📚 Fractions, Decimals and Percentages Mastery | 分数、小数和百分数全掌握
Welcome to your comprehensive guide on fractions, decimals and percentages. These three concepts are closely linked and form the foundation of many areas in Key Stage 3 mathematics. By the end of this article, you will be able to convert between them, order them, perform basic operations and apply them to everyday situations with confidence.
欢迎来到分数、小数和百分数的全面指南。这三个概念紧密相连，构成了关键阶段三（KS3）数学许多领域的基础。读完本文后，你将能够在它们之间轻松转换、排序、进行基本运算，并自信地将其应用于日常场景中。
1. What are Fractions? | 什么是分数？
A fraction represents a part of a whole or, more generally, any number of equal parts. The top number is called the numerator, which tells us how many parts we have. The bottom number is the denominator, which shows into how many equal parts the whole is divided. For example, if a pizza is cut into 8 equal slices and you eat 3, you have eaten 3/8 of the pizza. Here, 3 is the numerator and 8 is the denominator.
分数表示整体的一部分，或者更一般地说，任意数量的等份。上面的数称为分子，它告诉我们有多少份。下面的数是分母，表示整体被分成多少等份。例如，一个披萨被切成8等份，你吃了3块，那么你就吃了这个披萨的3/8。这里3是分子，8是分母。
- A proper fraction has a numerator smaller than the denominator, such as ⅔ or ½. 真分数的分子比分母小，例如⅔或½。
- An improper fraction has a numerator equal to or larger than the denominator, such as 5/4 or 7/3. 假分数的分子等于或大于分母，例如5/4或7/3。
- A mixed number combines a whole number and a proper fraction, such as 1½ . 带分数由整数和真分数组合而成，例如1½。
2. Equivalent Fractions | 等值分数
Equivalent fractions have different numerators and denominators but represent the same value. You can obtain an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number. For instance, ½ is equal to 2/4, 3/6 and 4/8. All these fractions show exactly half of the whole.
等值分数具有不同的分子和分母，但表示相同的值。你可以通过将分子和分母同时乘以或除以同一个非零数来得到等值分数。例如，½等于2/4、3/6和4/8。所有这些分数都恰好表示整体的一半。
To check if two fractions are equivalent, cross-multiply. If the products are equal, the fractions are equivalent. For ½ and 3/6: 1 × 6 = 6 and 2 × 3 = 6, so they are equal.
要检查两个分数是否等值，可以交叉相乘。如果乘积相等，则分数等值。对于½和3/6：1×6=6，2×3=6，因此它们相等。
a/b = c/d only if a × d = b × c
3. Simplifying Fractions | 分数化简
Simplifying a fraction means writing it in its lowest terms. To do this, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by that number. For example, to simplify 8/12, the GCD of 8 and 12 is 4. Dividing top and bottom by 4 gives 2/3. So 8/12 simplifies to ⅔.
化简分数意味着将其写成最简形式。要做到这一点，先找到分子和分母的最大公约数（GCD），然后将两者同时除以这个数。例如，要化简8/12，8和12的GCD是4。上下同时除以4得到2/3。因此8/12化简为⅔。
If a fraction is in its simplest form, the numerator and denominator have no common factor other than 1. This makes it easier to compare and perform operations. Always check if your answer can be simplified further.
如果一个分数已经是最简形式，分子和分母除了1之外没有其他公因数。这会让比较和运算更加容易。务必检查你的答案是否可以进一步化简。
4. Converting Fractions to Decimals | 分数转为小数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, ¼ becomes 1 ÷ 4 = 0.25. If the denominator is a power of 10 like 10, 100 or 1000, the conversion is simple: 3/10 = 0.3, 47/100 = 0.47.
要将分数转换为小数，用分子除以分母。例如，¼变成1÷4=0.25。如果分母是10、100或1000这样的10的幂，转换就很简单：3/10=0.3，47/100=0.47。
Some fractions give terminating decimals, such as ⅛ = 0.125. Others produce recurring decimals, like ⅓ = 0.333… which we write as 0.3̅. Recognising these patterns helps in quick conversions.
有些分数会得到有限小数，例如⅛=0.125。另一些则产生循环小数，比如⅓=0.333……我们记作0.3̅。识别这些模式有助于快速转换。
| Fraction | Decimal |
|---|---|
| ½ | 0.5 |
| ⅓ | 0.333… (0.3̅) |
| ⅔ | 0.666… (0.6̅) |
| ¼ | 0.25 |
| ¾ | 0.75 |
| ⅕ | 0.2 |
5. Converting Decimals to Fractions | 小数转分数
To change a decimal into a fraction, write the decimal as the numerator over a denominator that is a power of 10, then simplify. For a terminating decimal, count the number of decimal places. For example, 0.7 has one decimal place, so write it as 7/10. 0.25 has two decimal places, so it becomes 25/100, which simplifies to ¼.
要将小数转换为分数，把小数写成分子，分母为10的幂，然后化简。对于有限小数，数一数小数位数。例如，0.7有一位小数，因此写成7/10。0.25有两位小数，所以变成25/100，化简为¼。
For recurring decimals, a slightly different approach is needed. For instance, 0.3̅ (0.333…) equals ⅓. You can prove this by letting x = 0.333…, then 10x = 3.333… Subtract x to get 9x = 3, so x = 3/9 = ⅓.
对于循环小数，需要略微不同的方法。例如，0.3̅（0.333…）等于⅓。你可以这样证明：设x = 0.333…，那么10x = 3.333…。相减得到9x=3，因此x=3/9=⅓。
6. Understanding Percentages | 理解百分数
A percentage is a fraction with a denominator of 100. The symbol % means ‘out of 100’. So 25% literally means 25 per 100, or 25/100, which simplifies to ¼. Percentages are used everywhere: scores, discounts, interest rates and statistics. Being comfortable with percentages allows you to quickly compare different quantities.
百分数是分母为100的分数。符号%表示“每一百”，即“百分之”。因此25%的字面意思是25/100，化简为¼。百分数到处都有使用：分数、折扣、利率和统计数据。熟练掌握百分数可以让你快速比较不同的数量。
To find a percentage of an amount, multiply the amount by the percentage divided by 100. For example, 20% of 60 is (20 ÷ 100) × 60 = 0.2 × 60 = 12. You can also use the fraction equivalent: 20% = 1/5, so 1/5 of 60 = 12.
要计算一个数量的百分之几，用该数量乘以百分数除以100。例如，60的20%是(20÷100)×60=0.2×60=12。你也可以使用等值分数：20%=1/5，所以60的1/5=12。
7. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数之间的转换
The three representations are interconvertible. The key relationships to remember are: decimal to percent – multiply by 100; percent to decimal – divide by 100; fraction to percent – convert to decimal first, then multiply by 100. For example, ⅖ as a decimal is 0.4, and as a percentage it is 40%.
这三种表示形式可以相互转换。需要记住的关键关系是：小数转百分数——乘以100；百分数转小数——除以100；分数转百分数——先转为小数，再乘以100。例如，⅖作为小数是0.4，作为百分数则是40%。
The table below provides common equivalents that are worth memorising to speed up your work.
下表提供了一些值得记忆的常见等值关系，有助于提高做题速度。
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅓ | 0.333… | 33⅓% |
| ⅔ | 0.666… | 66⅔% |
| ⅕ | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
8. Comparing and Ordering | 比较和排序
When you have a mix of fractions, decimals and percentages, the easiest way to compare them is to convert all numbers into the same form – usually decimals. Place them on a number line or simply compare their decimal values. For example, order ⅗, 0.55, and 52% from smallest to largest. ⅗ = 0.6, 52% = 0.52, so the order is 52% (0.52), 0.55, ⅗ (0.6).
当你遇到分数、小数和百分数混合在一起时，最简单的比较方法是将所有数转换成同一种形式——通常是小数。把它们放在数轴上，或者直接比较它们的小数值。例如，将⅗、0.55和52%从小到大排序。⅗=0.6，52%=0.52，因此顺序是52% (0.52)、0.55、⅗ (0.6)。
When comparing fractions without a calculator, convert them to have a common denominator. For ⅔ and ¾, the common denominator is 12: ⅔ = 8/12 and ¾ = 9/12, so ¾ is larger. This method is especially useful in KS3 exams.
当不用计算器比较分数时，可以将它们通分。对于⅔和¾，公分母是12：⅔=8/12，¾=9/12，因此¾更大。这种方法在KS3考试中尤其有用。
9. Adding and Subtracting Fractions | 分数的加减法
To add or subtract fractions, they must have the same denominator. If the denominators are already the same, simply add or subtract the numerators while keeping the denominator unchanged. For instance, ⅛ + 3/8 = 4/8, which simplifies to ½.
要加减分数，它们必须有相同的分母。如果分母已经相同，只需将分子相加或相减，分母保持不变。例如，⅛ + 3/8 = 4/8，化简为½。
If the denominators are different, find the least common denominator (LCD). The LCD is the lowest common multiple of the denominators. For ⅓ + ¼ , the LCD of 3 and 4 is 12. Rewrite as 4/12 + 3/12 = 7/12. Always simplify your final answer if possible.
如果分母不同，找到最小公分母（LCD）。LCD是各分母的最小公倍数。对于⅓ + ¼，3和4的LCD是12。重写为4/12 + 3/12 = 7/12。如果可能，始终将最终答案化简。
For mixed numbers, add the whole parts and fraction parts separately, or convert to improper fractions first. Example: 1½ + 2⅓ = (1+2) + (½+⅓) = 3 + (3/6+2/6) = 3 5/6.
对于带分数，可以分别将整数部分和分数部分相加，或者先转换为假分数。例如：1½ + 2⅓ = (1+2) + (½+⅓) = 3 + (3/6+2/6) = 3 5/6。
a/b + c/d = (a×d + b×c) / (b×d)
10. Real-life Problems | 实际生活中的问题
Fractions, decimals and percentages appear in many everyday contexts. In a sale, a 20% discount on a £40 item can be calculated by finding 10% first (£4) and doubling it to get £8 off, so you pay £32. Alternatively, 20% = 0.2, so discount = 0.2 × 40 = £8.
分数、小数和百分数出现在许多日常场景中。在一次打折活动中，一件40英镑的商品打20%的折扣，可以先算出10%是4英镑，再翻倍得到优惠8英镑，因此你支付32英镑。或者，20%=0.2，那么折扣=0.2×40=8英镑。
Recipes often use fractions: adding ½ cup of sugar to ⅓ cup of milk requires finding the total volume as 5/6 cup. Measurements in construction use decimals and fractions together, making fluency essential.
食谱中经常用到分数：加入½杯糖和⅓杯牛奶需要求出总体积为5/6杯。建筑中的测量同时使用小数和分数，因此熟练转换至关重要。
In statistics, probabilities are often written as fractions, decimals or percentages. For instance, the chance of rolling a 3 on a fair dice is 1/6 = 0.1666… ≈ 16.7%. Understanding all three forms ensures you interpret data correctly.
在统计学中，概率通常写成分数、小数或百分数。例如，掷一个公平的骰子得到3点的机会是1/6 = 0.1666… ≈ 16.7%。理解这三种形式能确保你正确解读数据。
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