📚 Fractions, Decimals and Percentages | 分数、小数和百分数
Fractions, decimals and percentages are three different ways to represent parts of a whole. At Key Stage 3, you need to be able to convert between these forms fluently, compare them, and solve real-life problems. This article will guide you through each conversion method with clear steps, examples, and common pitfalls to avoid. Whether you are preparing for a test or simply want to strengthen your number skills, mastering these links will make many topics in mathematics much easier.
分数、小数和百分数是表示整体中部分数量的三种不同方式。在 Key Stage 3 阶段，你需要能够熟练地在这些形式之间进行转换，比较它们的大小，并解决实际生活问题。本文将用清晰的步骤、例子和需要避免的常见错误，引导你掌握每一种转换方法。无论你是在准备考试，还是只想加强自己的数字技能，掌握这些联系都会让数学中的很多主题变得更加简单。
1. Understanding Fractions | 理解分数
A fraction has a numerator (top number) and a denominator (bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have. For example, 3/4 means three out of four equal parts.
分数由分子（上面的数）和分母（下面的数）组成。分母表示整体被分成了多少等份，分子表示你拥有多少份。例如，3/4 表示四等份中的三份。
Fractions can be proper (numerator smaller than denominator, e.g. 2/5), improper (numerator larger than denominator, e.g. 7/4) or mixed numbers (whole number and a proper fraction, e.g. 1½). Improper fractions and mixed numbers are often used interchangeably.
分数可以是真分数（分子比分母小，例如 2/5）、假分数（分子比分母大，例如 7/4）或带分数（整数加真分数，例如 1½）。假分数和带分数经常可以相互转换使用。
oth the numerator and denominator by the same number. For instance, 1/2 = 2/4 = 3/6 = 0.5.
等值分数看起来不同，但数值相等。你可以通过将分子和分母同时乘以或除以同一个数，得到等值分数。例如，1/2 = 2/4 = 3/6 = 0.5。
2. Understanding Decimals | 理解小数
A decimal number uses a decimal point to separate the whole number part from the fractional part. Each digit after the point represents tenths, hundredths, thousandths, and so on. For example, 0.75 means 7 tenths and 5 hundredths, which is the same as 75/100.
小数使用小数点将整数部分和小数部分分开。小数点后的每一位数字依次表示十分位、百分位、千分位等。例如，0.75 表示 7 个十分之一和 5 个百分之一，等同于 75/100。
Decimals can be terminating, like 0.25 or 1.6, or recurring, like 0.333... (often written with a dot above the repeating digit). Understanding place value is the key to converting decimals to fractions.
小数可以是有限小数，如 0.25 或 1.6，也可以是循环小数，如 0.333...（通常在重复的数字上方标点表示）。理解位值是小数转换为分数的关键。
When reading a decimal like 0.125, think of it as 125 thousandths. This immediately tells you the fraction form: 125/1000, which can then be simplified.
在读出像 0.125 这样的小数时，可以把它看作 125 个千分之一。这直接给出了分数形式：125/1000，然后再进行化简。
3. Understanding Percentages | 理解百分数
Percent means 'out of 100'. The symbol % is used to show a number as a fraction of 100. For example, 45% means 45 out of 100, or 45/100. Percentages are very useful for comparing proportions because they always have the same denominator (100).
百分数表示“每一百份中的数量”，用符号 % 表示。例如，45% 就是 100 份中的 45 份，也就是 45/100。百分数在比较比例时非常有用，因为它们总是以相同的分母（100）为基础。
A percentage greater than 100% represents a value larger than the whole. For instance, 150% of a quantity means 1.5 times the quantity. Percentages less than 1% are also possible, like 0.5% which is equal to 0.005.
大于 100% 的百分数表示超过整体的数值。例如，某个量的 150% 就是这个量的 1.5 倍。小于 1% 的百分数也是存在的，如 0.5% 等于 0.005。
etween percentages, fractions and decimals very straightforward once you remember that 100% = 1 whole.
你可以把百分数看作分母为 100 的分数。一旦你记住 100% = 1 个整体，百分数、分数和小数之间的联系就变得非常直接了。
4. Converting Fractions to Decimals | 分数转小数
To change a fraction to a decimal, divide the numerator by the denominator. Use short division or a calculator. For simple fractions, you can also find an equivalent fraction with a denominator of 10, 100 or 1000 first.
把分数化成小数，用分子除以分母即可。可以使用短除法或计算器。对于简单的分数，也可以先找到一个分母为 10、100 或 1000 的等值分数。
Or convert 3/5 to a decimal: multiply numerator and denominator by 2 to get 6/10, which is 0.6.
例子：把 7/8 化成小数。用 7 除以 8：7 ÷ 8 = 0.875，所以 7/8 = 0.875。或者把 3/5 化成小数：将分子分母同时乘以 2 得到 6/10，即 0.6。
nd 2/11 = 0.181818... In KS3, you may be asked to write these using dot notation, e.g. 0.3̇ for 1/3 and 0.1̇8̇ for 2/11.
有些分数会得到循环小数，例如 1/3 = 0.333...，以及 2/11 = 0.181818...。在 KS3 中，你可能会被要求用点标记法写出循环小数，如 1/3 写作 0.3̇，2/11 写作 0.1̇8̇。
Use the division method confidently: 1/6 = 1 ÷ 6 = 0.1666..., which becomes 0.16̇ (only the 6 repeats).
要自信地使用除法：1/6 = 1 ÷ 6 = 0.1666...，写作 0.16̇（只有 6 循环）。
5. Converting Decimals to Fractions | 小数转分数
To write a terminating decimal as a fraction, look at the place value of the last digit. For example, 0.7 is seven tenths, so 7/10. 0.25 ends in the hundredths place, so it is 25/100, which simplifies to 1/4.
要把一个有限小数写成分数，找出最后一位数字所在的位值。例如，0.7 是十分之七，所以是 7/10。0.25 的最后一位在百分位，所以是 25/100，化简后得到 1/4。
1000. Simplify by dividing numerator and denominator by 125 (the greatest common factor) to get 3/8.
0.375 的分步转换：最后一位 5 在千分位 → 375/1000。分子分母同时除以 125（最大公因数），得到 3/8。
ole number: 2 + 0.6 = 2 + 6/10 = 2 3/5 as a mixed number, or 13/5 as an improper fraction.
对于如 2.6 这样的带小数，把整数分开：2 + 0.6 = 2 + 6/10 = 2 3/5（带分数），或者化成假分数 13/5。
you are mainly expected to know common ones: 0.333... = 1/3, 0.666... = 2/3, 0.999... = 1.
循环小数可以用代数方法转换为分数，但在 KS3 阶段你主要需要知道一些常见的：0.333... = 1/3，0.666... = 2/3，0.999... = 1。
6. Converting Fractions to Percentages | 分数转百分数
To convert a fraction to a percentage, first make the denominator equal to 100 if possible. Multiply the numerator and denominator by the same number so the denominator becomes 100. The new numerator is the percentage.
要把分数转换为百分数，如果可以，先让分母等于 100。将分子分母同时乘以相同的数，使分母变为 100。新的分子就是百分数值。
Example: 3/5 is equivalent to 60/100 (multiply top and bottom by 20). So 3/5 = 60%. Another example: 7/10 = 70/100 = 70%.
例子：3/5 等于 60/100（分子分母同乘 20），所以 3/5 = 60%。另一个例子：7/10 = 70/100 = 70%。
ator) and then multiply by 100. For instance, 5/8 = 5 ÷ 8 = 0.625, then 0.625 × 100 = 62.5%.
如果分母不容易变成 100，可以先将分数转化为小数（用分子除以分母），再乘以 100。例如，5/8 = 5 ÷ 8 = 0.625，然后 0.625 × 100 = 62.5%。
Remember that 1/3 = 0.333... so as a percentage it is 33.3̇%. This is often rounded to 33.3% or 33 1/3%. The exact percentage is 33 1/3%.
记住 1/3 = 0.333...，所以化成百分数是 33.3̇%。通常四舍五入为 33.3% 或写作 33 1/3%。精确的百分数是 33 1/3%。
7. Converting Percentages to Fractions | 百分数转分数
a fraction, write the percentage number over 100 and simplify if possible. For example, 40% = 40/100 = 2/5 (divide top and bottom by 20).
把百分数化成分数，将百分数的数字写在分母 100 上方，然后进行化简。例如，40% = 40/100 = 2/5（分子分母同除以 20）。
If the percentage has a decimal, like 12.5% erator and denominator by 10 to clear the decimal: 125/1000, then simplify to 1/8. So 12.5% = 1/8.
如果百分数包含小数，例如 12.5%，先写成 12.5/100。将分子分母同时乘以 10 去掉小数：125/1000，然后化简得到 1/8。所以 12.5% = 1/8。
Percentages with fractions, like 33 1/3%, can be written as an improper fraction over 100: (100/3) / 100 = 100/300 = 1/3. It's often easier to recognise common equivalents.
带有分数的百分数，如 33 1/3%，可以写成假分数除以 100：(100/3) / 100 = 100/300 = 1/3。通常更容易的是熟记常见等价关系。
Always simplify the fraction to its lowest terms. For example, 75% = 75/100 = 3/4. 5% = 5/100 = 1/20. Knowing these off by heart speeds up problem solving.
一定要把分数化简到最简。例如，75% = 75/100 = 3/4。5% = 5/100 = 1/20。把这些记下来会加快解题速度。
8. Converting Decimals and Percentages | 小数与百分数互转
Converting between decimals and percentages is very straightforward because percentages are just hundredths. To change a decimal to a percentage, multiply by 100 (move the decimal point two places to the right). To change a percentage to a decimal, divide by 100 (move the decimal point two places to the left).
小数与百分数的互相转换非常直接，因为百分数就是百分之几。要把小数变成百分数，乘以 100（把小数点向右移动两位）。要把百分数变成小数，除以 100（把小数点向左移动两位）。
Examples: 0.23 × 100 = 23% ; 0.6 × 100 = 60% ; 1.25 × 100 = 125% ; 5% ÷ 100 = 0.05 ; 0.4% = 0.004.
例子：0.23 × 100 = 23% ；0.6 × 100 = 60% ；1.25 × 100 = 125% ；5% ÷ 100 = 0.05 ；0.4% = 0.004。
Be careful with single-digit percentages: 7% as a decimal is 0.07, not 0.7. The zero before the 7 is needed as a place holder. Always check that the decimal point has moved two places correctly.
要小心一位数的百分数：7% 转化成小数是 0.07，而不是 0.7。7 前面的零是必要的，用来占位。始终检查小数点是否正确移动了两位。
This two-way conversion is extremely common in questions about discounts, tax, and data. Mentally moving the decimal point is a skill worth practising.
这种双向转换在关于折扣、税收和数据的题目中极为常见。在脑中移动小数点的技巧值得多加练习。
9. Comparing Fractions, Decimals and Percentages | 比较大小
When you need to compare quantities given in different forms, the easiest strategy is to convert all of them to the same form – usually decimals or percentages. Then put them in order.
当你需要比较以不同形式给出的量时，最简单的策略是把它们都转换成同一种形式——通常是小数或百分数。然后再排序。
Example: Arrange 2/5, 0.35, 38% from smallest to largest. Convert all to decimals: 2/5 = 0.4, 0.35 stays, 38% = 0.38. Order: 0.35, 0.38, 0.4. So the order is 0.35, 38%, 2/5.
例子：把 2/5、0.35、38% 从小到大排列。全部转换为小数：2/5 = 0.4，0.35 不变，38% = 0.38。排序：0.35、0.38、0.4。所以顺序是 0.35、38%、2/5。
mon denominator of 100 is often handy for percentages. For example, 17/20 compared with 85%: 17/20 = 85/100 = 85%, so they are equal.
对于分母不同的分数，要么转换成小数，要么找一个公分母。用 100 作为公分母对于百分数来说往往很方便。例如，比较 17/20 与 85%：17/20 = 85/100 = 85%，所以二者相等。
When using a number line, marking 0, 0.5 and 1 as benchmarks helps. 1/3 is just over 0.3, 2/3 is about 0.67, etc. Familiarity with these benchmarks makes estimation much faster.
在数轴上进行比较时，用 0、0.5 和 1 作为基准点很有帮助。1/3 略大于 0.3，2/3 约为 0.67 等。熟悉这些基准点能使估算变得更快。
10. Common Equivalent Values to Memorize | 常用等值记忆表
Having a mental reference table of common fractions, decimals and percentages can save time in exams and real life. Below is a key table for KS3 students.
在脑海中有一张常见分数、小数和百分数的参考表，可以在考试和日常生活中节省时间。下面是一份针对 KS3 学生的关键表格。
| 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% |
| 7/8 | 0.875 | 87.5% |
| 1/3 | 0.3̇ | 33 1/3% |
| 2/3 | 0.6̇ | 66 2/3% |
| 1/10 | 0.1 | 10% |
Practise writing these equivalences from memory. They form the building blocks for more complex calculations involving ratios, probability, and financial maths.
练习默写这些等值关系。它们是涉及比例、概率和金融数学的更复杂计算的基础。
11. Word Problems and Real-life Applications | 应用题与实际应用
you to interpret a scenario and work with fractions, decimals or percentages. For example, "In a sale, all prices are reduced by 30%. If a jacket originally cost £45, what is the sale price?" To solve, find 30% of £45 (0.3 × 45 = 13.5) and subtract: 45 − 13.5 = £31.50. Alternatively, recognise that you pay 70%, so 0.7 × 45 = 31.5.
很多考题要求你理解一个情景，并运用分数、小数或百分数。例如：“一件上衣原价45英镑，打七折出售，打折后的价格是多少？”（原文是“减价30%”，即支付70%）。计算30% of £45 (0.3 × 45 = 13.5)，然后减去：45 − 13.5 = £31.50。或者直接意识到你支付70%，所以 0.7 × 45 = 31.5。
Word problems might involve fractions of amounts: "A class has 28 students. 3/7 of them are boys. How many girls are there?" Then girls = 28 − 12 = 16. You could also work with the fraction of girls (4/7) directly.
应用题可能涉及一个数量的几分之几：“一个班级有28名学生，其中3/7是男生。女生有多少人？”首先求出 28 的 3/7：28 ÷ 7 × 3 = 12 名男生。然后女生 = 28 − 12 = 16。你也可以直接用女生的分数 (4/7) 来计算。
Combining multiple conversions is common: "Tom scored 17 out of 25 in a test. What is his score as a percentage?" 17/25 = 68/100 = 68%. This type of question is used frequently in school reports and statistics.
经常需要结合多种转换：“汤姆在测验中得了17分（满分25分）。他的成绩用百分数表示是多少？”17/25 = 68/100 = 68%。这类问题在学校报告和统计中经常出现。
Percentages greater than 100% appear in profit and increase problems: "The price of a house increased by 120% over ten years. If it originally cost £200,000, what is the new price?" 120% of 200,000 = 1.2 × 200,000 = 240,000, so new price = 200,000 + 240,000 = £440,000.
大于100%的百分数出现在利润和增长问题中：“一栋房子的价格在十年内上涨了120%。如果原价是20万英镑，新价格是多少？”120% of 200,000 = 1.2 × 200,000 = 240,000，所以新价格 = 200,000 + 240,000 = £440,000。
12. Common Mistakes to Avoid | 常见错误需避免
Mistake 1: Forgetting that percentages are out of 100. Students sometimes write 0.5 as 5% instead of 50%. Remember, multiply by 100 to go from decimal to percentage.
错误一：忘记了百分数是基于100的。学生有时会把 0.5 写成 5% 而不是 50%。记住，从小数转换为百分数要乘以100。
Mistake 2: Confusing the order of operations when finding a fraction of an amount. '1/3 of 60' ÷ 3 × 1, not 60 × 3 ÷ 1. Always divide by the denominator, then multiply by the numerator.
错误二：在求一个数的几分之几时搞错运算顺序。“1/3 of 60”是指 60 ÷ 3 × 1，而不是 60 × 3 ÷ 1。始终先除以分母，再乘以分子。
Mistake 3: Not simplifying fractions fully. Writing 50% as 50/100 is correct, but you should simplify to 1/2 unless otherwise stated. Always check if your fraction can be reduced.
错误三：分数没有化简到底。把50%写成50/100是正确的，但除非题目另有要求，否则应该化简为1/2。始终检查分数是否可以约分。
Mistake 4: Incorrectly applying percent increase or decrease. A 20% increase on £50 is not the same as finding 20% of £ the multiplier method (1.2 × 50) avoids missing the original amount. For a decrease of 15%, use a multiplier of 0.85.
错误四：错误地应用百分比增减。在50英镑的基础上增加20%，不等于先求出50英镑的20%再加回去；使用乘数法 (1.2 × 50) 可以避免遗漏原始数额。若减少15%，则使用乘数 0.85。
Mistake 5: Mixing up recurring decimals. Always indicate which digit(s) repeat using a dot above the first and last repeating digit, not every digit if it can be grouped. Paper practice helps build accuracy.
错误五：混淆循环小数的表示方法。总是用点在首个和末个重复数字上方标出重复的位数，如果能分组就别在每个数字上都点。书面练习有助于提高准确性。
Reviewing these common errors will strengthen your overall understanding and help you avoid losing marks on what should be straightforward questions.
复习这些常见错误会加强你的整体理解，并帮助你在本应直接了当的题目上避免丢分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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