📚 Cambridge KS3 Mathematics: Fractions, Decimals and Percentages | 剑桥初中数学：分数、小数与百分比
hole. Practice material such as resource p108_2.pdf asks you to move fluently between these three forms, compare their sizes, and use them in calculations. This guide explains each skill with clear worked 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。
| Type 类型 | Example 例子 | Key feature 特征 |
| Proper fraction 真分数 | 2/7 | numerator < denominator 分子小于分母 |
| Improper fraction 假分数 | 9/4 | numerator ≥ denominator 分子大于或等于分母 |
| Mixed number 带分数 | 2¼ | whole number + fraction 整数加分数的形式 |
A fraction can also act as an operator on a quantity. To find 3/5 of 20, divide 20 by 5 to get 4, then multiply by 3 to get 12. This "divide by the denominator, multiply by the numerator" routine is used constantly in word problems.
分数也可以作为一个作用于数量的运算符。求 20 的 3/5，先用 20 除以 5 得 4，再乘以 3 得 12。这种"先除以分母、再乘以分子"的方法在应用题中反复出现。
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 can make an equivalent fraction by multiplying both the numerator and the denominator by the same non-zero number, or by dividing both by a common factor.
等值分数看起来不同，但表示完全相同的值。把分子和分母同时乘以同一个非零数，或同时除以一个公因数，就得到等值分数。
Multiplying or dividing the top and bottom together keeps the ratio unchanged, in the same way that 1 ÷ 2 and 50 ÷ 100 both give 0.5.
例如，1/2 = 2/4 = 5/10 = 50/100。上下同时乘除，比值保持不变，就像 1 ÷ 2 和 50 ÷ 100 都得 0.5 一样。
A fraction is in its simplest form when no number greater than 1 divides both the numerator and the denominator. To simplify, divide both parts by their highest common factor (HCF).
当分子与分母没有大于 1 的公因数时，分数就处于最简形式。化简时，用最大公因数（HCF）同时除分子和分母。
ify 18/24. The HCF of 18 and 24 is 6, so 18 ÷ 6 = 3 and 24 ÷ 6 = 4. Therefore 18/24 = 3/4.
示例：化简 18/24。18 与 24 的最大公因数是 6，所以 18 ÷ 6 = 3，24 ÷ 6 = 4。因此 18/24 = 3/4。
18/24 ÷ 6/6 = 3/4
iven in simplest form unless the question says otherwise, because mark schemes often deduct a mark for an unsimplified fraction.
也可以分步约分：18/24 = 9/12 = 3/4。除非题目另有要求，答案必须写成最简形式，因为评分标准常会为未化简的分数扣分。
3. Converting Fractions to Decimals | 分数化为小数
A fraction is a division waiting to happen. To change a fraction into a decimal, divide the numerator by the denominator, using short division or a written method.
分数本质上是一个尚未完成的除法。要把分数变成小数，只需用分子除以分母，可以笔算或使用短除法。
cause 8 × 0.375 = 3. Check your answer by multiplying the decimal back by the denominator.
例如：3/8 = 3 ÷ 8 = 0.375，因为 8 × 0.375 = 3。可以用小数乘以分母来检验结果。
prime factors, for example 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125 and 1/20 = 0.05.
有些分数得到有限小数，即小数位有限并会终止。当分母化为最简形式后质因数只有 2 和 5 时就会如此，例如 1/2 = 0.5、1/4 = 0.25、1/5 = 0.2、1/8 = 0.125、1/20 = 0.05。
/11 = 0.0909... In exams a dot is placed over the first and last repeating digit instead of writing the digits out.
另一些分数得到循环小数，即某个数字或数字组无限重复。例如 1/3 = 0.3333...、5/6 = 0.8333...，而 1/11 = 0.0909...。考试中通常在首位和末位循环数字上加一个点，而不把数字全部写出来。
When a recurring decimal must be used in a calculation, round to a sensible number of places, such as 3 decimal places, and only round at the final step to avoid accumulating errors.
如果必须把循环小数用于计算，请四舍五入到合理的位数，例如保留三位小数，并且只在最后一步取近似值，以免误差累积。
4. Converting Decimals to Fractions | 小数化为分数
To turn a decimal into a fraction, use place value. Read the decimal aloud, write it as a fraction with a power of ten as the denominator, then simplify.
把小数化为分数要利用数位值。读出这个小数，先写成分母为 10 的幂的分数，再化简。
- 0.7 = 7/10 (one decimal place → tenths 一位小数 → 十分位)
- 0.35 = 35/100 = 7/20 (two decimal places → hundredths 两位小数 → 百分位)
- 0.125 = 125/1000 = 1/8 (three decimal places → thousandths 三位小数 → 千分位)
- 2.4 = 24/10 = 12/5 = 2⅖ (keep the whole-number part 保留整数部分)
Remember that the number of decimal places tells you the denominator: one place means tenths, two places means hundredths, three places means thousandths, and so on.
记住：小数位数决定分母。一位小数对应十分位，两位对应百分位，三位对应千分位，依此类推。
Always simplify the result. 0.35 becomes 35/100, and dividing both parts by 5 gives 7/20, which cannot be reduced further because 7 and 20 share no common factor.
一定要化简结果。0.35 写成 35/100，分子分母同除以 5 得 7/20，由于 7 与 20 没有公因数，无法继续约分。
5. Percentages as Fractions and Decimals | 百分比与分数、小数
The word percent means "per hundred", so a percentage is simply a fraction with a denominator of 100. 45% means 45 out of every 100, that is 45/100, which simplifies to 9/20 and equals 0.45 as a decimal.
percent 一词意为"每一百"，所以百分比就是以 100 为分母的分数。45% 表示每一百中有 45，即 45/100，化简得 9/20，化成小数是 0.45。
To convert a decimal to a percentage, multiply by 100 and add the % sign. To convert a percentage to a decimal, divide by 100. To convert a percentage to a fraction, write it over 100 and simplify.
小数化百分比：乘以 100 并加上 % 号。百分比化小数：除以 100。百分比化分数：写成分母为 100 的分数再化简。
fraction → decimal: ÷ top and bottom | decimal → percentage: × 100
Certain conversions appear so often that they should be memorised, because they save time and reduce careless errors in the exam.
有些换算出现得极为频繁，应该背下来，因为这样可以节省时间并减少考试中的粗心错误。
| Fraction 分数 | Decimal 小数 | Percentage 百分比 |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33⅓% |
| 1/10 | 0.1 | 10% |
A very common mistake is to confuse 0.3 with 3%. In fact 0.3 = 30% and 3% , so always check whether the decimal point has moved two places in the correct direction.
一个常见错误是把 0.3 与 3% 混淆。事实上 0.3 = 30%，而 3% = 0.03。所以要始终检查小数点是否朝正确方向移动了两位。
6. Finding a Percentage of an Amount | 求一个数的百分比
e most reliable method is to change the percentage into a decimal and multiply. To find 15% of 240, work out 0.15 × 240 = 36.
最可靠的方法是把百分比化成小数再相乘。求 240 的 15%，计算 0.15 × 240 = 36。
A second method builds the answer from easy percentages. To find 35% of 240, take 10% = 24, so 30% = 72, then 5% 72 + 12 = 84. This is useful for mental arithmetic and for checking your calculator work.
第二种方法用容易算的百分比逐步搭建答案。求 240 的 35%，先算 10% = 24，于是 30% = 72，5% = 12，最后 72 + 12 = 84。这种方法适合口算，也适合检验计算器结果。
- 10% → divide by 10 除以 10
- 5% → half of 10% 10% 的一半
- 1% → divide by 100 除以 100
- 20% → 10% × 2 或 除以 5
- 25% → divide by 4 除以 4
- 50% → divide by 2 除以 2
Using simple fraction equivalents can be even faster: 25% of 68 is 68 ÷ 4 = 17, and 75% of 68 is 17 × 3 = 51.
使用简单的分数等价形式往往更快：68 的 25% 是 68 ÷ 4 = 17，68 的 75% 就是 17 × 3 = 51。
7. Expressing One Quantity as a Percentage of Another | 求一个量占另一个量的百分比
When a question asks you to write one amount as a percentage of another, form a fraction and then convert it. The formula is straightforward and works for every context.
当题目要求把一个量表示成另一个量的百分比时，先写出分数再转换即可。这个公式非常直接，适用于所有情境。
percentage = (part ÷ whole) × 100%
a student scores 18 out of 24 in a test. The percentage is 18 ÷ 24 × 100 = 0.75 × 100 = 75%.
例如：某学生在一场考试中 24 分得 18 分。百分比为 18 ÷ 24 × 100 = 0.75 × 100 = 75%。
Identify the whole carefully. In questions about marks, the whole is the total mark; in questions about money, the whole is the original or total amount, not the change. Reading the question twice prevents most mistakes of this type.
要仔细确认"整体"是什么。分数题中整体是总分；金钱题中整体是原价或总额，而不是找零。把题目读两遍可以避免这类错误的大部分。
Answers above 100% an the whole, for example selling an item for 75 when it cost 60 gives 75 ÷ 60 × 100 = 125%, a 25% profit.
答案超过 100% 是完全可能的，只表示部分大于整体。例如 60 元买进、75 元卖出得 75 ÷ 60 × 100 = 125%，即获利 25%。
8. Comparing and Ordering the Three Forms | 三种形式的比较与排序
To compare numbers written in different forms, convert them all into the same form. Decimals are usually the easiest common language, because comparing 0.625 with 0.62 is immediate.
要比较写成不同形式的数，请把它们统一成同一种形式。小数通常是最方便的公共语言，因为比较 0.625 与 0.62 一目了然。
Example: arrange 0.6, 5/8, 62% and 3/5 in ascending order. Convert everything to decimals: 0.6 = 0.600, 5/8 = 0.625, 62% = 0.62 and 3/5 = 0.6. So the order is 0.6 = 3/5, then 62%, then 5/8.
例如：把 0.6、5/8、62% 和 3/5 按从小到大排列。全部化成小数：0.6 = 0.600，5/8 = 0.625，62% = 0.62，3/5 = 0.6。因此顺序是 0.6 = 3/5，然后 62%，最后 5/8。
When writing the final answer, keep the form used in the question. If the question gives fractions and percentages, answer with fractions and percentages, not with your working decimals.
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