

To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. To convert a fraction to a percentage, multiply the fraction by 100. To convert a percentage to a fraction, write it over 100 and simplify.

将小数化为百分数时乘以 100；将百分数化为小数时除以 100。将分数化为百分数时用分数乘以 100；将百分数化为分数时写成分母为 100 的分数并化简。

• 0.75 × 100 = 75%
• 40% ÷ 100 = 0.40
• 3/5 × 100 = 60%
• 65% = 65/100 = 13/20

These conversions are useful when you need to compare a test score, a discount or a data proportion quickly.

当需要快速比较考试成绩、折扣或数据比例时，这些转换非常有用。

3. Finding a Percentage of a Quantity | 求一个量的百分数

To find a percentage of an amount, use the rule: percentage of amount = (percentage / 100) × amount. For example, 15% of £80 is calculated as 0.15 × 80 = £12.

要求一个量的百分数，使用规则：量的百分数 = (百分数 / 100) × 总量。例如，80 英镑的 15% 计算为 0.15 × 80 = 12 英镑。

15% of 80 = (15 / 100) × 80 = £12

This method is tested in questions about discounts, test marks, tax and commission.

该方法常用于折扣、考试分数、税费和佣金类问题。

4. Using Non-Calculator Methods: 10%, 5%, 1% and Multiples | 非计算器方法：10%、5%、1%及倍数

You can build harder percentages from simple benchmarks. 10% = amount ÷ 10, 5% = 10% ÷ 2, and 1% = amount ÷ 100. For example, to find 35% of 240, first find 10% = 24, then 5% = 12, so 30% = 3 × 24 = 72 and 35% = 72 + 12 = 84.

你可以从简单的基准构造更复杂的百分数。10% = 总量 ÷ 10，5% = 10% ÷ 2，1% = 总量 ÷ 100。例如，求 240 的 35%：先求 10% = 24，再求 5% = 12，因此 30% = 3 × 24 = 72，35% = 72 + 12 = 84。

• 10% of 240 = 24
• 5% of 240 = 12
• 35% of 240 = 72 + 12 = 84

This non-calculator approach is often required in Cambridge KS3 mental and written papers.

这种非计算器方法在剑桥 KS3 的口算和笔试中经常要求使用。

5. Percentage Increase and Decrease | 百分数增加与减少

To increase an amount by r%, multiply by (1 + r / 100). To decrease an amount by r%, multiply by (1 – r / 100). Example: £60 increased by 15% = 60 × 1.15 = £69. £80 decreased by 25% = 80 × 0.75 = £60.

将数量增加 r%，乘以 (1 + r / 100)；减少 r%，乘以 (1 – r / 100)。例如，60 英镑增加 15% = 60 × 1.15 = 69 英镑；80 英镑减少 25% = 80 × 0.75 = 60 英镑。

New value = Original × (1 ± r / 100)

Using multipliers is faster and less prone to error than finding the change and then adding or subtracting.

使用乘数比先求出变化量再加减更快，也更不容易出错。

6. Finding the Original Value After a Percentage Change | 百分数变化后求原值

If a final amount includes a known percentage change, divide by the multiplier. After a 20% increase, final value = original × 1.20, so original = final ÷ 1.20. After a 15% decrease, final value = original × 0.85, so original = final ÷ 0.85.

如果最终数量包含已知的百分数变化，除以乘数即可。增加 20% 后，最终值 = 原值 × 1.20，因此原值 = 最终值 ÷ 1.20；减少 15% 后，最终值 = 原值 × 0.85，因此原值 = 最终值 ÷ 0.85。

Example: A coat costs £69 after a 15% increase. Original price = 69 ÷ 1.15 = £60.

例如：一件外套涨价 15% 后为 69 英镑。原价 = 69 ÷ 1.15 = 60 英镑。

This reverse percentage skill is common in KS3 word problems and checkpoints.

这种反向百分数技能在 KS3 应用题和 checkpoint 考试中很常见。

7. Percentage Change Calculations | 百分数变化计算

Percentage change = (new value – original value) / original value × 100%. If a price rises from £40 to £50, the change is £10, so percentage change = 10/40 × 100 = 25%. If it falls from £50 to £45, percentage change = -5/50 × 100 = -10%, a 10% decrease.

百分数变化 = (新值 – 原值) / 原值 × 100%。如果价格从 40 英镑涨到 50 英镑，变化量为 10 英镑，因此百分数变化 = 10/40 × 100 = 25%；如果从 50 英镑降到 45 英镑，百分数变化 = -5/50 × 100 = -10%，即减少 10%。

Percentage change = (New – Original) / Original × 100%

Remember that the denominator is always the original value, not the new value.

注意分母始终是原值，而不是新值。

8. Percentages in Real-Life Contexts | 百分数在实际生活中的应用

Percentages appear in discounts, VAT, profit and loss, simple interest, test results and survey data. For simple interest, use I = P × r × t / 100. For a discount, multiply the original price by (1 – discount% / 100).

百分数出现在折扣、增值税、利润与亏损、单利、考试成绩和调查数据中。单利公式为 I = P × r × t / 100。计算折扣时，用原价乘以 (1 – 折扣率 / 100)。

Always read the question carefully to decide whether you need a percentage of an amount, a percentage change or an original value.

仔细审题，判断需要求的是量的百分数、百分数变化还是原值。

9. Comparing Quantities Using Percentages | 用百分数比较数量

Percentages standardise comparisons. For example, 17 out of 25 is 68%, while 19 out of 30 is about 63.3%, so the first is a better score. Always write both as percentages before comparing to avoid misleading by different totals.

百分数使比较标准化。例如，25 题中对 17 题是 68%，30 题中对 19 题约为 63.3%，因此前者表现更好。比较前应先将两者写成百分数，以免因总数不同而产生误导。

This technique is especially useful in data and statistics questions where different sample sizes appear.

当数据与统计问题中出现不同的样本量时，这种技巧尤其有用。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Do not confuse percentage increase with percentage of amount. Do not add or subtract percentages directly when the bases are different. For example, a 20% increase followed by a 10% decrease is not a net 10% increase.

不要混淆“百分数增加”与“求一个量的百分数”。当基数不同时，不要直接相加或相减百分数。例如，先增加 20% 再减少 10%，净变化并不是增加 10%。

Show clear working, use multipliers, and check that the final answer is sensible. In exams, look for wording such as ‘of’, ‘increased by’, ‘decreased by’, ‘original value’ and ‘percentage change’.

展示清晰步骤，使用乘数，并检查最终答案是否合理。考试中注意“的”、“增加”、“减少”、“原值”和“百分数变化”等关键词。

Practising these question types from Cambridge KS3 past papers will help you avoid common traps and gain confidence.

练习剑桥 KS3 历年真题中的这些题型，有助于避开常见陷阱并增强信心。

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📚 Fractions, Decimals and Percentages | KS3 数学：分数、小数与百分数

Fractions, decimals and percentages are three different ways of describing the same value. They appear constantly in everyday life, from measuring ingredients in a recipe to calculating discounts in a shop.

分数、小数和百分数是描述同一数值的三种不同方式。它们在日常生活中随处可见，从按食谱称量食材到在商店里计算折扣都离不开它们。

In this KS3 revision guide, you will learn how to convert between the three forms, compare them and use them to solve problems confidently.

在这份 KS3 复习指南中，你将学习如何在这三种形式之间进行转换、比较它们，并自信地运用它们解决问题。

1. Understanding Fractions | 理解分数

A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts are taken.

分数写作 a/b，其中 a 是分子，b 是分母。分母表示整体被分成多少等份，分子表示取了多少份。

For example, 3/5 means the whole is split into 5 equal parts and 3 of them are selected. A unit fraction has a numerator of 1, such as 1/2 or 1/8.

例如，3/5 表示整体被分成 5 等份，并选取其中的 3 份。单位分数的分子为 1，如 1/2 或 1/8。

Proper fractions have a numerator smaller than the denominator, improper fractions have a numerator greater than or equal to the denominator, and mixed numbers combine an integer with a proper fraction.

真分数的分子小于分母，假分数的分子大于或等于分母，带分数由一个整数和一个真分数组成。

Fractions are useful when a value is not a whole number, but decimals and percentages are often easier to compare in practical situations.

当数值不是整数时，分数非常有用，但在实际情境中，小数和百分数通常更容易比较。

2. Equivalent Fractions and Simplifying | 等值分数与化简

Equivalent fractions have different numerators and denominators but represent the same value. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero number.

等值分数的分子和分母不同，但表示相同的值。你可以通过将分子和分母同时乘以或除以同一个非零数来得到等值分数。

For example, 2/3 = 4/6 = 8/12 because each form is obtained by multiplying the top and bottom by the same factor.

例如，2/3 = 4/6 = 8/12，因为每种形式都是将分子和分母同时乘以相同的因数得到的。

To simplify a fraction, divide both terms by their highest common factor (HCF). The fraction 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6.

化简分数时，将分子和分母同时除以它们的最大公因数 (HCF)。分数 18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6。

You can check if two fractions are equivalent by cross-multiplying. For example, 2/3 and 4/6 are equivalent because 2 × 6 = 3 × 4.

你可以通过交叉相乘来检验两个分数是否等值。例如，2/3 和 4/6 等值，因为 2 × 6 = 3 × 4。

Always simplify your final answer unless the question specifically asks for an unsimplified fraction.

除非题目明确要求保留未化简的分数，否则最终答案一定要化简。

3. Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. A fraction is simply a division statement.

将分数转换为小数，用分子除以分母即可。分数本质上就是除法。

For example, 3/8 means 3 ÷ 8 = 0.375, so 3/8 = 0.375. You can use short division or a calculator if allowed.

例如，3/8 表示 3 ÷ 8 = 0.375，因此 3/8 = 0.375。你可以使用短除法，或者如果允许的话使用计算器。

Some fractions produce terminating decimals, such as 1/4 = 0.25, while others produce recurring decimals, such as 1/3 = 0.333… which can be written as 0.3 with a dot over the 3.

有些分数产生有限小数，如 1/4 = 0.25；另一些则产生循环小数，如 1/3 = 0.333…，可写作 0.3，在 3 上方加点。

If the denominator is 10, 100 or 1000, conversion is immediate: 7/10 = 0.7, 23/100 = 0.23, and 9/1000 = 0.009.

如果分母是 10、100 或 1000，转换非常直接：7/10 = 0.7，23/100 = 0.23，9/1000 = 0.009。

Remember that the decimal point separates the whole number part from the fractional part, so 2/5 = 0.4 means 4 tenths, not 4 hundredths.

记住小数点把整数部分和小数部分分开，所以 2/5 = 0.4 表示 4 个十分之一，而不是 4 个百分之一。

4. Decimals to Fractions | 小数转分数

To convert a terminating decimal to a fraction, write the decimal as the numerator over a power of 10, then simplify if possible.

将有限小数转换为分数时，把小数作为分子，分母为 10 的幂，然后尽可能化简。

For example, 0.65 = 65/100. Dividing the numerator and denominator by 5 gives 13/20.

例如，0.65 = 65/100。将分子和分母同时除以 5，得到 13/20。

The number of decimal places determines the denominator: one decimal place gives 10, two decimal places gives 100, three decimal places gives 1000, and so on.

小数位数决定分母：一位小数对应 10，两位对应 100，三位对应 1000，依此类推。

For mixed decimals such as 2.4, write the integer part separately: 2.4 = 2 + 4/10 = 2 2/5.

对于像 2.4 这样的带小数，先写出整数部分：2.4 = 2 + 4/10 = 2 2/5。

Always check if the fraction can be simplified. For example, 0.25 = 25/100 = 1/4, so 0.25, 25/100 and 1/4 all describe the same value.

一定要检查分数是否可以化简。例如，0.25 = 25/100 = 1/4，所以 0.25、25/100 和 1/4 都表示相同的值。

5. Percentages as Fractions and Decimals | 百分数作为分数与小数

A percentage is a fraction with a denominator of 100. The symbol ‘%’ means ‘out of 100’, so 45% = 45/100 = 0.45.

百分数是分母为 100 的分数。符号 ‘%’ 表示“每 100”，因此 45% = 45/100 = 0.45。

To convert a percentage to a fraction, write it over 100 and simplify. For example, 25% = 25/100 = 1/4.

将百分数转换为分数时，把它写在 100 之上并化简。例如，25% = 25/100 = 1/4。

To convert a percentage to a decimal, divide by 100. This is the same as moving the decimal point two places to the left: 7% = 0.07 and 130%

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📚 Solving Linear Equations: Cambridge KS3 Maths p124_2 | 解一元一次方程：剑桥 KS3 数学 p124_2

Linear equations are the foundation of algebra in the Cambridge Lower Secondary programme. The exercise in p124_2.pdf focuses on equations that contain brackets, variables on both sides, and small fractional coefficients. This article explains the complete method, using the style of question 2 as a worked model.

一元一次方程是剑桥初中数学代数的基础。p124_2.pdf 中的练习重点考查含有括号、两边含有未知数以及简单分数系数的方程。本文以第 2 题的题型为例，完整讲解解题方法。

1. Understanding the Question | 理解题意

The question in p124_2.pdf asks you to solve an equation for the unknown x. A typical example is 2(3x – 1) = 4(x + 2). Solving means finding every value of x that makes the left-hand side equal to the right-hand side.

p124_2.pdf 中的题目要求解出未知数 x。一个典型例子是 2(3x – 1) = 4(x + 2)。解方程意味着找出所有使左边等于右边的 x 值。

Read the question twice. Underline the brackets and any like terms before starting. This helps you choose the correct first step.

请把题目读两遍。开始

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📚 Cambridge KS3 Maths: Fractions, Decimals and Percentages | 剑桥 KS3 数学：分数、小数和百分比

Working confidently with fractions, decimals and percentages is a core skill in the Cambridge KS3 Mathematics curriculum. This topic often appears on pages such as 123 of the learner’s book, where you are asked to convert, compare and calculate with all three forms. This revision guide explains the key ideas step by step and gives you clear examples.

在剑桥 KS3 数学课程中，熟练处理分数、小数和百分比是一项核心技能。这类内容经常出现在教材第 123 页等位置，要求同学们在三种形式之间进行转换、比较和计算。本复习指南将逐步讲解关键概念，并给出清晰的示例。

1. Understanding Fractions | 理解分数

A fraction is written as a numerator over a denominator, for example 3/4. The denominator tells you how many equal parts make one whole, and the numerator tells you how many of those parts are being considered.

分数写作分子在上、分母在下的形式，例如 3/4。分母表示一个整体被平均分成多少份，分子表示我们选取了多少份。

A proper fraction has a numerator smaller than its denominator, such as 2/5. An improper fraction has a numerator equal to or greater than its denominator, such as 7/4, and a mixed number combines a whole number with a proper fraction, such as 1 3/4.

真分数的分子小于分母，例如 2/5。假分数的分子大于或等于分母，例如 7/4。带分数由一个整数和一个真分数组成，例如 1 3/4。

2. Equivalent Fractions and Simplest Form | 等值分数与最简形式

Equivalent fractions represent the same quantity even though they look different. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.

等值分数表示相同的量，尽管形式不同。将分子和分母同时乘以或除以同一个非零数，就可以得到等值分数。

To write a fraction in its simplest form, divide the numerator and denominator by their highest common factor (HCF). For example, 12/16 becomes 3/4 because the HCF of 12 and 16 is 4.

要把分数化为最简形式，用分子和分母的最大公因数 (HCF) 同时去除它们。例如，12/16 化简为 3/4，因为 12 和 16 的最大公因数是 4。

3. Converting Between Improper Fractions and Mixed Numbers | 假分数与带分数互转

To change a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, 2 3/5 becomes (2 × 5 + 3)/5 = 13/5.

将带分数化为假分数时，用整数乘以分母，加上分子，再把结果放在原分母上。例如，2 3/5 变为 (2 × 5 + 3)/5 = 13/5。

To change an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part, the remainder is the new numerator, and the denominator stays the same.

将假分数化为带分数时，用分子除以分母。商是整数部分，余数是新的分子，分母保持不变。

4. Understanding Decimals | 理解小数

Decimals are another way of writing fractions whose denominators are powers of 10. The first decimal place is tenths, the second is hundredths, and the third is thousandths.

小数是分母为 10 的幂的分数的另一种写法。小数点后第一位是十分位，第二位是百分位，第三位是千分位。

For example, 0.3 means 3/10, 0.25 means 25/100, and 0.375 means 375/1000. This place-value link makes converting between decimals and fractions easier.

例如，0.3 表示 3/10，0.25 表示 25/100，0.375 表示 375/1000。这种位值联系使小数与分数的转换更加容易。

5. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator: 3/8 = 3 ÷ 8 = 0.375.

要将分数转换为小数，用分子除以分母。可以使用短除法或计算器：3/8 = 3 ÷ 8 = 0.375。

Some fractions produce terminating decimals, such as 1/4 = 0.25. Others produce recurring decimals, such as 1/3 = 0.333… where the 3 repeats forever. A recurring decimal can be written with a dot above the repeating digit.

有些分数会得到有限小数，例如 1/4 = 0.25。另一些则得到循环小数，例如 1/3 = 0.333…，其中 3 无限重复。循环小数可以在重复数字上方加点表示。

6. Converting Decimals to Fractions | 小数转分数

To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000 and so on, depending on the number of decimal places. Then simplify if possible.

将有限小数转换为分数时，根据小数位数，将小数写成分母为 10、100、1000 等的分数，然后尽可能化简。

For example, 0.45 = 45/100 = 9/20. For a recurring decimal like 0.444…, the conversion uses a special algebraic method: let x = 0.444…, then 10x = 4.444…, subtracting gives 9x = 4, so x = 4/9.

例如，0.45 = 45/100 = 9/20。对于 0.444… 这样的循环小数，转换时使用一种特殊的代数方法：设 x = 0.444…，则 10x = 4.444…，相减得 9x = 4，所以 x = 4/9。

7. Understanding Percentages | 理解百分比

A percentage is a fraction with denominator 100. The symbol % means ‘out of 100’, so 35% = 35/100 = 0.35.

百分比是分母为 100 的分数。百分号 % 表示 “每一百”，因此 35% = 35/100 = 0.35。

Percentages are useful for comparing quantities because they convert different totals to a common scale of 100. For example, a test score of 17 out of 20 is 17/20 = 0.85 = 85%.

百分比便于比较数量，因为它把不同的总量都换算成以 100 为统一标准。例如，20 分中获得 17 分就是 17/20 = 0.85 = 85%。

8. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比互转

To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify.

将百分比化为小数，除以 100；将小数化为百分比，乘以 100；将百分比化为分数，写成百分号下的数除以 100 再化简。

The table below shows common conversions that are worth memorising for Cambridge KS3 assessments.

下表列出了一些值得在剑桥 KS3

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📚 KS3 Maths: Percentage Increase and Decrease | KS3 数学：百分数的增加与减少

Percentage increase and decrease are essential skills in the Cambridge KS3 Mathematics curriculum. They appear in word problems about prices, discounts, interest, profit, tax and data changes. This article explains the core methods, common formulas and exam strategies step by step.

百分数增加与减少是剑桥 KS3 数学课程中的基本技能。它们经常出现在关于价格、折扣、利息、利润、税收和数据变化的文字题中。本文将逐步讲解核心方法、常用公式和考试策略。

1. Understanding Percentages | 理解百分数

A percentage is a number expressed as a fraction of 100. The symbol % means ‘out of 100’, so 45% means 45 out of 100, or 45/100. In decimal form, 45% is 0.45.

百分数是以 100 为分母的分数。符号 % 表示“每一百份中”，所以 45% 表示 100 份中的 45 份，即 45/100。写成小数就是 0.45。

Being able to switch quickly between percentages, decimals and fractions makes all percentage calculations easier.

能够快速在百分数、小数和分数之间转换，会让所有百分数计算变得更加简单。

2. Finding a Percentage of a Quantity | 求一个数量的百分之几

To find p% of a quantity N, first write p% as a decimal by dividing by 100. Then multiply the decimal by N. For example, 15% of 240 is 0.15 × 240 = 36.

求某个量 N 的 p%，先把 p% 除以 100 写成小数。然后用这个小数乘以 N。例如，240 的 15% 是 0.15 × 240 = 36。

p% of N = (p ÷ 100) × N

Always check that your answer makes sense. A useful benchmark is 10%. For 240, 10% is 24, so 15% should be 24 + 12 = 36.

始终检查答案是否合理。10% 是一个有用的基准。对于 240，10% 是 24，因此 15% 应该是 24 + 12 = 36。

3. Percentage Increase | 百分数增加

Percentage increase describes how much a value grows compared with its original amount. The increase is always measured against the original value, not the new value.

百分数增加描述一个值相对于原始值增长了多少。增加量始终是相对于原始值来衡量的，而不是相对于新值。

Percentage increase = (increase ÷ original) × 100%

To find a new value directly, add the increase to the original. If a quantity increases by r%, its new value can also be written as original × (1 + r/100).

要直接求新值，将增加量加到原始值上。如果一个量增加了 r%，它的新值也可以写成 原始值 × (1 + r/100)。

Example: a school club had

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📚 KS3 Cambridge Maths: Fractions, Decimals and Percentages (p120 Q2) | 剑桥中学数学：分数、小数与百分数（p120 第2题）

In this article we review a core KS3 Cambridge maths topic: fractions, decimals and percentages. This topic appears frequently in Checkpoint tests and in homework pages such as page 120, question 2. You will see how the three forms are linked, how to convert between them, and how to apply them to real problems.

本文复习剑桥初中数学的核心专题：分数、小数与百分数。该专题经常出现在 Checkpoint 测试以及类似第 120 页第 2 题的作业中。你将看到这三种形式之间的联系、如何互相转换，以及如何将它们应用于实际问题。

1. Understanding the Big Picture | 理解整体图景

Fractions, decimals and percentages are three different ways of describing the same idea: a part of a whole. A fraction uses a numerator and denominator, a decimal uses place value, and a percentage is always out of 100.

分数、小数和百分数是描述同一个概念的三种不同方式：整体的一部分。分数用分子和分母表示，小数使用位值，百分数则总是以 100 为基准。

For example, one half can be written as ½, 0.5 or 50%. The ability to move quickly between these forms is a key skill in the Cambridge KS3 curriculum.

例如，一半可以写成 ½、0.5 或 50%。在这些形式之间快速转换是剑桥 KS3 课程中的关键技能。

Before practising conversions, make sure you understand that the three representations have equal value even though they look different.

在练习转换之前，请确保你理解这三种表示虽然看起来不同，但数值相等。

2. Equivalent Fractions | 等值分数

An equivalent fraction is created by multiplying or dividing both the numerator and denominator by the same non-zero number. This does not change the value of the fraction.

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而得到的分数。这样做不会改变分数的值。

For instance, 1/2 = 2/4 = 4/8 because both parts of the fraction have been scaled by the same factor.

例如，1/2 = 2/4 = 4/8，因为分数的分子和分母都按相同的倍数进行了缩放。

In a typical exercise such as page 120 question 2, you may be asked to write a fraction in its simplest form. To simplify, divide the numerator and denominator by their highest common factor.

在类似第 120 页第 2 题的典型练习中，你可能需要把分数化为最简形式。化简分数的方法是分子和分母同时除以它们的最大公因数。

For example, 12/16 simplifies to 3/4 because the highest common factor of 12 and 16 is 4.

例如，12/16 化简为 3/4，因为 12 和 16 的最大公因数是 4。

3. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator if allowed.

将分数转换为小数的方法是分子除以分母。可以使用短除法，或者在允许的情况下使用计算器。

fraction → decimal: numerator ÷ denominator

A fraction like 3/8 becomes 0.375 because 3 ÷ 8 = 0.375.

像 3/8 这样的分数转换为 0.375，因为 3 ÷ 8 = 0.375。

Some fractions produce terminating decimals, such as 1/4 = 0.25, while others produce recurring decimals, such as 1/3 = 0.333… In KS3 you need to recognise both forms.

有些分数可以化为有限小数，例如 1/4 = 0.25；有些则化为循环小数，例如 1/3 = 0.333…… 在 KS3 中你需要识别这两种形式。

If you see a fraction like 7/20, multiply the denominator to 100 first: 7/20 = 35/100 = 0.35. This method is often quicker when the denominator is a factor of 100.

如果遇到 7/20 这样的分数，可以先把分母变成 100：7/20 = 35/100 = 0.35。当分母是 100 的因数时，这种方法通常更快。

4. Converting Decimals to Percentages | 小数转百分数

To convert a decimal to a percentage, multiply the decimal by 100 and add the % symbol. This is equivalent to moving the decimal point two places to the right.

将小数转换为百分数的方法是小数乘以 100 并加上百分号。这相当于把小数点向右移动两位。

decimal → percentage: × 100%

For example, 0.35 becomes 35%, and 0.075 becomes 7.5%.

例如，0.35 变成 35%，0.075 变成 7.5%。

If the decimal is greater than 1, the percentage is greater than 100%. For instance, 1.25 = 125%. This is useful for increase problems.

如果小数大于 1，百分数就大于 100%。例如，1.25 = 125%。这在增长问题中很有用。

A common mistake is to write 0.5 as 0.50% instead of 50%. Remember that 0.5 is half of one, and half of 100 is 50.

一个常见错误是把 0.5 写成 0.50% 而不是 50%。请记住，0.5 是一的一半，而 100 的一半是 50。

5. Converting Percentages to Fractions | 百分数转分数

To convert a percentage to a fraction, write the percentage over 100 and then simplify if possible.

将百分数转换为分数的方法是先把百分数写成以 100 为分母的分数，然后在可能的情况下进行化简。

percentage → fraction: percentage ÷ 100, then simplify

For example, 25% = 25/100 = 1/4, and 62.5% = 62.5/100 = 625/1000 = 5/8.

例如，25% = 25/100 = 1/4；62.5% = 62.5/100 = 625/1000 = 5/8。

If the percentage is a mixed number such as 33⅓%, write it as an improper fraction first: 33⅓% = (100/3)/100 = 100/300 = 1/3.

如果百分数是带分数，例如 33⅓%，先写成假分数：33⅓% = (100/3)/100 = 100/300 = 1/3。

KS3 questions often ask you to match fractions, decimals and percentages, so memorising common conversions such as 1/4 = 25% = 0.25 will save time.

KS3 题目经常要求匹配分数、小数和百分数，因此记住常见转换如 1/4 = 25% = 0.25 可以节省时间。

6. Comparing and Ordering | 比较与排序

To compare fractions, decimals and percentages, convert all values into the same form. Decimals are usually the easiest for ordering.

要比较分数、小数和百分数，先将所有数值转换成同一种形式。小数通常最容易排序。

<

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📚 Sharing in a Ratio: Cambridge KS3 Maths | 按比例分配：剑桥 KS3 数学

Sharing quantities in a given ratio is a core skill in the Cambridge KS3 mathematics curriculum. It appears regularly in written exercises, including the practice set on page 119 question 2, and is essential for later work on proportion, percentages and real-life financial problems.

按给定比例分配数量是剑桥 KS3 数学课程的核心技能。它经常出现在书面练习中，包括第119页第2题的练习，并且对后续的比例、百分数和实际财务问题都至关重要。

1. What is a Ratio? | 什么是比？

A ratio compares two or more quantities in the same unit. We write ratios using a colon, for example 3 : 2. This means for every 3 parts of the first quantity, there are 2 parts of the second quantity. The total number of equal parts is found by adding the numbers in the ratio.

比用来比较两个或两个以上相同单位的量。我们用冒号写比，例如 3 : 2。这表示第一种量每占 3 份，第二种量就占 2 份。总份数等于比中各项数字之和。

For the ratio 3 : 5, there are 3 + 5 = 8 equal parts in total. Understanding this idea is the foundation for all sharing problems.

对于比 3 : 5，总共有 3 + 5 = 8 个相等的份。理解这个概念是解决所有分配问题的基础。

• 3 : 5 means total parts = 8; 3 : 5 表示总份数 = 8
• A ratio has no units; 比没有单位
• The order of the ratio matters; 比的顺序很重要

2. Reading a Ratio Question | 阅读比例题

When a question says “Share £120 between A and B in the ratio 3 : 5”, it is asking you to split the whole amount into equal parts according to the ratio. Identify the total amount and the ratio terms carefully before you start.

当题目说“按 3 : 5 的比例在 A 和 B 之间分配 120 英镑”时，是要求你按照该比例将总量分成相等的份。开始之前请仔细识别总量和比中的各项。

Useful steps for reading the question:

阅读题目的有用步骤：

• Find the total amount to be shared; 找到要分配的总量
• Find the ratio and check its order; 找到比并检查其顺序
• Add the ratio terms to get total parts; 将比中各项相加得到总份数

3. The Bar Model Method | 条形模型法

A bar model is a visual way to represent sharing in a ratio. Draw one bar, divide it into equal sections matching the total parts, then label the known total. Each section represents one part.

条形模型是一种按比例分配的可视化方法。画一个条形，把它分成与总份数相等的若干段，然后标出已知总量。每一段代表一份。

For £120 shared in the ratio 3 : 5, draw a bar with 8 equal sections. The first 3 sections represent A, and the remaining 5 sections represent B. The whole bar equals £120.

对于按 3 : 5 分配 120 英镑，画一个分成 8 个等份的条形。前 3 段代表 A，后 5 段代表 B。整个条形等于 120 英镑。

Total parts = 3 + 5 = 8 | One part shown by one section | 总份数 = 3 + 5 = 8 | 一份用一段表示

4. The Unit-Method | 单位法

The unit-method uses three steps: find the total number of parts, divide the given amount by the total parts to find the value of one part, then multiply by each ratio term.

单位法使用三个步骤：求出总份数，用已知数量除以总份数得到一份的值，再分别乘以比中的各项。

This method works for money, lengths, masses, volumes and any other quantity that can be divided into equal parts.

该方法适用于金钱、长度、质量、体积以及任何可以分成相等份的量。

One part = Total amount ÷ Total parts | 一份 = 总量 ÷ 总份数

• Step 1: Find total parts; 步骤1：求总份数
• Step 2: Divide total amount by total parts; 步骤2：总量除以总份数
• Step 3: Multiply one part by each ratio term; 步骤3：一份乘以比中各项

5. Worked Example: Sharing Money | 示例：分钱问题

Work through this example: Share £120 between A and B in the ratio 3 : 5.

完成这个示例：按 3 : 5 的比例在 A 和 B 之间分配 120 英镑。

• Total parts = 3 + 5 = 8; 总份数 = 3 + 5 = 8
• One part = £120 ÷ 8 = £15; 一份 = 120 ÷ 8 = 15 英镑
• A = 3 × £15 = £45; A = 3 × 15 = 45 英镑
• B = 5 × £15 = £75; B = 5 × 15 = 75 英镑
• Check: £45 + £75 = £120; 检查：45 + 75 = 120 英镑

The answer is £45 for A and £75 for B. Always check that the amounts add back to the original total.

答案是 A 得 45 英镑，B 得 75 英镑。一定要检查各数量相加是否等于原总量。

6. Using Algebra to Share in a Ratio | 用代数按比例分配

You can also set up an equation. Let one part be x. Then the amounts are 3x and 5x, and the sum equals the total.

你也可以设方程。设一份为 x。那么数量就是 3x 和 5x，它们的和等于总量。

3x + 5x = 120 → 8x = 120 → x = 15

Then A = 3 × 15 = 45 and B = 5 × 15 = 75. This algebraic approach links ratio sharing to solving linear equations.

那么 A = 3 × 15 = 45，B = 5 × 15 = 75。这种代数方法把按比例分配与解一元一次方程联系起来。

7. Ratio and Fractions | 比与分数

A ratio can be converted into fractions of the whole. In the ratio 3 : 5, the first part receives 3/8 of the total and the second receives 5/8 of the total.

比可以转化为整体的分数。在 3 : 5 中，第一部分得到总量的 3/8，第二部分得到总量的 5/8。

A = (3/8) × £120 = £45 | B = (5/8) × £120 = £75

This is useful when a question asks for one share as a fraction of the whole, or when comparing two related ratio problems.

当题目要求求某一份占整体的几分之几，或者比较两个相关的比例问题时，这种方法很有用。

8. Common Mistakes | 常见错误

Students often forget to add the ratio terms before dividing. Another mistake is using the ratio term as the actual amount. Always calculate the value of one part first.

学生常常在除法之前忘记将比中各项相加。另一个错误是把比中的项直接当作实际数量。一定要先计算一份的值。

• Mistake: dividing by 3 instead of by total parts 8; 错误：除以 3 而不是总份数 8
• Mistake: leaving the answer as 3 : 5 without finding amounts; 错误：只写出 3 : 5 而没有求出实际数量
• Mistake: mixing units, such as pounds and pence; 错误：单位混用，例如英镑和便士混用
• Mistake: swapping the order of shares; 错误：调换份额顺序

Careful reading and a clear layout will help you avoid these errors.

认真读题和清晰的书写布局能帮助你避免这些错误。

9. Exam-Style Practice | 考试型练习

Try these quick questions to check your understanding:

尝试以下快速练习来检查理解：

• Share £200 in the ratio 3 : 7. 按 3 : 7 分配 200 英镑。
• Divide 360 ml in the ratio 1 : 5. 按 1 : 5 分配 360 毫升。
• The ratio of boys to girls in a class is 4 : 5. If there are 27 students in total, how many boys are there? 班级男女比例为 4 : 5。如果共有 27 名学生，男生有多少人？

For the last question, total parts = 4 + 5 = 9, one part = 27 ÷ 9 = 3, so boys = 4 × 3 = 12.

最后一题中，总份数 = 4 + 5 = 9，一份 = 27 ÷ 9 = 3，所以男生 = 4 × 3 = 12。

10. Key Takeaways | 核心要点

To share in a ratio, always find the total number of parts, divide to find one part, then multiply. Use a bar model if you need a visual check. Verify that the amounts add back to the original total.

按比例分配时，总是先求总份数，除以总量得到一份，再乘以各项。如果需要直观检查，可以用条形图。验证各数量相加等于原总量。

This skill is tested regularly at KS3 and forms the basis for harder proportion questions later in Cambridge IGCSE mathematics.

这门技能在 KS3 阶段经常考查，也为剑桥 IGCSE 数学后续更复杂的比例问题打下基础。

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📚 KS3 Cambridge Maths: Page 111 Q2 – Percentages to Fractions | KS3 剑桥数学：第111页第2题——百分数化分数

In this KS3 Cambridge Mathematics revision article, we focus on a typical Page 111 Question 2 task: converting percentages into fractions in their simplest form. This topic is at the heart of the Cambridge Key Stage 3 curriculum and builds essential number skills for ratio, proportion, probability and everyday problem solving.

在这篇 KS3 剑桥数学复习文章中，我们聚焦第 111 页第 2 题这一典型任务：把百分数化为最简分数。该主题是剑桥 Key Stage 3 课程的核心，为比、比例、概率和日常问题解决打下重要的数字技能基础。

1. Understanding the Page 111 Q2 Task | 理解第111页第2题任务

Question 2 usually presents a list of percentages such as 30%, 45%, 12.5% and 150%, and asks you to write each one as a fraction in its lowest terms.

第 2 题通常会列出一组百分数，如 30%、45%、12.5% 和 150%，要求你把每个百分数写成最简分数。

The instruction “in its simplest form” means you must reduce the fraction until the numerator and denominator have no common factor except 1.

“最简形式”意味着你必须约分，直到分子和分母除了 1 之外没有公因数。

You should show clear working, as Cambridge examiners award method marks for the conversion and simplification steps.

你应当展示清晰的步骤，因为剑桥考官会为转换和约分步骤给予方法分。

2. What Is a Percentage? | 什么是百分数？

The word “percentage” comes from the Latin “per centum”, meaning “out of 100”. Therefore, x% simply means x out of 100, or x/100.

“百分数（percentage）”一词来自拉丁语 “per centum”，意思是“每一百”。因此，x% 就是每一百中的 x，或者 x/100。

This definition gives us a direct rule: to change a percentage to a fraction, write the percentage number over 100 and then simplify.

这个定义给了我们一条直接规则：要把百分数化为分数，先把百分数的数值写在 100 的上方，然后进行约分。

For example, 25% = 25/100, which simplifies to 1/4.

例如，25% = 25/100，约分后得到 1/4。

3. Step-by-Step Conversion Method | 分步转换方法

Follow these three steps for every percentage in Question 2: first, write the percentage as a fraction with denominator 100; second, find the greatest common factor of the numerator and denominator; third, divide both by that factor.

对于第 2 题中的每个百分数，按以下三步操作：第一，把百分数写成分母为 100 的分数；第二，找出分子和分母的最大公因数；第三，用该公因数同时除分子和分母。

If the percentage has a decimal part, such as 12.5%, first eliminate the decimal by multiplying numerator and denominator by 10, 100 or 1000 as needed.

如果百分数含有小数部分，例如 12.5%，

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📚 Cambridge KS3 Mathematics: Fractions, Decimals and Percentages | 剑桥初中数学：分数、小数与百分比

In the Cambridge Lower Secondary Mathematics course, fractions, decimals and percentages are treated as connected ways of describing parts of a whole. The exercises in resource p108_2.pdf require you to move flexibly between these three forms, compare their sizes, and use them in calculations. This guide explains each skill with clear examples so that you can answer similar questions accurately.

在剑桥初中数学课程中，分数、小数和百分比被视为描述整体的一部分的三种相互关联的方式。资源 p108_2.pdf 中的练习要求你在三种形式之间灵活转换、比较大小并用于计算。本指南通过清晰的示例逐一讲解每项技能，帮助你准确解答类似问题。

1. The Meaning of Fractions | 分数的意义

A fraction is written as a numerator over a denominator. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts are being considered.

分数写作分子在上、分母在下。分母表示整体被平均分成多少份，分子表示你正在考虑其中的几份。

For example, the fraction 3/5 means 3 parts out of 5 equal parts. It can also be read as the division 3 ÷ 5, which gives the decimal 0.6.

例如，分数 3/5 表示 5 等份中的 3 份。它也可以读作除法 3 ÷ 5，结果是小数 0.6。

A proper fraction has a numerator smaller than its denominator, such as 2/7. An improper fraction has a numerator equal to or greater than its denominator, such as 9/4, and can be written as a mixed number 2¼.

真分数的分子小于分母，如 2/7。假分数的分子大于或等于分母，如 9/4，它可以写成带分数 2¼。

2. Equivalent Fractions and Simplest Form | 等值分数与最简形式

Equivalent fractions look different but represent exactly the same value. You

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📚 Cambridge KS3 Mathematics: Solving Linear Equations with Brackets and Unknowns on Both Sides | 剑桥KS3数学：解含括号及两边含未知数的一元一次方程

Linear equations are the foundation of algebra at Key Stage 3. In this article, you will learn how to solve equations that contain brackets, fractions, and unknown terms on both sides, using the balancing method.

一元一次方程是 KS3 代数的基础。本文将介绍如何运用天平法解含有括号、分数以及两边都含未知数的方程。

1. What Is a Linear Equation? | 什么是一元一次方程？

A linear equation has only one unknown, and the highest power of the unknown is 1. For example, 3x + 2 = 14 and 2(x − 5) = 3x + 1 are linear equations.

一元一次方程只有一个未知数，且未知数的最高次数为 1。例如，3x + 2 = 14 和 2(x − 5) = 3x + 1 都是一元一次方程。

The solution is the value of x that makes the equation true.

方程的解就是使等式成立的 x 值。

2. The Balancing Method | 天平法

Think of an equation as a balance. Whatever you do to one side, you must also do to the other side.

把方程想象成天平。你对一边做的运算，另一边也必须做同样的运算。

Key operations:

关键运算：

• Add or subtract the same amount from both sides.
• Multiply or divide both sides by the same non-zero number.

两边同时加上或减去同一个数；两边同时乘或除以同一个非零数。

Example: Solve x + 5 = 12.

例题：解 x + 5 = 12。

x = 12 − 5 = 7

2x = 14, so x = 7

Use inverse operations in reverse order to isolate x.

按相反顺序使用逆运算，把 x 单独留在等号一边。

3. Expanding Brackets First | 先去括号

If an equation contains brackets, expand them before collecting like terms. Use the distributive law: a(b + c) = ab + ac.

如果方程含有括号，先展开再合并同类项。使用分配律：a(b + c) = ab + ac。

Example: Solve 3(x + 2) = 21.

例题：解 3(x + 2) = 21。

3x + 6 = 21

3x = 15

x = 5

Be careful with negative signs: −2(x − 3) = −2x + 6.

注意负号：−2(x − 3) = −2x + 6。

4. Unknowns on Both Sides | 两边含未知数

When x appears on both sides, collect the x terms on one side and the numbers on the other.

当 x 出现在等式两边时，把含 x 的项移到一边，数字移到另一边。

Example: Solve 5x − 2 = 3x + 8.

例题：解 5x − 2 = 3x + 8。

5x − 3x = 8 + 2

2x = 10

x = 5

Moving a term across the equals sign changes its sign.

项移过等号时要变号。

5. Solving Equations with Fractions | 解含分数的方程

Multiply every term by the lowest common denominator to clear fractions.

将每一项乘以最小公分母，消去分母。

Example: Solve (x + 4) ÷ 3 = 6.

例题：解 (x + 4) ÷ 3 = 6。

x + 4 = 18

x = 14

If the equation is x/2 + 3 = 7, multiply all terms by 2 first: x + 6 = 14, so x = 8.

如果方程是 x/2 + 3 = 7，先每一项乘 2：x + 6 = 14，所以 x = 8。

6. Checking Your Answer by Substitution | 代入检验答案

Always substitute your answer back into the original equation to check.

一定要把答案代回原方程检验。

For 5x − 2 = 3x + 8, if x = 5, left side = 5 × 5 − 2 = 23 and right side = 3 × 5 + 8 = 23, so both sides match.

对于 5x − 2 = 3x + 8，若 x = 5，左边 = 5 × 5 − 2 = 23，右边 = 3 × 5 + 8 = 23，两边相等。

7. Common Mistakes to Avoid | 常见错误及避免方法

• Forgetting to multiply every term when clearing fractions.
• Expanding brackets with a negative multiplier incorrectly.
• Changing the sign incorrectly when moving terms across the equals sign.

常见错误包括：去分母时漏乘某项；括号外是负数时展开出错；移项时忘记变号。

Take one step at a time and write each stage clearly.

一步一步来，每一步都写清楚。

8. Word Problems Leading to Equations | 由文字题建立方程

Translate the words into an equation by letting x represent the unknown quantity.

设未知量为 x，把文字翻译成方程。

Example: Three times a number, increased by 4, equals 19. Let the number be x.

例题：一个数的 3 倍再加 4 等于 19。设这个数为 x。

3x + 4 = 19

3x = 15

x = 5

Check: 3 × 5 + 4 = 19.

检验：3 × 5 + 4 = 19。

9. Practice Questions | 练习题

Try these, then check your answers.

尝试以下题目，然后核对答案。

10. Summary and Key Takeaways | 小结与要点

Linear equations are solved by keeping the equation balanced, expanding brackets, clearing fractions, and collecting like terms.

解一元一次方程的关键是保持等式平衡、去括号、去分母并合并同类项。

• Use inverse operations in order.
• Always check your solution in the original equation.

按顺序使用逆运算；一定要把解代回原方程检验。

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📚 Solving Linear Equations: KS3 Cambridge Maths | 解一元一次方程：KS3剑桥数学

Solving linear equations is one of the most important skills in the KS3 Cambridge mathematics curriculum. It builds on number operations and prepares students for more advanced algebra, graphs and problem-solving. This article reviews key methods, common question styles and exam tips, closely matched to tasks such as those on p92_2.pdf.

解一元一次方程是 KS3 剑桥数学课程中最重要的技能之一。它以数的运算为基础，为学生进一步学习代数、图像和问题解决打下基础。本文回顾核心方法、常见题型和考试技巧，紧扣 p92_2.pdf 等典型练习。

1. What Is a Linear Equation? | 什么是线性方程？

A linear equation is an algebraic statement in which two expressions are equal, and the unknown variable appears only to the power of 1. For example, 3x + 2 = 11 is a linear equation because the variable x is not squared or cubed.

线性方程是一个代数等式，其中两个表达式相等，未知数的次数仅为 1。例如 3x + 2 = 11 是线性方程，因为变量 x 没有被平方或立方。

In KS3, the main goal is to find the value of the unknown that makes the equation true. This value is called the solution or root.

在 KS3 阶段，主要目标是求出使等式成立的未知数的值。这个值称为方程的解或根。

2. The Balance Method | 天平法（等式平衡法）

The balance method treats an equation like a set of scales. Whatever you do to one side, you must do to the other side to keep the equation balanced.

天平法将方程视为一架天平。对等式一边施加任何运算，必须对另一边做同样运算，才能保持等式平衡。

For example, if you add 3 to the left side, you must also add 3 to the right side. If you divide the left side by 5, you must divide the right side by 5.

例如，如果在左边加 3，右边也必须加 3；如果将左边除以 5，右边也必须除以 5。

x + 3 = 10 → x + 3 – 3 = 10 – 3 → x = 7

3. Solving One-Step Equations | 解一步方程

A one-step equation needs only one inverse operation to find the solution. The four basic types are addition, subtraction, multiplication and division.

一步方程只需一步逆运算即可求出解。四种基本类型是加法、减法、乘法和除法。

Remember to use the inverse operation: addition undoes subtraction, multiplication undoes division, and vice versa.

记住使用逆运算：加法抵消减法，乘法抵消除法，反之亦然。

4. Solving Two-Step Equations | 解两步方程

A two-step equation involves two operations. The standard method is to undo the addition or subtraction first, then undo the multiplication or division.

两步方程包含两种运算。标准方法是先消去加或减，再消去乘或除。

2x + 5 = 17 → 2x = 12 → x = 6

Sometimes the order is reversed if the variable is divided before a term is added: x/3 – 4 = 8 → x/3 = 12 → x = 36.

有时顺序相反，例如变量先除以 3 再减去 4：x/3 – 4 = 8 → x/3 = 12 → x = 36。

5. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law, then simplify and solve as usual.

当方程含有括号时，先用分配律展开括号，然后化简并照常求解。

3(x + 2) = 21 → 3x + 6 = 21 → 3x = 15 → x = 5

Always multiply every term inside the bracket by the number outside. A common sign error is to forget the negative sign, as in -2(x – 3) = -2x + 6.

必须用括号外的数乘以括号内的每一项。常见符号错误是忘记负号，例如 -2(x – 3) = -2x + 6。

6. Equations with Variables on Both Sides | 含两边变量的方程

If the unknown appears on both sides of the equation, first collect the variable terms on one side and the number terms on the other side.

如果未知数出现在等式两边，先将含变量的项集中到一边，将常数项集中到另一边。

5x + 3 = 2x + 12 → 3x = 9 → x = 3

Aim to keep the variable coefficient positive to reduce sign errors. In the above case, subtracting 2x from both sides is easier than subtracting 5x.

尽量保持变量系数为正，以减少符号错误。在上例中，两边减去 2x 比减去 5x 更容易。

7. Equations with Fractions | 含分数的方程

To solve equations involving fractions, multiply every term on both sides by the least common denominator (LCD) to clear the fractions.

解含分数的方程时，先将等式两边每一项都乘最小公分母（LCD），以消去分母。

x/4 + 1 = 3 → x/4 = 2 → x = 8

A more complex example: (2x)/3 = 4 → multiply both sides by 3: 2x = 12 → x = 6.

更复杂的例子：(2x)/3 = 4 → 两边乘以 3：2x = 12 → x = 6。

8. Translating Word Problems into Equations | 将文字题转化为方程

Many KS3 problems ask you to form and solve an equation from a real-life situation. Use a letter for the unknown and build the equation step by step.

许多 KS3 题目要求根据实际情境建立并解方程。用字母表示未知数，并逐步建立方程。

Example: I think of a number, multiply it by 4, then subtract 7. The answer is 25. Let n be the number: 4n – 7 = 25, so n = 8.

例如：我想一个数，将它乘以 4，再减去 7，结果是 25。设这个数为 n：4n – 7 = 25，因此 n = 8。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

The most common errors include doing an operation to only one side, forgetting to change signs when expanding brackets, and mixing up the order of inverse operations.

最常见的错误包括只对等式一边进行运算、展开括号时忘记变号、混淆逆运算的顺序。

Always use a written check by substituting your solution back into the original equation. This takes a few seconds and can catch many mistakes.

始终通过代入原方程进行书面检验。这只需几秒钟，却能发现许多错误。

10. Practice Questions and Checkpoint Tips | 练习题与 Checkpoint 应考提示

Try these questions to consolidate your skills: 7x – 2 = 26, 4(x – 3) = 32, 3x + 5 = 2x + 17, and x/5 + 2 = 7.

尝试以下题目以巩固技能：7x – 2 = 26，4(x – 3) = 32，3x + 5 = 2x + 17，x/5 + 2 = 7。

In the Cambridge Checkpoint test, show every step clearly. Even if your final answer is wrong, method marks can be awarded for correct balance and inverse operations.

在剑桥 Checkpoint 测试中，要清楚地写出每一步。即使最终答案错误，正确的等式平衡和逆运算步骤也能获得方法分。

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📚 Fractions, Decimals and Percentages for KS3 Cambridge Maths | KS3剑桥数学：分数、小数与百分比

In the Cambridge KS3 Mathematics curriculum, fractions, decimals and percentages are treated as three connected ways of describing the same idea: parts of a whole. This topic builds fluency in converting between forms, comparing values, performing calculations and applying these skills to real-world problems. A strong grasp of these number forms is essential for later work in ratio, proportion, algebra and probability.

在剑桥 KS3 数学课程中，分数、小数和百分比被视为同一概念的三种相互关联的表达方式：整体的部分。本专题帮助学生熟练地在三种形式之间转换、比较数值、进行计算，并将这些技能应用于实际问题。扎实掌握这些数的形式对于后续学习比、比例、代数和概率至关重要。

1. Understanding Fractions | 理解分数

A fraction is written as a numerator over a denominator. The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts are being considered. For example, in the fraction 3/4, the whole is divided into 4 equal parts and 3 parts are selected.

分数写作分子在上、分母在下。分母表示整体被分成多少等份，分子表示取其中的几份。例如，在分数 3/4 中，整体被分成 4 等份，取其中的 3 份。

Fractions can represent parts of a shape, parts of a set, or a point on a number line. A proper fraction has a numerator smaller than the denominator, such as 5/8. An improper fraction has a numerator greater than or equal to the denominator, such as 9/4. A mixed number combines a whole number with a proper fraction, such as 2 1/3.

分数可以表示图形的一部分、集合的一部分或数轴上的一个点。真分数的分子小于分母，例如 5/8。假分数的分子大于或等于分母，例如 9/4。带分数由一个整数和一个真分数组成，例如 2 1/3。

Proper: 3/5    Improper: 7/4    Mixed: 1 3/4

2. Equivalent Fractions and Simplifying | 等值分数与化简

Equivalent fractions look different but have the same value. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero whole number. For example, 1/2 = 2/4 = 3/6 = 4/8.

等值分数看似不同但数值相同。你可以将分子和分母同时乘以或除以同一个非零整数来得到等值分数。例如，1/2 = 2/4 = 3/6 = 4/8。

Simplifying a fraction means dividing the numerator and denominator by their highest common factor until no common factor remains except 1. The fraction 12/18 simplifies to 2/3 because the highest common factor of 12 and 18 is 6.

化简分数是指用分子和分母的最大公因数同时除以它们，直到除 1 以外没有其他公因数为止。分数 12/18 化简为 2/3，因为 12 和 18 的最大公因数是 6。

12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3

3. Decimals and Place Value | 小数与位值

A decimal is another way to write a fraction whose denominator is a power of 10. The digits after the decimal point represent tenths, hundredths, thousandths and so on. For example, 0.35 means 3 tenths plus 5 hundredths, which is equal to 35/100.

小数是分母为 10 的幂的分数的另一种写法。小数点后的数字分别表示十分位、百分位、千分位等。例如，0.35 表示 3 个十分之一加 5 个百分之一，等于 35/100。

Place value is important when comparing and rounding decimals. The decimal 0.7 is greater than 0.699 because 7 tenths is larger than 6 tenths plus 9 hundredths plus 9 thousandths. Rounding 3.276 to two decimal places gives 3.28 because the digit in the thousandths place is 6, so the hundredths digit is rounded up.

位值在比较和四舍五入小数时非常重要。小数 0.7 大于 0.699，因为 7 个十分之一大于 6 个十分之一加 9 个百分之一加 9 个千分之一。将 3.276 四舍五入到两位小数得到 3.28，因为千分位上的数字是 6，所以百分位数字向上进位。

4. Converting Between Fractions and Decimals | 分数与小数互化

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375. Some fractions produce terminating decimals, while others produce recurring decimals such as 1/3 = 0.333… which can be written with a dot above the repeating digit.

要将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。一些分数产生有限小数，而另一些产生循环小数，例如 1/3 = 0.333…，可以在重复数字上方加点表示。

To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100 or 1000 depending on the number of decimal places, then simplify. For example, 0.45 = 45/100 = 9/20.

要将有限小数转换为分数，根据小数位数将小数写成分母为 10、100 或 1000 的分数，然后化简。例如，0.45 = 45/100 = 9/20。

3/8 = 0.375    and    0.45 = 9/20

5. Percentages and Their Meaning | 百分比及其意义

A percentage is a fraction with denominator 100. The symbol % means ‘out of 100’. For example, 47% means 47 out of 100, which is equal to 47/100 or 0.47. Percentages are useful for comparing proportions because they give a common scale.

百分比是分母为 100 的分数。符号 % 表示“每 100 份中的份数”。例如，47% 表示 100 份中的 47 份，等于 47/100 或 0.47。百分比对于比较比例非常有用，因为它们提供了统一的尺度。

Percentages can be greater than 100 when a quantity increases beyond its original value. For example, 150% means 150 out of 100, or 1.5 times the original amount. Percentages less than 1 are also possible, such as 0.5% which equals 0.005.

当某个量超过原值时，百分比可以大于 100。例如，150% 表示 100 份中的 150 份，即原值的 1.5 倍。小于 1 的百分比也存在，例如 0.5% 等于 0.005。

6. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比的互化

The three forms are interchangeable. To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, 0.85 = 85%. To convert a percentage to a decimal, divide by 100. For example, 32% = 0.32.

这三种形式可以相互转换。要将小数转换为百分比，乘以 100 并加上 % 符号。例如，0.85 = 85%。要将百分比转换为小数，除以 100。例如，32% = 0.32。

To convert a fraction to a percentage, first divide the numerator by the denominator to obtain a decimal, then multiply by 100. For example, 3/5 = 0.6 = 60%. To convert a percentage to a fraction, write it as a fraction over 100 and simplify. For example, 65% = 65/100 = 13/20.

要将分数转换为百分比，先用分子除以分母得到小数，再乘以 100。例如，3/5 = 0.6 = 60%。要将百分比转换为分数，将其写成分母为 100 的分数并化简。例如，65% = 65/100 = 13/20。

7. Comparing and Ordering | 比较和排序

To compare fractions, decimals and percentages, convert all values to the same form. Often decimals are the easiest to compare because you can compare place values directly. For example, to order 2/5, 0.44 and 39%, convert them all to decimals: 0.4, 0.44 and 0.39. In ascending order they are 0.39, 0.4, 0.44, which is 39%, 2/5, 0.44.

要比较分数、小数和百分比，将所有数值转换为同一种形式。通常小数最容易比较，因为可以直接比较位值。例如，要对 2/5、0.44 和 39% 排序，将它们都转换为小数：0.4、0.44 和 0.39。按升序排列为 0.39、0.4、0.44，即 39%、2/5、0.44。

When comparing fractions with different denominators, find a common denominator, convert each fraction and compare the numerators. For example, to compare 2/3 and 5/8, use 24 as the common denominator: 2/3 = 16/24 and 5/8 = 15/24, so 2/3 is greater than 5/8.

比较不同分母的分数时，先找到公分母，转换每个分数，然后比较分子。例如，比较 2/3 和 5/8，用 24 作为公分母：2/3 = 16/24，5/8 = 15/24，所以 2/3 大于 5/8。

8. Adding and Subtracting Fractions | 分数的加减法

To add or subtract fractions with the same denominator, simply add or subtract the numerators and keep the denominator. For example, 2/9 + 4/9 = 6/9 = 2/3, and 7/10 – 3/10 = 4/10 = 2/5.

要将同分母分数相加或相减，只需将分子相加或相减，分母保持不变。例如，2/9 + 4/9 = 6/9 = 2/3，7/10 – 3/10 = 4/10 = 2/5。

If the denominators are different, find the lowest common denominator first, rewrite each fraction as an equivalent fraction, and then add or subtract the numerators. For example, 1/4 + 2/5 = 5/20 + 8/20 = 13/20. For mixed numbers, add or subtract the whole number parts and fraction parts separately, borrowing if necessary.

如果分母不同，先找出最小公分母，将每个分数改写为等值分数，然后再进行分子加减。例如，1/4 + 2/5 = 5/20 + 8/20 = 13/20。对于带分数，分别加减整数部分和分数部分，必要时进行借位。

1/4 + 2/5 = 5/20 + 8/20 = 13/20

9. Multiplying and Dividing Fractions | 分数的乘除法

To multiply fractions, multiply the numerators together and multiply the denominators together, then simplify if possible. For example, 3/4 × 2/5 = 6/20 = 3/10. Mixed numbers should first be converted to improper fractions before multiplying.

要将分数相乘，将分子相乘、分母相乘，然后尽可能化简。例如，3/4 × 2/5 = 6/20 = 3/10。带分数在相乘前应先转换为假分数。

To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.

要除以一个分数，乘以它的倒数。分数的倒数通过交换其分子和分母得到。例如，5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4。

5/6 ÷ 2/3 = 5/6 × 3/2 = 5/4 = 1 1/4

10. Percentage of Amounts and Problems | 百分数应用与问题解决

To find a percentage of an amount, convert the percentage to a decimal and multiply by the amount. For example, to find 15% of 240, calculate 0.15 × 240 = 36. Alternatively, find 10% first and then adjust: 10% of 240 is 24, 5% is 12, so 15% is 24 + 12 = 36.

要求一个数的百分之几，将百分比转换为小数再乘以这个数。例如，求 240 的 15%，计算 0.15 × 240 = 36。也可以先求 10% 再调整：240 的 10% 是 24，5% 是 12，所以 15% 是 24 + 12 = 36。

Percentage increase and decrease problems are common. To increase 80 by 25%, find 25% of 80 which is 20, then add it to the original amount: 80 + 20 = 100. Alternatively, multiply by 1.25. To decrease 60 by 30%, find 30% of 60 which is 18, then subtract: 60 – 18 = 42, or multiply by 0.70.

百分比增减问题很常见。将 80 增加 25%，先求 80 的 25% 是 20，然后加到原数上：80 + 20 = 100。也可以乘以 1.25。将 60 减少 30%，先求 60 的 30% 是 18，然后相减：60 – 18 = 42，或者乘以 0.70。

• Find 20% of 150: 0.20 × 150 = 30
• Increase 40 by 15%: 40 × 1.15 = 46
• Decrease 72 by 25%: 72 × 0.75 = 54

• 求 150 的 20%：0.20 × 150 = 30
• 将 40 增加 15%：40 × 1.15 = 46
• 将 72 减少 25%：72 × 0.75 = 54

In word problems, identify the original amount, the percentage change and whether the change is an increase or decrease. Then choose the correct multiplication or division operation. Drawing a bar model can help visualise the relationship between the part and the whole.

在应用题中，要确定原量、百分比变化以及变化是增加还是减少。然后选择正确的乘法或除法运算。画条形模型有助于直观表示部分与整体的关系。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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