📚 Converting Fractions, Decimals and Percentages | 分数、小数和百分数转换
Numbers can be expressed in different forms depending on the context. In mathematics and daily life, three representations appear most often: fractions, decimals, and percentages. A fraction shows a part of a whole using a numerator and a denominator. A decimal uses a decimal point and place values. A percentage represents a number out of 100 using the % symbol. These three forms are deeply connected, and being able to switch between them is a fundamental skill in IGCSE Edexcel Mathematics. This article will take you through every conversion step by step, with clear examples and common pitfalls to avoid.
数字可以根据上下文以不同的形式表示。在数学和日常生活中，最常见的三种表示法是分数、小数和百分数。分数用分子和分母表示整体的一部分。小数使用小数点与数位值。百分数则用 % 符号表示一个数占 100 的比例。这三种形式紧密相连，能否在它们之间自由转换是 IGCSE Edexcel 数学的一项基础技能。本文将逐步带你走完每一种转换方法，配以清晰的示例和需要避免的常见错误。
1. Understanding the Three Forms | 理解三种表示形式
A fraction is written as a numerator over a denominator, such as 3/4. It tells us how many equal parts of a whole we have. A decimal uses a decimal point to separate the whole part from the fractional part, for example 0.75. Each digit after the point has a place value (tenths, hundredths, thousandths, etc.). A percentage is always out of 100; 75% means 75 out of 100. All three describe the same proportion and can be converted seamlessly.
分数写作分子在分母之上，例如 3/4，它告诉我们整体被分成多少等份以及我们取了多少份。小数使用小数点将整数部分和小数部分隔开，例如 0.75，小数点后的每一位都有其数位（十分位、百分位、千分位等）。百分数始终以 100 为基准，75% 表示 100 份中的 75 份。这三者描述的是同一个比例，并可以无缝转换。
2. Converting Fractions to Decimals | 分数化小数
git or a group of digits repeats forever. It is essential to recognise common fractions that produce recurring decimals, for instance 1/3 = 0.333... and 2/11 = 0.181818... . Use long division or a calculator, but you should be able to spot patterns manually for exam questions that require exact values.
要把分数化成小数，用分子（上方）除以分母（下方）。对于像 3/8 这样的简单分数，计算 3 ÷ 8 = 0.375。有些分数会得到有限小数，有些则会得到循环小数。有限小数意味着小数在几位后终止；循环小数则意味着一个数字或一组数字永远重复出现。认识常见的产生循环小数的分数很重要，例如 1/3 = 0.333… 以及 2/11 = 0.181818…。你可以使用长除法或计算器，但在需要精确值的考题中，你应该能够手工发现规律。
3. Converting Decimals to Fractions | 小数化分数
st common factor: 65/100 ÷ 5/5 = 13/20. For mixed decimals like 2.4, write 2.4 = 2 + 0.4 = 2 + 4/10 = 2 + 2/5 = 12/5 as an improper fraction if needed. Always simplify fully — examiners expect fractions in their simplest form.
首先看小数最后一位的数位值。将该小数写作以 10 的幂为分母的分数。例如，0.65 的最后一位在百分位：0.65 = 65/100。然后将分子与分母同时除以它们的最高公因数进行约分：65/100 ÷ 5/5 = 13/20。对于像 2.4 这样的带小数，可写作 2.4 = 2 + 0.4 = 2 + 4/10 = 2 + 2/5，化成假分数则为 12/5。务必彻底约分——考官期望看到最简分数。
4. Converting Fractions to Percentages | 分数化百分数
To convert a fraction to a percentage, multiply the fraction by 100 and attach the % because a percentage is a fraction with denominator 100. For 3/5: (3/5) × 100 = 60, so 60%. If the fraction is improper, the percentage can be greater than 100: 7/4 × 100 = 175% ady a factor of 100, you can use an equivalent fraction: 9/20 = (9×5)/(20×5) = 45/100 = 45% check that your multiplication is correct and that you’ve expressed the result accurately.
要将分数化为百分数，将分数乘以 100 并加上 % 符号。这样做的原理在于百分数本身就是分母为 100 的分数。以 3/5 为例：(3/5) × 100 = 60，故为 60%。如果分数是假分数，百分数可以大于 100：7/4 × 100 = 175%。当分母恰好是 100 的因数时，你也可以利用等值分数：9/20 = (9×5)/(20×5) = 45/100 = 45%。无论采用哪种方法，都要确保乘法正确，并且结果表达准确。
5. Converting Percentages to Fractions | 百分数化分数
e as a fraction with denominator 100, then simplify. For a whole-number percentage like 65% tor and denominator by 5, giving 13/20. If the percentage includes a decimal, such as 12.5% 0, then simplify to 1/8. For percentages over 100, the fraction will be improper, e.g. 150% 150/100 = 3/2. Always present fractions in their lowest terms unless instructed otherwise.
将百分数写成分母为 100 的分数，然后约分。对于整数百分数如 65%，写作 65/100，并将分子分母同除以 5，得到 13/20。如果百分数中含有小数，如 12.5%，可先将分子分母同乘以 10 以消除小数：12.5/100 = 125/1000，然后约分为 1/8。对于超过 100 的百分数，分数将为假分数，例如 150% = 150/100 = 3/2。除非另有说明，始终将分数化为最简形式呈现。
6. Converting Decimals to Percentages | 小数化百分数
Multiply the decimal by 100 and add the % k method is simply to shift the point. For 0.47, move the point two places right to get 47%. For 0.03, you get 3%. For a decimal greater than 1, the percentage will exceed 100: 1.6 → 160%. For very small decimals, careful with zeros: 0.005 becomes 0.5%. If your decimal has many digits, multiply exactly: 0.875 × 100 = 87.5%. Never forget to include the % symbol — a percentage without % is just a number.
将小数乘以 100 并加上百分号。因为乘以 100 相当于将小数点向右移动两位，一种快捷方法就是移动小数点。对于 0.47，右移两位得到 47%。对于 0.03，得到 3%。对于大于 1 的小数，百分数将超过 100：1.6 → 160%。对于非常小的小数，要注意零的处理：0.005 变成 0.5%。若小数位数较多，应精确相乘：0.875 × 100 = 87.5%。永远不要忘记添加 % 符号——没有百分号的百分数只是一个普通数字。
7. Converting Percentages to Decimals | 百分数化小数
by 100, which is the opposite operation. Move the decimal point two places to the left. 78% becomes 0.78; 5% becomes 0.05; 0.7% becomes 0.007. If a percentage already has a decimal point, like 34.5% y 100 to get 0.345. For percentages over 100, the decimal will be greater than 1, e.g. 250% s, always convert a percentage to a decimal or fraction first to avoid mistakes in multiplication and division.
将百分数除以 100，这是之前的逆运算。将小数点向左移动两位。78% 变为 0.78；5% 变为 0.05；0.7% 变为 0.007。如果百分数本身带有小数点，如 34.5%，除以 100 后得到 0.345。对于超过 100 的百分数，小数结果大于 1，例如 250% → 2.5。在实际计算中使用百分数时，务必先将其转换为小数或分数，以避免乘除运算中的错误。
8. Converting Recurring Decimals to Fractions | 循环小数化分数
3... → 9x = 3, so x = 3/9 = 1/3. For two-digit repeating patterns, such as 0.121212..., let x = 0.12... . Multiply by 100: 100x = 12.12... . Subtract x: 100x − x = 99x = 12, giving x = 12/99 = 4/33. If the recurring part does not start immediately after the decimal point, the technique needs an extra step: for 0.1666... (0.16 with the 6 recurring), let x = 0.1666..., multiply by 100 to shift two places: 100x = 16.66..., then multiply by 10: 10x = 1.666... . Subtract to isolate the repeating part: 100x − 10x = 90x = 15, so x = 15/90 = 1/6. This method is examinable and you must show clear algebraic steps.
循环小数具有重复出现的数字模式，通常在重复数字上方加点或划线表示。对于简单的循环小数如 0.333…（通常记作 0.3，3 上加点），设 x = 0.3… 。乘以 10 得：10x = 3.3… 。将第二个方程减去第一个方程：10x − x = 3.3… − 0.3… → 9x = 3，则 x = 3/9 = 1/3。对于两位循环节，如 0.121212…，设 x = 0.12… 。乘以 100：100x = 12.12… 。减去 x：100x − x = 99x = 12，得 x = 12/99 = 4/33。如果循环部分不是紧接在小数点后开始，需要额外步骤：对于 0.1666…（0.16 且 6 循环），设 x = 0.1666…，乘以 100 以移动两位：100x = 16.66…，再乘以 10：10x = 1.666… 。相减以隔离循环部分：100x − 10x = 90x = 15，故 x = 15/90 = 1/6。这一方法属于考试范围，作答时必须清晰地展示代数步骤。
9. Common Equivalents Table | 常见等价表
Memorising these common conversions will save you time in your IGCSE exam. The table below lists fractions that frequently appear, along with their decimal and percentage equivalents. Pay special attention to thirds, quarters, fifths, eighths, and tenths.
熟记下列常见转换可为你 IGCSE 考试节省时间。下表列出了经常出现的分数及其对应的小数和百分数。请尤其注意三分之一、四分之一、五分之一、八分之一和十分之一。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333... (0.3) | 33⅓% |
| 2/3 | 0.666... (0.6) | 66⅔% |
| 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% |
| 1/10 | 0.1 | 10% |
| 1/20 | 0.05 | 5% |
10. Tips and Common Mistakes | 提示与常见错误
One of the most frequent errors is forgetting to simplify a fraction fully. After writing 0.75 as 75/100, you must reduce it to 3/4. Another common slip is misplacing the decimal point when multiplying or dividing by 100: always double-check whether the point moves left or right. When converting a percentage like 0.3% to a decimal, many students mistakenly write 0.3 instead of 0.003. When handling recurring decimals, carefully check how many digits repeat and multiply by the appropriate power of 10; using the wrong multiplier will ruin the algebra. In the exam, if you need to order a mixture of fractions, decimals and percentages, convert them all to the same form first. Also, keep an eye on mixed numbers: convert them to improper fractions before performing most conversions. Finally, use your calculator to verify your handwritten work, but always show complete working for method marks.
最常见的错误之一是忘记将分数彻底约分。在把 0.75 写成 75/100 之后，必须再约分为 3/4。另一个常见失误是在乘以或除以 100 时点错小数点：务必反复确认小数点究竟是向右还是向左移动。把类似 0.3% 的百分数化为小数时，许多学生错误地写成 0.3 而不是 0.003。处理循环小数时，应仔细查看几位数字在循环，并乘以恰当的 10 的幂；用错乘数会让代数过程彻底出错。在考试中，若需要将分数、小数和百分数混合排序，应先将它们全部转换成同一种形式。此外，请留意带分数：先将其化为假分数，再进行大多数转换。最后，可以用计算器验证纸上的结果，但一定要展示完整的解题步骤以获得方法分。
11. Real-World Applications and Checking Your Work | 实际应用与结果检验
Converting between fractions, decimals, and percentages is not just an exam skill — it is used in finance, data interpretation, and science. A 20% discount on a $45 item means you can find 20% of 45 by converting 20% to 0.2, then multiplying: 0.2 × 45 = $ tage greater than 100 always corresponds to a decimal greater than 1 or an improper fraction. If your conversion produces a percentage like 250% have likely misplaced something. Always review your answer in the context of the problem.
分数、小数和百分数之间的转换不仅是一项考试技能——它们还广泛应用于金融、数据解读和科学领域。一件 45 美元的商品打 20% 的折扣，意味着你可以把 20% 转化为 0.2，再计算 0.2 × 45 = 9 美元折扣。在统计学中，比例常以小数或百分数表示，你必须在它们之间来回切换。一个快速检验方法：大于 100 的百分数总是对应大于 1 的小数或假分数。如果你的转换得出了一个 250% 这样的百分数，但你预计的只是一小部分，那么很可能某些地方搞错了。一定要结合问题的上下文审视你的答案。
12. Summary and Revision Checklist | 总结与复习清单
To master this topic, ensure you can:
要掌握这一主题，请确保你能做到以下各点：
- Convert between fractions, decimals and percentages fluently in both directions. 能在分数、小数和百分数之间双向流利转换。
- Identify and correctly handle terminating and recurring decimals. 能识别并正确处理有限小数和循环小数。
- Use the algebraic method to convert any recurring decimal to a fraction in simplest form. 能运用代数方法将任何循环小数化为最简分数。
- Memorise the common equivalences provided in the table. 熟记表格中提供的常见等价关系。
- Avoid the typical mistakes outlined above, especially decimal-place misplacement. 避免上述典型错误，尤其是小数点移位的失误。
Practice with past IGCSE questions, and time yourself to ensure you can perform conversions accurately under exam pressure. Confidence in this area builds a strong foundation for percentages, ratio, proportion, and many other mathematical concepts.
用历年 IGCSE 真题进行练习，并给自己计时，以确保能在考试压力下准确完成转换。在这一领域的自信会为你学好百分数、比、比例以及许多其他数学概念奠定坚实的基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图