📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数
Numbers can be written in many different forms, and three of the most important are fractions, decimals and percentages. In the Cambridge Lower Secondary Mathematics curriculum, you are expected to move fluently between these forms, understand their equivalence, and be able to compare quantities expressed in them. This article will take you through the core skills needed to convert, compare and apply fractions, decimals and percentages with confidence.
数字可以用多种不同的形式来表示，其中最重要的三种是分数、小数和百分数。在剑桥初中数学课程中，你需要能够在这几种形式之间灵活转换，理解它们之间的等价关系，并能比较用不同形式表示的数量。本文将带你学习转换、比较以及灵活运用分数、小数和百分数的核心技能。
1. Introduction: Why Fractions, Decimals and Percentages Matter | 引言：分数、小数与百分数为何重要
Whether you are sharing a cake, calculating a discount in a shop, or measuring ingredients in a recipe, fractions, decimals and percentages are everywhere in everyday life. In KS3 mathematics, being secure in these number forms allows you to solve problems in geometry, probability, statistics and algebra. Cambridge Checkpoint tests often include questions that ask you to order a mixed list like 0.4, 30% and ⅖, or to find a percentage of an amount using decimal multipliers.
无论你是在分蛋糕、计算商店折扣，还是在食谱中称量原料，分数、小数和百分数在日常生活里无处不在。在 KS3 数学中，扎实掌握这些数形式能够帮助你解决几何、概率、统计和代数中的问题。剑桥 Checkpoint 测试中经常出现这类题目：要求你对 0.4、30% 和 ⅖ 这样的混合列表排序，或者用小数倍数求一个数的百分之几。
2. What is a Fraction? | 什么是分数？
A fraction represents a part of a whole. It is written with a numerator (top number) and a denominator (bottom number), separated by a vinculum. For example, ⅗ means 3 equal parts out of a total of 5 equal parts. The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many of those parts you have. Fractions can be proper (numerator < denominator), improper (numerator > denominator), or mixed numbers combining an integer and a proper fraction.
分数表示一个整体的一部分。它由分子（上方的数）和分母（下方的数）组成，中间用分数线隔开。例如，⅗ 表示从一个被分成 5 等份的整体中取出 3 份。分母告诉你整体被分成了多少等份，分子告诉你取了多少份。分数可以是真分数（分子小于分母）、假分数（分子大于分母），或是由整数和真分数组成的带分数。
Fraction = Numerator ÷ Denominator
分数 = 分子 ÷ 分母
3. Decimals: Place Value and Tenths, Hundredths, Thousandths | 小数：数位与十分位、百分位、千分位
A decimal uses a decimal point to separate the whole number part from the fractional part. The first place after the point is the tenths place, the second is hundredths, and the third is thousandths. So 0.7 means seven tenths, 0.25 means twenty-five hundredths, and 0.008 means eight thousandths. Understanding place value is essential for reading decimals correctly and for converting between decimals and fractions.
小数利用小数点把整数部分与小数部分隔开。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。因此，0.7 表示七个十分之一，0.25 表示二十五个百分之一，0.008 表示八个千分之一。理解数位对于正确读出小数以及在小数与分数之间进行转换至关重要。
4. Percentages: Meaning of ‘Per Cent’ | 百分数：“百分比”的含义
A percentage is a fraction with a denominator of 100. The word ‘per cent’ comes from Latin and means ‘out of a hundred’. So 45% literally means 45 out of every 100, or 45/100. Percentages are useful for comparing proportions because they always use the same base of 100. In many real-life situations, such as test scores or interest rates, percentages give a quick, clear picture of a relative amount.
百分数实际上就是分母为 100 的分数。“百分比”一词源于拉丁语，意思是“每一百”。因此，45% 的字面含义就是每 100 份中的 45 份，即 45/100。百分数之所以便于比较比例，是因为它们总以 100 为统一基准。在考试成绩、利率等许多现实场景中，百分数能让人快速、清晰地把相对量表示出来。
5. Converting Fractions to Decimals | 分数化为小数
To convert any fraction to a decimal, divide the numerator by the denominator. For a simple fraction like ⅜, calculate 3 ÷ 8 = 0.375. Some fractions give terminating decimals (e.g. ⅕ = 0.2, ¾ = 0.75), while others give recurring decimals (e.g. ⅓ = 0.333…, ²⁄₇ = 0.285714285714…). In KS3 you are expected to recognise common fraction–decimal equivalences without always needing a calculator.
要把任何分数化成小数，只需用分子除以分母。对于像 ⅜ 这样的简单分数，计算 3 ÷ 8 = 0.375 即可。有些分数会得到有限小数（如 ⅕ = 0.2，¾ = 0.75），另一些则会得到循环小数（如 ⅓ = 0.333…，²⁄₇ = 0.285714285714…）。在 KS3 阶段，你需要在不依赖计算器的情况下记住一些常见的分数–小数等价关系。
6. Converting Decimals to Fractions | 小数化为分数
To write a terminating decimal as a fraction, use the place value of the last digit. For example, 0.7 ends in the tenths place, so 0.7 = 7/10. 0.25 ends in the hundredths place, so 0.25 = 25/100, which simplifies to ¼. For recurring decimals, a special algebraic method is used, but at KS3 you mainly work with terminating decimals and common recurring ones like 0.3̇ = ⅓.
要把一个有限小数写成分数，可以利用最后一位数字的数位。例如，0.7 落在十分位，所以 0.7 = 7/10。0.25 落在百分位，所以 0.25 = 25/100，再化简得到 ¼。对于循环小数，要使用一种专门的代数方法，不过在 KS3 阶段你主要接触的是有限小数以及像 0.3̇ = ⅓ 这样的常见循环小数。
7. Converting Fractions to Percentages | 分数化为百分数
Since a percentage is just a fraction with denominator 100, you can convert a fraction to a percentage by finding an equivalent fraction out of 100, or by dividing the numerator by the denominator and then multiplying by 100%. For example, ⅗: 3 ÷ 5 = 0.6, then 0.6 × 100% = 60%. Alternatively, ⅗ = 60/100 = 60%. Both methods give the same result, but the decimal method works for any fraction.
既然百分数就是分母为 100 的分数，你可以把分数先化成分母为 100 的等值分数，也可以用分子除以分母再乘 100% 来把分数化成百分数。例如，⅗：3 ÷ 5 = 0.6，然后 0.6 × 100% = 60%。或者，⅗ = 60/100 = 60%。两种方法结果相同，但小数法对任意分数都适用。
8. Converting Percentages to Fractions | 百分数化为分数
To convert a percentage to a fraction, write the percentage as a fraction with denominator 100, then simplify if possible. For example, 75% = 75/100 = ¾ (dividing numerator and denominator by 25). For percentages with decimals, such as 12.5%, write it as 12.5/100, multiply numerator and denominator by 10 to get 125/1000, and then simplify to ⅛. Always express your final answer in simplest form.
将百分数化为分数时，先把百分数写成分母为 100 的分数，再尽可能化简。例如，75% = 75/100 = ¾（分子分母同时除以 25）。对于含有小数的百分数，如 12.5%，可先写成 12.5/100，分子分母同乘 10 得到 125/1000，再化简为 ⅛。最终答案一定要用最简分数表示。
9. Converting Decimals to Percentages and Back | 小数与百分数的相互转换
Converting between decimals and percentages is straightforward: multiply the decimal by 100 to get the percentage, and divide the percentage by 100 to get the decimal. For example, 0.47 as a percentage is 0.47 × 100% = 47%. To convert 8% to a decimal, divide 8 by 100 to get 0.08. Because these two operations are inverses, you can think of moving the decimal point two places to the right (to make a percentage) or two places to the left (to make a decimal).
小数与百分数之间的转换非常直接：把小数乘 100 就得到百分数，把百分数除以 100 就得到小数。例如，0.47 用百分数表示就是 0.47 × 100% = 47%。要把 8% 化成小数，将 8 除以 100 得到 0.08。因为这两步运算是互逆的，你可以想象成把小数点向右移两位（得到百分数）或向左移两位（得到小数）。
10. Comparing and Ordering Fractions, Decimals and Percentages | 比较与排序
To compare quantities given in different forms, it is usually best to convert them all to the same form – often decimals or percentages. For example, to order 0.45, ⅖ and 30% from smallest to largest, first convert ⅖ = 0.4 and 30% = 0.3. Now compare 0.45, 0.4 and 0.3, giving 0.3, 0.4, 0.45, or 30%, ⅖, 0.45. When working without a calculator, you can also convert everything to fractions with a common denominator, which is especially useful when comparing several fractions.
要比较用不同形式表示的数量，最好先把它们都转化成同一种形式——通常是小数或百分数。例如，要把 0.45、⅖ 和 30% 从小到大排列，先把 ⅖ 化成 0.4，把 30% 化成 0.3。现在比较 0.45、0.4 和 0.3，得到 0.3、0.4、0.45，也就是 30%、⅖、0.45。在不使用计算器时，你也可以把所有数都化成分母相同的分数，尤其是在比较多个分数时这一方法非常有用。
11. Using a Number Line to Compare Values | 用数轴比较数值
A number line is a powerful visual tool for ordering fractions, decimals and percentages. Draw a line labelled from 0 to 1, mark the positions of the values you are comparing, and then simply read off the order from left to right. For example, placing 0.2, ¼, 50% and 0.05 on a number line immediately shows that 0.05 < 0.2 < ¼ < 50%. Number lines help you develop a strong sense of the relative size of different representations, which is particularly helpful when estimating.
数轴是排序分数、小数和百分数的强大视觉工具。画一条标有 0 到 1 的直线，把你需要比较的值在线上标出位置，然后从左到右依次读取顺序即可。例如，把 0.2、¼、50% 和 0.05 标在数轴上，马上就能看出 0.05 < 0.2 < ¼ < 50%。数轴能够帮助你建立起对不同表示形式相对大小的直观感觉，这对估算尤其有帮助。
12. Common Equivalences to Remember | 需要记住的常见等价关系
Memorising a set of basic equivalences will save you time in exams and build your number fluency. Below is a table of the most frequently used fractions, decimals and percentages in the Cambridge KS3 syllabus.
熟记一组基本的等价关系不仅能让你在考试中节省时间，还能增强你的数感。下表列出了剑桥 KS3 课程中最常出现的分数、小数和百分数的对应关系。
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅕ | 0.2 | 20% |
| ⅖ | 0.4 | 40% |
| ⅗ | 0.6 | 60% |
| ⅘ | 0.8 | 80% |
| ⅛ | 0.125 | 12.5% |
| ⅜ | 0.375 | 37.5% |
| ⅝ | 0.625 | 62.5% |
| ⅞ | 0.875 | 87.5% |
| ⅓ | 0.333… (or 0.3̇) | 33⅓% |
| ⅔ | 0.666… (or 0.6̇) | 66⅔% |
Using this table as a reference, practise writing these equivalences from memory until you can recall them instantly. This foundation will make more advanced topics, such as percentage increase and decrease or probability calculations, much easier.
以上表为参考，反复练习默写这些等价关系，直到能够瞬间回忆起来。有了这一基础，百分比增减、概率计算等更高阶的主题就会变得简单许多。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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