📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分数
In Key Stage 3 mathematics, understanding the relationship between fractions, decimals, and percentages is absolutely essential. This foundational knowledge not only unlocks many other areas of maths such as ratio, proportion, and algebra, but it also prepares you for everyday calculations like discounts, interest rates, and data interpretation. In this article, we will explore how to convert between these three forms, perform operations with them, and apply them to real-world problems. By the end, you should feel confident moving seamlessly between fractions, decimals, and percentages.
在 KS3 数学中，理解分数、小数和百分数之间的关系至关重要。这一基础知识不仅能帮助你掌握比例、比率和代数等数学领域，还为你进行折扣、利率和数据分析等日常计算做好准备。本文将探讨如何在这三种形式之间进行转换、如何对它们进行运算，以及如何将其应用于实际问题。通过本文的学习，你将能够自信地在分数、小数和百分数之间灵活切换。
1. What Are Fractions, Decimals, and Percentages? | 什么是分数、小数和百分数？
A fraction represents a part of a whole and is written as a numerator over a denominator, such as ½ or ⅜. The denominator shows how many equal parts the whole is divided into, and the numerator shows how many of those parts we have.
分数表示整体的一部分，写作分子除以分母的形式，例如 ½ 或 ⅜。分母表示整体被分成多少等份，分子表示我们拥有其中的几份。
A decimal uses a decimal point and place value to represent parts of a whole. For example, 0.5 means 5 tenths, and 0.25 means 25 hundredths. Every fraction can be written as a decimal by dividing the numerator by the denominator.
小数使用小数点及其数位来表示整体的一部分。例如，0.5 表示十分之五，0.25 表示百分之二十五。每个分数都可以通过分子除以分母转化为小数。
A percentage is a special way of expressing a number as a fraction of 100. The symbol % means ‘per hundred’. So 50% means 50 out of 100, or ½. Percentages are useful for comparing proportions quickly.
百分数是一种将数字表示为 100 分比的特殊方式。符号 % 意为“每一百”。所以 50% 表示 100 份中的 50 份，即 ½。百分数便于快速比较比例大小。
2. Converting Fractions to Decimals | 分数转化为小数
To convert any fraction to a decimal, simply divide the numerator by the denominator using short or long division. For instance, ⅜ becomes 3 ÷ 8 = 0.375. Some fractions result in terminating decimals, while others produce recurring decimals like ⅓ = 0.333… (written with a dot or bar over the repeating digit).
要将分数化为小数，只需用分子除以分母，可以使用短除法或长除法。例如，⅜ 等于 3 ÷ 8 = 0.375。有些分数会得到有限小数，而有些则会产生循环小数，如 ⅓ = 0.333…（在重复数字上方加点或横线表示）。
Common fractions that you should memorise include: ½ = 0.5, ¼ = 0.25, ¾ = 0.75, ⅕ = 0.2, ⅖ = 0.4, ⅗ = 0.6, ⅘ = 0.8, ⅛ = 0.125, and so on. Knowing these by heart speeds up calculations significantly.
你应该熟记的常见分数包括：½ = 0.5、¼ = 0.25、¾ = 0.75、⅕ = 0.2、⅖ = 0.4、⅗ = 0.6、⅘ = 0.8、⅛ = 0.125 等。熟记它们可以显著提高计算速度。
If the denominator is not a factor of 10, 100, or 1000, performing division is the most reliable method. You can also convert the fraction to an equivalent one with a denominator of 10 or 100 first if possible. For example, ⁷⁄₂₀ can be multiplied by 5 to get ³⁵⁄₁₀₀ = 0.35.
如果分母不是 10、100 或 1000 的因数，那么进行除法是最可靠的方法。如果可能，也可以先将分数化为分母为 10 或 100 的等价分数。例如，⁷⁄₂₀ 乘以 5 得到 ³⁵⁄₁₀₀ = 0.35。
3. Converting Decimals to Fractions | 小数转化为分数
To turn a terminating decimal into a fraction, identify the place value of the last digit. For example, 0.6 has the last digit in the tenths place, so it is ⁶⁄₁₀, which simplifies to ⅗. 0.45 has the last digit in the hundredths place, giving ⁴⁵⁄₁₀₀ = ⁹⁄₂₀ after simplification.
要将有限小数化为分数，先确定最后一位数字所在的数位。例如，0.6 最后一位在十分位，所以是 ⁶⁄₁₀，化简后为 ⅗。0.45 最后一位在百分位，可得 ⁴⁵⁄₁₀₀，化简后为 ⁹⁄₂₀。
For recurring decimals, there is an algebraic method: Let x equal the recurring decimal, multiply by a power of 10 to place the repeating block to the left of the decimal point, then subtract the original equation to eliminate the repeating part. For example, if x = 0.333…, then 10x = 3.333…; subtracting gives 9x = 3, so x = ⅓. This technique works for any recurring pattern.
对于循环小数，可采用代数方法：令 x 等于循环小数，乘以 10 的某次幂使循环节移到小数点左侧，然后减去原方程以消去循环部分。例如，若 x = 0.333…，那么 10x = 3.333…；相减得 9x = 3，故 x = ⅓。该方法适用于任何循环节。
Practise writing decimals as fractions in simplest form. Always check if the fraction can be reduced by dividing the numerator and denominator by their greatest common divisor (GCD). This skill is vital for maintaining accuracy in proportions and probability.
练习将小数化为最简分数。务必检查分子分母是否可以除以它们的最大公因数（GCD）来约分。这项技能对于保持比例和概率问题中的准确性至关重要。
4. Converting Between Percentages and Fractions | 百分数与分数的相互转化
Since percent means ‘per hundred’, converting a percentage to a fraction is straightforward: write the percentage as a fraction with denominator 100 and simplify if possible. For example, 25% = ²⁵⁄₁₀₀ = ¼. If the percentage is a decimal like 12.5%, first write it as ¹²⁵⁄₁₀₀₀ (because 12.5/100 = 125/1000) and then simplify to ⅛.
由于百分数意为“每一百”，将百分数化为分数很简单：将百分数写成分母为 100 的分数，再尽可能化简。例如，25% = ²⁵⁄₁₀₀ = ¼。如果百分数是小数，比如 12.5%，先写成 ¹²⁵⁄₁₀₀₀（因为 12.5/100 = 125/1000），然后化简为 ⅛。
To convert a fraction to a percentage, find an equivalent fraction with denominator 100 if possible, or multiply the fraction by 100%. For ⅘, multiply 4/5 × 100% = 400/5 % = 80%. Alternatively, ⅘ = 0.8 and 0.8 × 100% = 80%.
要将分数化为百分数，尽可能找到分母为 100 的等价分数，或者将分数乘以 100%。对于 ⅘，计算 4/5 × 100% = 400/5% = 80%。或者，⅘ = 0.8，0.8 × 100% = 80%。
Mastering these conversions allows you to choose the most convenient form for solving problems. For instance, comparing ⅗ and 62% becomes easy if you convert both to percentages (60% vs 62%) or to decimals (0.6 vs 0.62).
掌握这些转换可以帮助你在解题时选择最方便的形式。例如，比较 ⅗ 和 62% 时，如果都转化为百分数（60% 对比 62%）或小数（0.6 对比 0.62）就非常容易。
5. Converting Between Decimals and Percentages | 小数与百分数的相互转化
Converting a decimal to a percentage is as simple as multiplying by 100 and adding the % sign: 0.45 × 100 = 45%. For decimals less than 1, you might need to move the decimal point two places to the right. Thus, 0.03 becomes 3%, and 0.175 becomes 17.5%.
将小数化为百分数只需乘以 100 并加上百分号：0.45 × 100 = 45%。对于小于 1 的小数，只需将小数点向右移动两位。因此，0.03 变成 3%，0.175 变成 17.5%。
To change a percentage back to a decimal, divide by 100, which is equivalent to moving the decimal point two places to the left. 65% = 0.65, 8% = 0.08, and 125% = 1.25. Remember that percentages can be greater than 100% if the part is larger than the whole.
要将百分数转回小数，除以 100，也就是将小数点向左移动两位。65% = 0.65、8% = 0.08、125% = 1.25。请记住，如果部分大于整体，百分数可以大于 100%。
Because this conversion is so direct, many problems involving percentage increase or decrease are best solved by first converting the percentage to a decimal multiplier, as we will see later.
由于这种转换非常直接，许多涉及百分数增减的问题通常可以先将其转化为小数乘数来求解，我们稍后将看到这一点。
6. Ordering and Comparing Fractions, Decimals, and Percentages | 分数、小数和百分数的排序与比较
To compare a mixed set of fractions, decimals, and percentages, convert all numbers to the same form—usually decimals or percentages—and then compare them digit by digit. For example, arrange 0.3, ⅖, and 30% in ascending order. ⅖ = 0.4, 30% = 0.3, so the order is 0.3 (or 30%), 0.3 (wait, both are 0.3) and 0.4, so both 0.3 and 30% are equal and smaller than ⅖. The ascending order is 0.3 = 30% < ⅖.
要比较一组混合的分数、小数和百分数，可先将所有数转换为同一种形式——通常为小数或百分数——然后逐位进行比较。例如，将 0.3、⅖ 和 30% 按升序排列。⅖ = 0.4，30% = 0.3，因此顺序为 0.3（即 30%）小于 0.4，即 0.3 = 30% < ⅖。
On a number line, these values represent the same position if they are equivalent. Drawing a number line from 0 to 1 marked with key fractions and their decimal/percentage equivalents helps build intuition. Mark 0, ½ (0.5/50%), 1. Practise plotting numbers like 0.25, 75%, and ⁴⁄₅ to develop fluency.
在数轴上，如果这些值相等，它们表示相同的位置。画一条从 0 到 1 的数轴，并标出关键分数及其小数/百分数等价形式，有助于建立直觉。标出 0、½（0.5/50%）、1。练习标注 0.25、75% 和 ⁴⁄₅ 等数字，以提高熟练度。
When the denominators are different, avoid converting to fractions with common denominators unless necessary; decimals are often faster. However, if fractions are simple, finding a common denominator is still a valid and visual method.
当分母不同时，除非必要，尽量避免转换为同分母分数；使用小数通常更快。不过，如果分数简单，寻找公分母仍然是一种有效且直观的方法。
7. Adding and Subtracting Fractions | 分数的加法和减法
To add or subtract fractions with the same denominator, simply add or subtract the numerators and keep the denominator the same. For example, ³⁄₈ + ²⁄₈ = ⁵⁄₈. Always simplify the result if possible. If the answer is an improper fraction, you may be asked to write it as a mixed number.
要对同分母分数进行加减，只需将分子相加或相减，分母保持不变。例如，³⁄₈ + ²⁄₈ = ⁵⁄₈。始终尽可能化简结果。如果答案为假分数，题目可能要求将其写为带分数。
When denominators are different, first find the least common denominator (LCD). For ⅓ + ¼, the LCD of 3 and 4 is 12. Rewrite each fraction: ⅓ = ⁴⁄₁₂ and ¼ = ³⁄₁₂. Then add: ⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂. The same steps apply for subtraction.
当分母不同时，首先找出最小公分母（LCD）。对于 ⅓ + ¼，3 和 4 的最小公分母是 12。重写每个分数：⅓ = ⁴⁄₁₂，¼ = ³⁄₁₂。然后相加：⁴⁄₁₂ + ³⁄₁₂ = ⁷⁄₁₂。减法遵循同样的步骤。
With mixed numbers, add the whole numbers and the fractions separately, or convert to improper fractions first. For example, 1⅛ + 2¼. Convert to improper: ⁹⁄₈ + ⁹⁄₄ = ⁹⁄₈ + ¹⁸⁄₈ = ²⁷⁄₈ = 3⅜. Choose the method you find most reliable and practise it until it becomes automatic.
对于带分数，可分别相加整数部分和分数部分，或者先转化为假分数。例如，1⅛ + 2¼。转化为假分数：⁹⁄₈ + ⁹⁄₄ = ⁹⁄₈ + ¹⁸⁄₈ = ²⁷⁄₈ = 3⅜。选择你认为最可靠的方法并反复练习，直至熟练掌握。
8. Multiplying and Dividing Fractions | 分数的乘法和除法
Multiplication of fractions is straightforward: multiply the numerators together and the denominators together. Simplify before or after multiplication to keep numbers manageable. For ⅔ × ⅘, multiply: 2 × 4 = 8 and 3 × 5 = 15, giving ⁸⁄₁₅. This cannot be simplified further.
分数乘法很简单：分子相乘、分母相乘。在乘之前或之后约分可以使数字更易处理。对于 ⅔ × ⅘，相乘得 2×4=8，3×5=15，得到 ⁸⁄₁₅。它无法再化简。
For mixed numbers, convert to improper fractions first: 1½ × 2⅔ becomes ³⁄₂ × ⁸⁄₃ = ²⁴⁄₆ = 4. Cancelling common factors before multiplying saves time: ³⁄₂ × ⁸⁄₃, cancel 3s and cancel 2 into 8, giving 1/1 × 4/1 = 4.
对于带分数，先转化为假分数：1½ × 2⅔ 变成 ³⁄₂ × ⁸⁄₃ = ²⁴⁄₆ = 4。乘之前约去公因数可节省时间：³⁄₂ × ⁸⁄₃，约去 3，并将 2 与 8 约分，得到 1/1 × 4/1 = 4。
To divide by a fraction, multiply by its reciprocal (flip the second fraction). So ⅜ ÷ ½ becomes ⅜ × ²⁄₁ = ⁶⁄₈ = ¾. If working with mixed numbers, convert them to improper fractions before finding the reciprocal. Practice a variety of problems to build confidence with the ‘keep-change-flip’ rule.
除以一个分数，等于乘以它的倒数（将第二个分数翻转）。因此，⅜ ÷ ½ 变成 ⅜ × ²⁄₁ = ⁶⁄₈ = ¾。如果涉及带分数，在求倒数之前先转化为假分数。练习各种题目，熟悉“保留-变号-翻转”法则。
9. Finding a Percentage of a Quantity | 求一个数量的百分之几
To find a percentage of a given amount, first convert the percentage to a decimal or fraction, then multiply. For example, 15% of 200 = 0.15 × 200 = 30. Using fractions can be elegant: 15% = ¹⁵⁄₁₀₀ = ³⁄₂₀, then ³⁄₂₀ × 200 = (3 × 200) ÷ 20 = 600 ÷ 20 = 30.
要找出一个数量的百分之几，先将百分数转化为小数或分数，然后相乘。例如，200 的 15% = 0.15 × 200 = 30。使用分数也很巧妙：15% = ¹⁵⁄₁₀₀ = ³⁄₂₀，然后 ³⁄₂₀ × 200 = (3×200) ÷ 20 = 600 ÷ 20 = 30。
For mentally calculating common percentages, learn the 10% rule: find 10% by dividing by 10, then scale up or down. 20% is double 10%, 5% is half of 10%. For 17.5%, you could combine 10%, 5%, and 2.5% (half of 5%). This method improves number sense.
要进行常见百分数的心算，可学习 10% 法则：先除以 10 得到 10%，再据此缩放。20% 是 10% 的两倍，5% 是 10% 的一半。对于 17.5%，你可以组合 10%、5% 和 2.5%（5% 的一半）。这种方法可以提升数感。
Word problems often involve finding percentages of quantities: discount prices, tax, tips, and interest. Set up the calculation methodically: determine the decimal multiplier, apply it, and confirm whether the result makes sense in context.
应用题中经常需要求某个数量的百分之几：折扣价格、税费、小费和利息。有条理地建立计算过程：确定小数乘数，应用乘数，并验证结果在相应情境下是否合理。
10. Percentage Increase and Decrease | 百分数的增加与减少
When a quantity is increased by a percentage, you add the percentage to 100% to find the new total as a percentage of the original. For example, a 15% increase means the new value is 100% + 15% = 115% of the original. The decimal multiplier is 115% ÷ 100 = 1.15.
当某个数量增加一个百分数时，你将此百分数与 100% 相加，得到新数相当于原数的百分比。例如，增加 15% 意味着新值是原数的 100% + 15% = 115%。小数乘数为 115% ÷ 100 = 1.15。
For a decrease, subtract the percentage from 100%. A 20% decrease leaves 80%, giving a multiplier of 0.8. To apply: original amount × multiplier = new amount. For a £50 shirt with a 20% discount, £50 × 0.8 = £40.
对于减少，则从 100% 中减去该百分数。减少 20% 后剩余 80%，乘数为 0.8。使用时：原数 × 乘数 = 新数。一件 50 英镑的衬衫打八折，£50 × 0.8 = £40。
To find the percentage change, use the formula: (change ÷ original) × 100%. If a price rises from £80 to £100, the change is £20, so percentage increase = (20 ÷ 80) × 100% = 25%. If it falls from £100 to £80, the percentage decrease = (20 ÷ 100) × 100% = 20%. Be careful with the original value in the denominator.
要计算百分比变化，可使用公式：(变化量 ÷ 原数) × 100%。若价格从 £80 涨到 £100，变化量为 £20，则百分比增加 = (20 ÷ 80) × 100% = 25%。如果从 £100 降到 £80，百分比减少 = (20 ÷ 100) × 100% = 20%。注意分母应使用原始值。
11. Reverse Percentages | 逆向百分数
Reverse percentage problems ask you to find the original amount before a percentage increase or decrease. If a price includes a 20% increase, the given amount represents 120% of the original. To find 100%, divide by 120 and multiply by 100, or simply divide by the multiplier 1.2.
逆向百分数问题要求你找出百分数增减之前的原始数量。如果价格包含了 20% 的涨幅，那么给出的金额相当于原始值的 120%。要找出 100%，可以除以 120 再乘以 100，或者直接除以乘数 1.2。
For example, a sale price of £54 is after a 10% discount. So £54 represents 90% of the original. Original = £54 ÷ 0.9 = £60. Avoid the common mistake of taking 10% of £54 and adding it back; that does not give the correct original because the discount was applied to a larger amount.
例如，折后价 £54 是在打九折后的价格。因此 £54 代表原始值的 90%。原始价格 = £54 ÷ 0.9 = £60。避免常见的错误，即先将 £54 的 10% 再加回去；这样做无法得出正确的原始值，因为折扣是在更大的金额上计算的。
Set up the problem by identifying the percentage the given amount corresponds to, then scale to 100%. Using unitary method (finding 1% first) is a reliable technique: divide by the percentage to get 1%, then multiply by 100.
解决此类问题时，先确定给定金额对应的百分比，再缩放至 100%。使用归一法（先求 1%）是一种可靠的技术：除以相应百分数得到 1%，再乘以 100。
12. Real‑World Applications and Problem‑Solving | 实际应用与问题解决
Fractions, decimals, and percentages appear everywhere—from cooking recipes to financial literacy. Being able to scale ingredients using fractions, calculate discounts and VAT, interpret statistics and probabilities, or understand interest rates are critical life skills developed in KS3 maths.
分数、小数和百分数无处不在——从烹饪食谱到金融知识。能够使用分数调整配料用量、计算折扣和增值税、解读统计和概率，或者理解利率，这些都是 KS3 数学培养的重要生活技能。
When solving word problems, follow a strategy: read carefully to identify what you know and what is asked; convert all numbers to a consistent form; decide which operation is needed; perform the calculation; and finally check your answer is reasonable. For example, a recipe for 8 people requires 250g of flour, but you need it for 20 people. Scaling factor = 20/8 = 2.5, so flour needed = 250g × 2.5 = 625g.
解应用题时可采用以下策略：仔细阅读，明确已知和未知信息；将所有数字转换为同一种形式；确定所需的运算；进行计算；最后检查答案是否合理。例如，一份供 8 人食用的食谱需要 250 克面粉，而你需要为 20 人准备。比例因子 = 20/8 = 2.5，因此所需面粉 = 250g × 2.5 = 625g。
Probability problems often use fractions, decimals, and percentages. If the probability of rain is 0.3, that is 30% or ³⁄₁₀. Understanding these equivalent representations helps you communicate risk and compare likelihoods more effectively.
概率问题常使用分数、小数和百分数。如果下雨的概率是 0.3，那就是 30% 或 ³⁄₁₀。理解这些等价表示有助于你更有效地沟通风险和比较可能性。
Regular practice with mixed problem sets will strengthen your ability to switch between these forms and apply them in unfamiliar contexts. Aim to solve a variety of problems from textbooks, online resources, and past papers to build speed and accuracy.
通过混合问题集的常规练习，可以加强你在不同形式间切换的能力，并将其应用于陌生情境。建议从教科书、在线资源和历年试卷中选取各种题目进行练习，以提高速度和准确性。
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