📚 Mastering Fractions, Decimals and Percentages | 精通分数、小数和百分比
Fractions, decimals and percentages are three connected ways of writing parts of a whole. In KS3 Mathematics, you need to move smoothly between these forms and apply them to solve real-world problems. This article covers all the essential skills and provides step-by-step methods to build your confidence.
分数、小数和百分比是表示部分与整体的三种相互关联的方式。在 KS3 数学中，你需要在这三种形式之间自如转换，并运用它们解决实际问题。本文涵盖所有核心技能，并提供循序渐进的讲解，帮助你建立信心。
1. Understanding the Three Forms | 理解三种表达形式
A fraction, such as 3/4, represents a number of equal parts out of a total number of parts. A decimal uses a point to separate whole numbers from tenths, hundredths and thousandths, for example 0.75. A percentage means “per hundred” and is written with the % symbol, so 75% literally means 75 out of 100.
分数（如 3/4）表示将整体等分后取出若干份。小数用小数点将整数部分与十分位、百分位和千分位分开，例如 0.75。百分比意为“每一百份”，写作带有 % 符号的形式，因此 75% 的字面意思就是 100 份中的 75 份。
All three express the same relative amount. 3/4 = 0.75 = 75%. Recognizing this equivalence is the foundation for conversion, comparison and problem solving.
这三种形式表达的是同一个相对量。3/4 = 0.75 = 75%。认识这种等价关系是进行转换、比较和解决问题的基础。
2. Converting Fractions to Decimals | 分数转换为小数
To convert any fraction to a decimal, divide the numerator by the denominator. For 3/8, perform the division 3 ÷ 8 = 0.375. With a calculator you can get the answer directly, but it is good practice to understand the process by long division for simpler fractions.
将分数转换为小数，用分子除以分母。对于 3/8，计算 3 ÷ 8 = 0.375。使用计算器可以直接得到结果，但对于简单的分数，理解长除法的过程是很好的练习。
If the denominator is 10, 100 or 1000, the conversion is immediate: 7/10 = 0.7, 23/100 = 0.23, 9/1000 = 0.009.
如果分母是 10、100 或 1000，转换是即时的：7/10 = 0.7，23/100 = 0.23，9/1000 = 0.009。
Fraction → Decimal: Numerator ÷ Denominator
分数 → 小数：分子 ÷ 分母
3. Converting Decimals to Fractions | 小数转换为分数
Look at the place value of the last digit. If the decimal is 0.6, the 6 is in the tenths place, so write it as 6/10, then simplify to 3/5. For 0.25, the last digit is in the hundredths place: 25/100 = 1/4. For 0.125, the last digit (5) is in the thousandths place: 125/1000 = 1/8.
观察最后一位数字的数位。如果小数是 0.6，6 在十分位，因此写成 6/10，然后化简为 3/5。对于 0.25，最后一位在百分位：25/100 = 1/4。对于 0.125，最后一位（5）在千分位：125/1000 = 1/8。
Always simplify the fraction by dividing the numerator and denominator by their highest common factor.
始终通过分子分母同除以它们的最大公因数来化简分数。
4. Converting Percentages to Decimals and Fractions | 百分比转换为小数和分数
Percent means “per 100”. To change a percentage to a decimal, divide by 100. This is the same as moving the decimal point two places to the left. 65% = 65 ÷ 100 = 0.65. For a percentage like 7%, write 7/100 = 0.07, and 2.5% becomes 0.025.
百分比意为“每一百”。将百分比转换为小数，除以 100。这与将小数点向左移动两位相同。65% = 65 ÷ 100 = 0.65。对于像 7% 这样的百分数，写作 7/100 = 0.07，而 2.5% 变成 0.025。
To write a percentage as a fraction, put the number over 100 and simplify. 80% = 80/100 = 4/5. If the percentage includes a fraction, such as 33⅓%, first convert to an improper fraction (100/3 %) then divide by 100: (100/3) / 100 = 1/3.
要将百分比写成分数，将数字放在 100 上面并化简。80% = 80/100 = 4/5。如果百分比包含分数，如 33⅓%，先转换为假分数（100/3 %），然后除以 100：(100/3) / 100 = 1/3。
5. Converting Decimals and Fractions to Percentages | 小数和分数转换为百分比
To change a decimal to a percentage, multiply by 100. This moves the decimal point two places to the right. 0.27 × 100 = 27%. For a decimal with more digits, like 0.375, multiply by 100 to get 37.5%.
要将小数转换为百分比，乘以 100。这将小数点向右移动两位。0.27 × 100 = 27%。对于位数更多的小数，如 0.375，乘以 100 得到 37.5%。
To turn a fraction into a percentage, first convert it to a decimal by dividing, then multiply by 100. 3/5 = 0.6 → 60%. Alternatively, find an equivalent fraction with denominator 100: 3/5 = 60/100 = 60%.
要将分数转换为百分比，先通过除法转换为小数，再乘以 100。3/5 = 0.6 → 60%。或者，找出分母为 100 的等价分数：3/5 = 60/100 = 60%。
6. Common Fraction, Decimal and Percentage Equivalents | 常见分数、小数和百分比对照
Memorising the most common equivalents helps you work faster and check your answers. The table below shows key values you will use repeatedly.
熟记最常见的对应关系可以帮助你更快解题并检查答案。下表列出了你会反复用到的重要数值。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333… (0.3̄) | 33⅓% |
| 1/8 | 0.125 | 12.5% |
7. Ordering and Comparing Values | 数值的排序与比较
When asked to place fractions, decimals and percentages in order, convert them all to the same form. Often decimals are easiest for comparison. For example, order 3/5, 0.57, 58% from smallest to largest. Convert: 3/5 = 0.6, 58% = 0.58. In decimals: 0.57, 0.58, 0.6 → 0.57, 58%, 3/5.
当需要将分数、小数和百分比排序时，先将它们转换为同一种形式。通常转换为小数最便于比较。例如，将 3/5、0.57、58% 从小到大排列。转换：3/5 = 0.6，58% = 0.58。用小数表示为：0.57、0.58、0.6 → 0.57、58%、3/5。
Always double-check by converting back to check the order makes sense.
始终通过回代检查顺序是否合理。
8. Finding a Percentage of a Quantity | 求一个量的百分之几
To find a percentage of an amount, change the percentage to a decimal and multiply. To find 15% of 240 kg, calculate 0.15 × 240 = 36 kg. You can also use a fraction method: 15% = 15/100 = 3/20, then (3/20) × 240 = 36.
求一个量的百分之几，将百分比转换为小数再相乘。计算 240 kg 的 15%，即 0.15 × 240 = 36 kg。也可以使用分数方法：15% = 15/100 = 3/20，然后 (3/20) × 240 = 36。
For more difficult percentages like 17.5%, break it into smaller known parts: 10% + 5% + 2.5%.
对于像 17.5% 这样较难的百分比，可拆分为较小的已知部分：10% + 5% + 2.5%。
9. Percentage Increase and Decrease | 百分比的增加与减少
When a quantity increases by a percentage, multiply by (1 + the decimal form of the percentage). An increase of 20% means the new amount is 120% of the original, so multiply by 1.2. For a decrease, multiply by (1 – percentage as a decimal). A 15% decrease uses a multiplier of 0.85.
当一个量增加某个百分比时，乘以（1 + 百分比的小数形式）。增加 20% 意味着新数量是原数量的 120%，因此乘以 1.2。对于减少，乘以（1 − 百分比的小数形式）。减少 15% 使用的乘数为 0.85。
Example: A coat costs £80 before a 30% reduction. New price = 80 × 0.70 = £56.
例题：一件大衣降价 30% 前售价为 80 英镑。新价格 = 80 × 0.70 = 56 英镑。
10. Solving Word Problems Using Percentages | 用百分比解文字题
Many real-life problems involve finding percentages, percentage change or reverse percentages. A typical reverse percentage question: After a 20% increase, a bicycle costs £360. What was the original price? The new price represents 120% of the original, so original = 360 ÷ 1.20 = £300.
许多实际问题涉及求百分比、百分比变化或逆向求原值。一个典型的逆向百分比题目：一辆自行车涨价 20% 后售价为 360 英镑，原价是多少？新价格代表原价的 120%，因此原价 = 360 ÷ 1.20 = 300 英镑。
Identify what the 100% represents in each problem and set up the correct multiplier or division.
在每个问题中确定 100% 代表什么，并建立正确的乘法或除法。
11. Relating Ratios, Fractions and Percentages | 将比、分数和百分比联系起来
A ratio compares parts, while fractions and percentages often compare a part to the whole. If orange juice and lemonade are mixed in the ratio 3 : 2, the fraction of orange juice in the mixture is 3/(3+2) = 3/5. As a percentage, that is 60%. This allows you to easily convert between different representations in proportion problems.
比是比较各部分之间的关系，而分数和百分比常将部分与整体进行比较。如果橙汁和柠檬水按 3 : 2 的比例混合，混合物中橙汁占的分数为 3/(3+2) = 3/5。转换为百分比，即 60%。这使你能在比例问题中轻松转换不同的表达方式。
You can also use a ratio table to scale quantities up or down while maintaining the same flavour strength or concentration.
你也可以运用比例表按比例放大或缩小数量，同时保持相同的风味浓度。
12. Practice and Summary | 练习与总结
Have a go at these practice questions:
试一试以下练习题：
- Convert 7/20 to a decimal and a percentage. (Answer: 0.35, 35%)
- 将 7/20 转换为小数和百分比。（答案：0.35，35%）
- Which is larger: 3/8 or 40%? (3/8 = 0.375, 40% = 0.4, so 40% is larger)
- 哪个更大：3/8 还是 40%？（3/8 = 0.375，40% = 0.4，所以 40% 更大）
- A shirt costs £32 after a 20% discount. What was the original price? (32 ÷ 0.8 = £40)
- 一件衬衫打八折后售价 32 英镑。原价是多少？（32 ÷ 0.8 = 40 英镑）
Mastering fractions, decimals and percentages is about knowing the conversion rules, practicing until they become automatic, and understanding the underlying idea of part-to-whole relationships. Keep a reference table of common equivalents and you will tackle KS3 problems with confidence.
精通分数、小数和百分比的关键在于掌握转换规则、反复练习直到熟练，并理解“部分与整体”这一核心思想。保存一份常见等价关系对照表，你就能自信地应对 KS3 中的各类题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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