📚 Year 7 Topic Series 400: Mastering Fractions, Decimals and Percentages | 七年级主题系列400：掌握分数、小数与百分数
Fractions, decimals and percentages are three connected ways to describe parts of a whole. In IGCSE Year 7 Mathematics, students need to convert between these forms, compare them and use them in calculations with confidence. This article covers the key skills, the methods that examiners expect to see, and the common exam-style questions step by step, so you can move from "I sort of get it" to "I can do this every time".
分数、小数和百分数是描述整体部分的三种相互关联的方式。在 IGCSE 七年级数学中，学生需要在这些形式之间熟练转换、比较大小并用于计算。本文逐步讲解关键技能、考官期望看到的标准方法，以及常见考试题型，帮助你从"大概懂了"提升到"每次都能做对"。
The core idea is simple: 1/2, 0.5 and 50% all mean exactly the same amount. Once you truly believe that, every conversion question becomes a language translation exercise rather than a memory test. Work through each section below, copy the worked examples into your notebook, and then try the practice questions at the end.
核心思想很简单：1/2、0.5 和 50% 表示完全相同的大小。一旦你真正理解这一点，所有转换题就变成了一次"语言翻译"，而不是背诵测试。请逐节阅读，把例题抄进笔记本，最后完成文末的练习题。
1. Understanding Equivalent Fractions | 理解等值分数
Equivalent fractions are fractions that have the same value even though their numerators and denominators are different. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero whole number. This works because you are really multiplying the fraction by a version of 1, such as 3/3 or 5/5.
等值分数是指数值相同但分子和分母不同的分数。将分子和分母同时乘以或除以同一个非零整数，就可以得到等值分数。其原理是你实际上把分数乘以了 1 的另一种写法，例如 3/3 或 5/5。
For example, if you start with 1/2 and multiply both parts by 3, you get 3/6, which is the same amount as 1/2. If you divide both parts of 10/15 by 5, you get 2/3. This skill is important when adding, subtracting or comparing fractions.
例如，从 1/2 开始，将分子和分母都乘以 3，就得到 3/6，它与 1/2 表示同样的大小；把 10/15 的分子和分母同时除以 5，就得到 2/3。这项技能在分数的加减法和比较大小中非常重要。
1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 50/100
- Multiplying builds larger terms: 2/5 × 4/4 = 8/20.
乘以同一个数会得到分子分母更大的等值分数：2/5 × 4/4 = 8/20。 - Dividing builds smaller terms: 12/18 ÷ 6/6 = 2/3.
除以同一个数会得到分子分母更小的等值分数：12/18 ÷ 6/6 = 2/3。 - The value never changes: only the way the fraction is written changes.
分数的数值始终不变，改变的只是书写形式。 - Never add: you cannot make an equivalent fraction by adding the same number to the numerator and denominator.
千万不要用加法：给分子分母同时加同一个数，得到的不是等值分数。
t. This cross-multiplication test also becomes the basis for solving equations later in IGCSE.
给七年级学生的一个快速检验方法：交叉相乘。如果 1 × 6 = 2 × 3，那么 1/2 与 3/6 就是等值分数。这个交叉相乘的检验方法，日后在 IGCSE 解方程时也会用到。
2. Simplifying Fractions to Lowest Terms | 把分数化为最简形式
A fraction is in its simplest form, or lowest terms, when the numerator and denominator have no common factor other than 1. To simplify, divide both parts by their highest common factor (HCF). If you cannot spot the HCF straight away, divide repeatedly by any common factor until nothing else divides.
当分子与分母除 1 以外没有公因数时，分数就处于最简形式（即最低项）。化简的方法是：把分子和分母同时除以它们的最大公因数（HCF）。如果一时看不出最大公因数，可以先反复除以任意公因数，直到不能再除为止。
8 ÷ 6 = 3 and 24 ÷ 6 = 4, giving 3/4. Checking: 3 and 4 share no factor except 1, so the answer is in lowest terms.
例题：化简 18/24。18 的因数是 1、2、3、6、9、18，24 的因数是 1、2、3、4、6、8、12、24，最大公因数是 6，所以 18 ÷ 6 = 3，24 ÷ 6 = 4，得到 3/4。检验：3 与 4 除 1 外没有公因数，因此结果已是最简形式。
18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4
answer; the HCF method is faster, but the repeated method is safer when the numbers are large and you are unsure of the HCF.
另一种方法是反复约分：18/24 ÷ 2/2 = 9/12，再 9/12 ÷ 3/3 = 3/4。两种方法结果相同：用最大公因数法更快，但当数字较大、你无法确定最大公因数时，反复约分法更稳妥。
Examiners almost always ask for the answer "in its simplest form". Even if your calculation is correct, an unsimplified final answer can lose a mark. Make simplification the last automatic step of every fraction question.
考官几乎总会要求答案写成"最简形式"。即使计算过程正确，未化简的答案也可能被扣分。请把"化简"变成每道分数题的最后一步自动动作。
3. Converting Fractions to Decimals | 把分数化为小数
There are two reliable methods for turning a fraction into a decimal. Method 1 works when the denominator is 10, 100, 1000 or can easily be changed into one of these. Method 2, dividing the numerator by the denominator, always works.
把分数化为小数有两种可靠方法。方法一适用于分母是 10、100、1000，或能方便地化成分母为 10、100、1000 的分数；方法二——用分子除以分母——则永远适用。
- Method 1 - change the denominator: 7/20 = (7 × 5)/(20 × 5) = 35/100 = 0.35.
方法一——改变分母：7/20 = (7 × 5)/(20 × 5) = 35/100 = 0.35。 - Method 2 - short division: 3/8 means 3 ÷ 8 = 0.375.
方法二——短除法：3/8 就是 3 ÷ 8 = 0.375。 - Watch the place value: 35/100 has two decimal places, so it is 0.35, not 3.5.
注意数位：35/100 有两位小数，所以是 0.35，而不是 3.5。 - Check with estimation: 3/8 is a bit less than 1/2, so 0.375 makes sense.
用估算检验：3/8 略小于 1/2，因此 0.375 是合理的。
Some fractions produce terminating decimals, which stop after a certain number of decimal places. This happens when the denominator in lowest terms has only 2 and 5 as prime factors, for example 4, 8, 20, 25 and 40.
有些分数产生有限小数，即小数点后到某一位就结束。当分数化为最简形式后，分母的质因数只有 2 和 5 时就会出现这种情况，例如分母为 4、8、20、25、40。
0.1666... and 5/11 = 0.454545.... The repeating part is shown with dots or a bar above the digits in some textbooks, but in Year 7 answers you may simply write three dots or state "0.45 recurring".
另一些分数产生循环小数，即某一位数字或某一组数字无限重复。例如 1/3 = 0.333...，2/3 = 0.666...，1/6 = 0.1666...，5/11 = 0.454545...。有些教材用点或横线标出循环节，但在七年级的作答中，写出三个点或注明"0.45 循环"即可。
1/3 = 0.333... (recurring) 3/8 = 0.375 (terminating)
4. Converting Decimals to Fractions | 把小数化为分数
Every decimal can be written as a fraction by reading its place value. The number of decimal places tells you the denominator: one decimal place means tenths, two means hundredths, three means thousandths.
任何小数都可以通过读它的数位写成分数。小数位数决定分母：一位小数是十分之几，两位小数是百分之几，三位小数是千分之几。
- 0.6 = 6/10 = 3/5
0.6 是十分之六，约分为五分之三。 - 0.75 = 75/100 = 3/4
0.75 是百分之七十五，约分为四分之三。 - 0.125 = 125/1000 = 1/8
0.125 是千分之一百二十五，约分为八分之一。 - 2.35 = 235/100 = 47/20 = 2 7/20
带小数先化成假分数，再约分或化成带分数。
After writing the fraction, always simplify it. A decimal such as 0.375 becomes 375/1000, which simplifies by dividing both parts by 125 to give 3/8. Dividing by 125 feels strange at first, but it is simply the largest number that divides both 375 and 1000 exactly.
写完分数后一定要化简。例如 0.375 写成 375/1000，分子分母同时除以 125 得到 3/8。除以 125 看起来有点奇怪，但它正是能同时整除 375 和 1000 的最大数。
x = 7, so x = 7/9. The same method turns 0.454545... into 45/99, which simplifies to 5/11.
循环小数也可以用一个小巧的代数技巧转换。对于 0.777...，设 x = 0.777...，则 10x = 7.777...，两式相减得 9x = 7，所以 x = 7/9。同样的方法可把 0.454545... 化为 45/99，再约简为 5/11。
10x − x = 9x → 9x = 7 → x = 7/9
5. Understanding Percentages | 理解百分数
The word "per cent" means "per hundred", so 45% simply means 45 out of every 100, or 45/100. Percentages are especially useful for comparing quantities because they always use the same base of 100.
"per cent" 意为"每一百"，所以 45% 就是每 100 份中的 45 份，即 45/100。百分数在比较数量时特别有用，因为它们总是以 100 为同一基准。
To convert a percentage to a fraction, write it over 100 and simplify: 45% tage to a decimal, divide by 100, which moves the decimal point two places to the left: 45% = 0.45, and 7% = 0.07.
把百分数化为分数：写成分母为 100 的分数再化简，例如 45% = 45/100 = 9/20。把百分数化为小数：除以 100，也就是把小数点向左移两位，例如 45% = 0.45，7% = 0.07。
Going the other way, multiply by 100 and add the % sign. A fraction or decimal becomes a percentage by multiplying by 100: 0.62 = 62%, 3/5 = 0.6 = 60%, and 3/8 = 0.375 = 37.5%.
反向转换时，乘以 100 并加上百分号。分数或小数乘以 100 就变成百分数：0.62 = 62%，3/5 = 0.6 = 60%，3/8 = 0.375 = 37.5%。
- Percentage to decimal: ÷ 100 (45% → 0.45).
百分数化小数：除以 100。 - Decimal to percentage: × 100 (0.08 → 8%).
小数化百分数：乘以 100。 - Fraction to percentage: convert to a decimal first, then × 100.
分数化百分数：先化成小数，再乘以 100。 - Common errors: 5% = 0.05, not 0.5; 120% = 1.2, which is greater than 1.
常见错误：5% = 0.05，不是 0.5；120% = 1.2，大于 1。
6. The Master Conversion Table | 三种形式互化总表
Learning the most common conversions by heart saves valuable time in tests and helps you estimate answers sensibly. The table below lists the conversions that appear most often in Year 7 and IGCSE papers.
熟记最常见的转换可以在考试中节省宝贵时间，也能帮助你合理估算答案。下表列出了七年级和 IGCSE 试卷中最常出现的转换。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 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% |
| 3/8 | 0.375 | 37.5% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33.3...% |
| 2/3 | 0.666... | 66.6...% |
When a question mixes forms, the safest strategy is to convert everything into decimals, because decimals are the easiest to line up and compare. Percentages are best for final answers about change or proportion, while fractions are best when the answer needs to be exact.
当题目混合了不同形式时，最安全的策略是把所有数都化成小数，因为小数最容易对齐和比较。若答案涉及变化或比例，百分数最直观；若答案需要精确值，分数则最合适。
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