📚 Cambridge KS3 Mathematics: Page 240 Question 1 – Fractions, Decimals and Percentages | 剑桥KS3数学：第240页第1题——分数、小数与百分数综合运用
Page 240 Question 1 brings together the three most common ways of describing parts of a whole: fractions, decimals and percentages. The question requires you to move confidently between these forms and then apply percentage thinking to a real school context.
第240页第1题综合了描述整体中部分的三种常用方式：分数、小数和百分数。该题要求你能够在这些形式之间灵活转换，然后将百分数思维应用到一个真实的学校情境中。
We will work through the problem step by step, highlight key conversions, and examine the common mistakes that cost marks in Cambridge KS3 assessments.
我们将逐步解答该题，强调关键转换，并分析在剑桥KS3测评中容易失分的常见错误。
1. Understanding the Problem Statement | 理解题意
Question 1 states: In a school of 240 students, 3/5 of the students are in Key Stage 3. Of these Key Stage 3 students, 45% are girls. Work out the number of girls in Key Stage 3.
第1题内容为：一所学校有240名学生，其中3/5的学生在第三学段。在这些第三学段学生中，45%是女生。计算第三学段女生的人数。
The word 'of' is very important here. In mathematics, 'of' often means multiplication. You first need to find 3/5 of 240, and then find 45% of that result.
这里的“的”字非常重要。在数学中，“的”通常表示乘法。你需要先求出240的3/5，然后再求出这个结果的45%。
Many students rush straight to 45% of 240. That is incorrect because the 45% applies only to the Key Stage 3 group, not to the whole school.
许多学生直接计算240的45%。这是不正确的，因为45%只适用于第三学段的学生，而不是全校学生。
2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
To solve this question efficiently, you need to be able to switch between fractions, decimals and percentages. The three forms are equivalent ways of writing the same proportion.
要高效地解答此题，你需要能够在分数、小数和百分数之间进行转换。这三种形式是表示同一比例的不同写法。
A fraction can be turned into a decimal by dividing the numerator by the denominator. A decimal can be turned into a percentage by multiplying by 100.
分数可以通过分子除以分母转化为小数。小数可以通过乘以100转化为百分数。
For example, 3/5 means 3 ÷ 5, which gives 0.6. Then 0.6 × 100 gives 60%. So 3/5, 0.6 and 60% all represent the same proportion.
例如，3/5表示3 ÷ 5，得到0.6。然后0.6 × 100得到60%。因此3/5、0.6和60%都表示同一个比例。
You should practise these conversions until they become automatic, because mixed-format questions are very common in Cambridge KS3 papers.
你应当练习这些转换直到熟练自如，因为混合格式的题目在剑桥KS3试卷中非常常见。
3. Key Equivalences to Memorise | 必须记住的常见等价关系
Some fraction-decimal-percentage equivalences appear so often that memorising them will save you valuable time in the exam.
一些分数、小数和百分数的等价关系出现得如此频繁，记住它们可以为你节省宝贵的考试时间。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33.3% |
The table above gives you a quick reference. In particular, 3/5 = 0.6 = 60% is central to this page 240 question.
上表为你提供了一个快速参考。特别是3/5 = 0.6 = 60%是这道第240页题目的核心。
4. Worked Example Part A: Fraction of an Amount | 示例A：求一个数量的几分之几
The first calculation is to find 3/5 of the total number of students. To find a fraction of an amount, divide by the denominator and multiply by the numerator.
第一个计算是求出学生总数中的3/5。要求一个数量的几分之几，先除以分母，再乘以分子。
Here the total is 240, so we calculate 3/5 of 240. You can do this in two stages: first divide 240 by 5, then multiply the result by 3.
这里总数是240，因此我们计算240的3/5。你可以分两步完成：先将240除以5，再将结果乘以3。
240 ÷ 5 = 48
Then multiply by the numerator:
然后乘以分子：
48 × 3 = 144
So there are 144 students in Key Stage 3.
因此第三学段共有144名学生。
give the same answer, but the divisibility of 240 by 5 makes the first method very clean.
你也可以将3/5转换为0.6并计算0.6 × 240。两种方法得到相同答案，但240能被5整除，因此第一种方法非常简便。
5. Worked Example Part B: Percentage of an Amount | 示例B：求一个数量的百分数
Now we need 45% of the 144 Key Stage 3 students. Percent means 'out of 100', so 45% means 45 out of every 100.
现在我们需要求出144名第三学段学生中的45%。百分数表示“每一百中的多少”，因此45%表示每一百中有45。
To find 45% of an amount, you can multiply by 45 and divide by 100. Equivalently, you can use the decimal form 0.45.
要求一个数量的45%，你可以乘以45再除以100。等价地，你也可以使用小数形式0.45。
Using the fraction method:
使用分数方法：
144 × 45 ÷ 100 = 6480 ÷ 100 = 64.8
Using the decimal method:
使用小数方法：
0.45 × 144 = 64.8
The answer is 64.8, but this represents a number of people. You cannot have 0.8 of a person, so the likely intended answer is 65 when rounded to the nearest whole number, or the question may expect an exact value if the numbers were chosen differently.
答案是64.8，但这里表示的是人数。人不能有0.8个，因此答案很可能是四舍五入到最接近的整数65，或者如果题目原本数据设计不同，也可能要求精确值。
In many real-life contexts, we interpret such a result as 'about 65 girls'. Always check whether the question asks for an exact number or a sensible rounded answer.
在许多实际情境中，我们会把这一结果解释为“约65名女生”。一定要检查题目是要求精确数值还是合理的取整答案。
6. Full Solution Summary | 完整解答总结
Let us bring the two stages together in a clear, exam-style layout. This is the kind of working that earns full marks in Cambridge KS3 assessments.
让我们将两个阶段以一种清晰、符合考试要求的格式整合起来。这正是剑桥KS3测评中能够获得满分的过程展示。
Step 1: Find the number of Key Stage 3 students.
步骤1：求第三学段学生人数。
3/5 × 240 = 144
Step 2: Find 45% of this group.
步骤2：求这一群体中的45%。
45% × 144 = 0.45 × 144 = 64.8
Therefore, the number of girls in Key Stage 3 is approximately 65.
因此，第三学段的女生人数约为65人。
If the question requires an exact whole number, the data would normally be chosen so that no rounding is needed. In this case, the working shows the correct mathematical process.
如果题目要求精确的整数，数据通常会设计成不需要取整。在本题中，解题过程展示了正确的数学方法。
7. Comparing Quantities Using a Common Format | 用统一格式比较数量
Many Cambridge KS3 questions ask you to compare two or more quantities given in different forms. The safest strategy is to convert everything into the same format.
许多剑桥KS3题目要求你比较以不同形式给出的两个或多个数量。最稳妥的策略是将所有数量转换成同一种格式。
For instance, if you are told that 3/5 of students walk to school and 58% travel by bus, which group is larger? Converting 3/5 to 60% immediately shows that the walking group is larger.
例如，如果题目告诉你3/5的学生步行上学，58%的学生乘公交上学，哪一组更多？将3/5转换为60%后立刻可以判断步行组更多。
Similarly, you can compare 0.45 and 2/5 by converting 2/5 to 0.4. Since 0.45 is greater than 0.4, 45% is greater than 2/5.
类似地，你可以通过将2/5转换为0.4来比较0.45和2/5。因为0.45大于0.4，所以45%大于2/5。
Choosing percentages is often the clearest method because most people find percentages easy to interpret in everyday situations.
选择百分数通常是最清晰的方法，因为大多数人在日常生活中更容易理解百分数。
8. Reverse Percentage Thinking | 逆向百分数思维
Sometimes you know the percentage and the result, but you need to find the original amount. This is called a reverse percentage problem.
有时你已知百分数和结果，但需要求出原来的数量。这被称为逆向百分数问题。
For example, if 20% of a number is 30, then 1% of the number is 30 ÷ 20 = 1.5, and 100% is 1.5 × 100 = 150.
例如，如果某个数的20%是30，那么这个数的1%是30 ÷ 20 = 1.5，而100%就是1.5 × 100 = 150。
In the page 240 question, we did not need reverse percentage thinking, but it is a natural extension that Cambridge examiners often include nearby in the same exercise.
在第240页的题目中，我们并不需要逆向百分数思维，但这是剑桥出题者常在同一练习附近加入的自然延伸内容。
Always ask yourself: am I finding the part from the whole, or the whole from the part? This tells you whether to multiply or divide.
始终问自己：我是从整体求部分，还是从部分求整体？这会告诉你该用乘法还是除法。
9. Common Errors and How to Avoid Them | 常见错误与避免方法
One very common error is applying the percentage to the wrong base. In this question, 45% must be taken from 144, not from 240.
一个非常常见的错误是把百分数应用于错误的基数。在本题中，45%必须从144中求取，而不是从240中求取。
Another common error is confusing 3/5 with 3.5 or with 35%. Remember that 3/5 is 60%, not 35%, because the denominator 5 means the whole is divided into five equal parts.
另一个常见错误是将3/5与3.5或35%混淆。记住3/5是60%，而不是35%，因为分母5表示整体被分成五等份。
Rounding too early can also cause problems. In multi-step calculations, keep the exact value until the final step, then round only if the question asks for it.
过早取整也会导致问题。在多步计算中，保持精确值直到最后一步，然后仅在题目要求时取整。
Finally, always include units or a short word statement in your final answer, such as '65 girls'. This makes your working clear and earns communication marks.
最后，始终在最终答案中包含单位或简短的文字说明，例如“65名女生”。这会使你的解题过程清晰，并赢得表达分。
10. Practice Question Walkthrough | 练习题详解
Try this extension question: In a school of 300 students, 2/5 are in Key Stage 3. Of these, 30% are boys. How many boys are in Key Stage 3?
试做这道拓展题：一所学校有300名学生，其中2/5在第三学段。这些学生中30%是男生。第三学段有多少名男生？
First find 2/5 of 300: divide 300 by 5 to get 60, then multiply by 2 to get 120. So there are 120 Key Stage 3 students.
首先求300的2/5：将300除以5得到60，再乘以2得到120。因此第三学段有120名学生。
Next find 30% of 120: 0.30 × 120 = 36. So there are 36 boys in Key Stage 3.
接下来求120的30%：0.30 × 120 = 36。因此第三学段有36名男生。
Notice that the two-step structure is identical to the page 240 question. Once you master this pattern, a whole family of problems becomes straightforward.
注意这一两步骤结构与第240页题目完全相同。一旦你掌握了这一模式，一整类问题都会变得简单直接。
11. Extension: Linking to Ratio and Proportion | 拓展：联系比和比例
Fractions, decimals and percentages are closely connected to ratio. For example, 3/5 means a ratio of 3 to 2 when comparing the part to the remaining part.
分数、小数和百分数与比密切相关。例如，3/5在比较部分与剩余部分时意味着3比2。
If 3/5 of students are in Key Stage 3, then 2/5 are not. The ratio of Key Stage 3 students to other students is 3:2.
如果3/5的学生在第三学段，那么2/5的学生不在。第三学段学生与其他学生的比是3:2。
Likewise, 45% can be written as 45/100, which simplifies to 9/20. So the ratio of girls to the whole Key Stage 3 group is 9:20.
同样，45%可以写成45/100，化简为9/20。因此女生与整个第三学段群体的比是9:20。
Seeing these connections helps you move flexibly between topic areas, which is a key skill in the Cambridge Lower Secondary Mathematics curriculum.
看到这些联系有助于你在不同主题之间灵活转换，这是剑桥初中数学课程中的一项关键能力。
12. Check Your Understanding | 检查理解
Before you finish, test yourself with these quick questions. Convert each fraction to a decimal and a percentage: 3/5, 1/4, 7/10, 2/3.
在结束之前，用这些快速问题测试自己。将以下分数分别转换为小数和百分数：3/5、1/4、7/10、2/3。
Then find 15% of 80, 40% of 250, and 5% of 960. Check that your answers are 12, 100 and 48.
然后计算80的15%、250的40%和960的5%。请核对自己的答案是否为12、100和48。
Finally, write down two different ways to find 3/5 of 240. If you can do all of this confidently, you are ready for any similar question in the Cambridge KS3 assessment.
最后，写出两种不同的方法来求240的3/5。如果你能自信地完成所有这些，你就为剑桥KS3测评中的任何类似题目做好了准备。
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