KS3 Cambridge Mathematics: Fractions, Decimals and Percentages | KS3 剑桥数学：分数、小数与百分数

Introduction | 引言

Fractions, decimals, and percentages are three different ways of representing parts of a whole. They form the foundation of numerical reasoning at KS3 level and appear throughout the Cambridge mathematics curriculum. Mastering the ability to convert between these three forms and apply them to real-world problems is essential for success in later topics such as ratio, proportion, algebra, and statistics.

分数、小数和百分数是表示整体的一部分的三种不同方式。它们构成了 KS3 阶段数值推理的基础，贯穿于剑桥数学课程的始终。掌握这三种形式之间的转换并能将其应用于实际问题，对于后续学习比例、代数、统计等课题至关重要。

What Are Fractions? | 什么是分数？

A fraction represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number). The denominator tells us how many equal parts the whole is divided into, and the numerator tells us how many of those parts we have. For example, in the fraction 3/4, the denominator is 4 (the whole is divided into 4 equal pieces) and the numerator is 3 (we have 3 of those pieces).

分数表示整体的一部分。它由分子（上面的数）和分母（下面的数）组成。分母告诉我们整体被分成了多少等份，分子告诉我们拥有其中多少份。例如，在分数 3/4 中，分母是 4（整体被分成 4 等份），分子是 3（我们拥有其中的 3 份）。

Types of Fractions | 分数的类型

Proper fractions: The numerator is smaller than the denominator (e.g., 2/5, 3/7). The value is always less than 1. Improper fractions: The numerator is larger than or equal to the denominator (e.g., 7/4, 9/3). The value is 1 or greater. Mixed numbers: A combination of a whole number and a proper fraction (e.g., 2 1/3, 5 3/8). Understanding these types helps you choose the right form for each calculation: improper fractions are better for multiplication and division, while mixed numbers are more readable for final answers.

真分数：分子小于分母（如 2/5、3/7），值始终小于 1。假分数：分子大于或等于分母（如 7/4、9/3），值为 1 或更大。带分数：由一个整数和一个真分数组成（如 2 1/3、5 3/8）。理解这些类型有助于你为每种计算选择合适的形式：假分数更适合乘法和除法，而带分数在最终答案中更易读。

Equivalent Fractions | 等值分数

Equivalent fractions have the same value even though they look different. For example, 1/2 = 2/4 = 3/6 = 4/8. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. This skill is crucial for simplifying fractions (reducing to lowest terms) and for finding common denominators when adding or subtracting fractions. To simplify 12/16, divide both numbers by their highest common factor (HCF) of 4 to get 3/4.

等值分数虽然看起来不同，但具有相同的数值。例如，1/2 = 2/4 = 3/6 = 4/8。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创建等值分数。这项技能对于化简分数（约分到最简形式）以及在加减分数时寻找公分母至关重要。要化简 12/16，将分子分母同时除以它们的最大公因数 4，得到 3/4。

Decimals: A Different Way to Show Parts | 小数：另一种表示部分的方式

Decimals use a decimal point and place value to represent parts of a whole. The digits after the decimal point represent tenths, hundredths, thousandths, and so on. For instance, 0.7 means 7 tenths, 0.25 means 25 hundredths (or 1/4), and 0.375 means 375 thousandths (or 3/8). Decimals are especially useful in real-world contexts where we need precise measurements, such as money (pounds and pence) and metric measurements (metres and centimetres).

小数使用小数点和位值来表示整体的一部分。小数点后的数字分别表示十分位、百分位、千分位等。例如，0.7 表示 7/10，0.25 表示 25/100（即 1/4），0.375 表示 375/1000（即 3/8）。小数在需要精确测量的实际场景中特别有用，比如货币（英镑和便士）和公制度量（米和厘米）。

Place Value for Decimals | 小数的位值

Hundreds	Tens	Units	.	Tenths	Hundredths	Thousandths
100	10	1	.	1/10	1/100	1/1000

Each column to the right of the decimal point is ten times smaller than the one before it. This consistency is what makes the decimal system (and the metric system) so powerful: you can express any quantity, no matter how small, by adding more decimal places.

小数点右侧的每一列都比前一列小十倍。这种一致性使得十进制系统（以及公制系统）非常强大：你可以通过添加更多小数位来表示任何数量，无论多么微小。

Percentages: Out of 100 | 百分数：以百为基

A percentage is simply a fraction with a denominator of 100. The symbol “%” means “per hundred” or “out of 100.” So 45% means 45 out of 100, which can be written as the fraction 45/100 or the decimal 0.45. Percentages are everywhere in daily life: test scores, discounts in shops, interest rates at banks, and statistics in the news all use percentages to make comparisons easy.

百分数就是分母为 100 的分数。符号 “%” 表示”每一百”或”百分之”。因此 45% 表示 100 份中的 45 份，可以写成分数 45/100 或小数 0.45。百分数在日常生活中无处不在：考试成绩、商店折扣、银行利率以及新闻中的统计数据都使用百分数来方便比较。

Converting Between the Three Forms | 三种形式之间的转换

Fraction to Decimal | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 means 3 divided by 8, which equals 0.375. For fractions with denominators that are powers of 10 (10, 100, 1000), you can simply write the numerator in the appropriate decimal places. For example, 7/100 = 0.07, and 23/1000 = 0.023. Common fractions like 1/2 = 0.5, 1/4 = 0.25, and 3/4 = 0.75 are worth memorising as they appear frequently in problems.

要将分数转换为小数，用分子除以分母。例如，3/8 表示 3 除以 8，等于 0.375。对于分母是 10 的幂（10、100、1000）的分数，你可以直接将分子写在相应的小数位上。例如，7/100 = 0.07，23/1000 = 0.023。像 1/2 = 0.5、1/4 = 0.25、3/4 = 0.75 这样的常见分数值得记住，因为它们在题目中频繁出现。

Decimal to Percentage | 小数转百分数

To convert a decimal to a percentage, multiply by 100 (move the decimal point two places to the right) and add the % sign. For example, 0.65 becomes 65%, 0.03 becomes 3%, and 1.5 becomes 150%. Note that percentages can be greater than 100% when you have more than the whole amount. To convert from a percentage back to a decimal, divide by 100 (move the decimal point two places to the left).

要将小数转换为百分数，乘以 100（将小数点向右移动两位）并加上 % 符号。例如，0.65 变为 65%，0.03 变为 3%，1.5 变为 150%。注意，当你拥有的数量超过整体时，百分数可以大于 100%。要从百分数转回小数，除以 100（将小数点向左移动两位）。

Percentage to Fraction | 百分数转分数

To convert a percentage to a fraction, write it as a fraction over 100 and simplify. For example, 75% = 75/100, which simplifies to 3/4 by dividing both numerator and denominator by 25. For decimal percentages such as 12.5%, first multiply by 10 to get 125/1000, then simplify to 1/8. Always reduce to the simplest form in your final answer.

要将百分数转换为分数，将其写成分母为 100 的分数，然后化简。例如，75% = 75/100，将分子分母同时除以 25 化简为 3/4。对于如 12.5% 这样带有小数的百分数，先乘以 10 得到 125/1000，然后化简为 1/8。最终答案中始终化简到最简形式。

Ordering and Comparing | 排序与比较

When you need to order a mix of fractions, decimals, and percentages, the easiest strategy is to convert them all to the same form. Usually, decimals are the most convenient for comparison because you can compare place values digit by digit. For example, to order 3/5, 0.58, and 56%, convert all to decimals: 3/5 = 0.6, 0.58 stays the same, 56% = 0.56. Now compare: 0.56 is smallest, then 0.58, then 0.6. So the order (ascending) is: 56%, 0.58, 3/5.

当你需要排序分数、小数和百分数的混合体时，最简单的策略是将它们全部转换为同一种形式。通常，小数进行比较最方便，因为你可以逐位比较位值。例如，要排序 3/5、0.58 和 56%，将它们全部转换为小数：3/5 = 0.6，0.58 不变，56% = 0.56。现在比较：0.56 最小，然后是 0.58，最后是 0.6。因此升序排列为：56%、0.58、3/5。

Finding a Percentage of an Amount | 求一个数的百分之几

This is one of the most practical skills in the topic. To find a percentage of an amount, first convert the percentage to a decimal (divide by 100), then multiply by the amount. For example, to find 15% of 200 pounds: 15% = 0.15, then 0.15 x 200 = 30 pounds. A useful shortcut: to find 10% of any number, simply divide by 10. Then 5% is half of that 10% value, 20% is double the 10% value, and so on. This “building up” method is especially helpful for mental calculations without a calculator.

这是本课题中最实用的技能之一。求一个数的百分之几，首先将百分数转换为小数（除以 100），然后乘以该数。例如，求 200 英镑的 15%：15% = 0.15，然后 0.15 x 200 = 30 英镑。一个有用的捷径：求任何数的 10% 只需除以 10。然后 5% 是该 10% 值的一半，20% 是 10% 值的两倍，以此类推。这种”逐步构建”的方法对于不用计算器的心算特别有帮助。

Percentage Increase and Decrease | 百分数的增加与减少

Percentage increase and decrease are essential for understanding price changes, population growth, and many other real-world applications. To increase an amount by a percentage, multiply by (1 + percentage as a decimal). For example, a 20% increase on 50 pounds: multiplier = 1 + 0.20 = 1.20, so the new amount is 50 x 1.20 = 60 pounds. To decrease by a percentage, multiply by (1 – percentage as a decimal). A 15% decrease on 80 pounds: multiplier = 1 – 0.15 = 0.85, so the new amount is 80 x 0.85 = 68 pounds. The multiplier method is much faster than finding the percentage amount first and then adding or subtracting.

百分数的增加与减少对于理解价格变化、人口增长以及许多其他实际应用至关重要。要使一个量增加某个百分比，乘以（1 + 小数形式的百分数）。例如，50 英镑增加 20%：倍数 = 1 + 0.20 = 1.20，所以新值为 50 x 1.20 = 60 英镑。要使一个量减少某个百分比，乘以（1 – 小数形式的百分数）。80 英镑减少 15%：倍数 = 1 – 0.15 = 0.85，所以新值为 80 x 0.85 = 68 英镑。乘法倍数法比先求百分数值再加或减要快得多。

Common Mistakes to Avoid | 常见错误与避免方法

Mistake 1: Forgetting to align decimal points when adding or subtracting. Always line up the decimal points vertically: 3.5 + 0.25 should be written with 3.50 above 0.25, not with numbers aligned on the right.

错误 1：加减时忘记对齐小数点。始终将小数点垂直对齐：3.5 + 0.25 应当写成 3.50 在上、0.25 在下，而不是将数字右对齐。

Mistake 2: Multiplying the numerator and denominator by different numbers when finding equivalent fractions. Whatever you do to the top, you must do the same to the bottom. 2/3 is equivalent to 6/9 (both multiplied by 3), not 6/12.

错误 2：求等值分数时将分子和分母乘以不同的数。对分子做了什么，对分母也必须做同样的操作。2/3 等值于 6/9（两者都乘以 3），而不是 6/12。

Mistake 3: Thinking 0.5 is the same as 0.05. Remember place value: 0.5 is five tenths (half), while 0.05 is five hundredths (one twentieth). The position of the zero matters greatly.

错误 3：认为 0.5 和 0.05 是一样的。记住位值：0.5 是十分之五（一半），而 0.05 是百分之五（二十分之一）。零的位置非常重要。

Mistake 4: Adding percentages directly in compound problems. A 10% increase followed by a 10% increase is NOT a 20% increase. The second 10% is calculated on the already-increased amount: 100 x 1.10 = 110, then 110 x 1.10 = 121. The total increase is actually 21%.

错误 4：在复合问题中直接加百分数。先增加 10% 再增加 10% 并不是增加 20%。第二个 10% 是在已经增加后的量上计算的：100 x 1.10 = 110，然后 110 x 1.10 = 121。总增加实际上是 21%。

Real-World Applications | 实际应用

The skills you learn in this topic are directly relevant to everyday situations. When you see “30% off” in a shop, you are using percentage decrease. When you split a pizza among friends, you are working with fractions. When you read your electricity meter or measure ingredients for cooking, you are using decimals. Banks use percentages to calculate interest on savings accounts. Sports statistics like batting averages in cricket or pass completion rates in football are expressed as decimals or percentages. Understanding these three forms fluently gives you the numeracy skills to navigate the world confidently.

你在本课题中学到的技能与日常场景直接相关。当你在商店看到”七折”时，你在使用百分数减少。当你和朋友分披萨时，你在使用分数。当你读电表或量取烹饪原料时，你在使用小数。银行用百分数计算储蓄账户的利息。体育统计数据如板球中的击球率或足球中的传球完成率都以小数或百分数表示。流利地理解这三种形式，赋予你自信地应对世界的数学素养。

Practice Questions | 练习题

Try these problems to test your understanding:

1. Convert 7/8 to a decimal and a percentage.
2. Find 35% of 240.
3. A jacket originally costs 45 pounds. In a sale, it is reduced by 20%. What is the sale price?
4. Order these from smallest to largest: 2/5, 45%, 0.38
5. A car’s value decreases by 15% each year. If it costs 12,000 pounds new, what is its value after one year?

尝试以下题目来检验你的理解：

1. 将 7/8 转换为小数和百分数。
2. 求 240 的 35%。
3. 一件夹克原价 45 英镑。打折时降价 20%。打折后价格是多少？
4. 将以下从小到大排序：2/5、45%、0.38
5. 一辆汽车的价值每年减少 15%。如果新车价格为 12,000 英镑，一年后价值多少？

Operations with Fractions | 分数的运算

Adding and Subtracting Fractions | 分数的加减

To add or subtract fractions, they must have the same denominator. If the denominators are already the same, simply add or subtract the numerators and keep the denominator unchanged. For example, 2/7 + 3/7 = 5/7, and 8/9 – 5/9 = 3/9 = 1/3 (always simplify your answer). When denominators are different, you must first find a common denominator. The most efficient approach is to find the lowest common multiple (LCM) of the denominators. For 1/4 + 2/5, the LCM of 4 and 5 is 20. Convert: 1/4 = 5/20, 2/5 = 8/20. Now add: 5/20 + 8/20 = 13/20. For mixed numbers, either convert to improper fractions first or work with the whole number and fractional parts separately.

要加减分数，它们必须有相同的分母。如果分母已经相同，只需加减分子，分母保持不变。例如，2/7 + 3/7 = 5/7，8/9 – 5/9 = 3/9 = 1/3（始终化简答案）。当分母不同时，你必须先找到公分母。最有效的方法是找到分母的最小公倍数（LCM）。对于 1/4 + 2/5，4 和 5 的 LCM 是 20。转换：1/4 = 5/20，2/5 = 8/20。现在相加：5/20 + 8/20 = 13/20。对于带分数，可以先转换为假分数，或者分别处理整数部分和分数部分。

Multiplying Fractions | 分数的乘法

Multiplying fractions is often easier than adding them because you do not need a common denominator. Simply multiply the numerators together and the denominators together: a/b x c/d = (a x c) / (b x d). For example, 2/3 x 4/5 = 8/15. Then simplify the result if possible. A useful shortcut: you can cancel common factors between any numerator and any denominator before multiplying. In 3/8 x 4/9, cancel 3 and 9 (divide both by 3) and cancel 4 and 8 (divide both by 4): (1/2) x (1/3) = 1/6. For mixed numbers, always convert to improper fractions first. 2 1/4 x 1 2/3 = 9/4 x 5/3 = 45/12 = 15/4 = 3 3/4.

分数乘法通常比加法更简单，因为你不需要公分母。只需分子相乘作为新分子，分母相乘作为新分母：a/b x c/d = (a x c) / (b x d)。例如，2/3 x 4/5 = 8/15。然后在可能的情况下化简结果。一个有用的捷径：你可以在乘法之前约去任何分子和任何分母之间的公因数。在 3/8 x 4/9 中，约去 3 和 9（同时除以 3），约去 4 和 8（同时除以 4）：(1/2) x (1/3) = 1/6。对于带分数，始终先转换为假分数。2 1/4 x 1 2/3 = 9/4 x 5/3 = 45/12 = 15/4 = 3 3/4。

Dividing Fractions | 分数的除法

To divide by a fraction, multiply by its reciprocal (flip the second fraction upside down). The phrase “Keep, Change, Flip” is a helpful memory aid: Keep the first fraction, Change the division sign to multiplication, Flip the second fraction. For example, 3/5 divided by 2/3 becomes 3/5 x 3/2 = 9/10. For mixed numbers, convert to improper fractions first. 1 1/2 divided by 3/4 = 3/2 divided by 3/4 = 3/2 x 4/3 = 12/6 = 2. Remember: never flip the first fraction, only the one you are dividing by.

分数除法：除以一个分数等于乘以它的倒数（将第二个分数上下颠倒）。”保留、改变、翻转”是一个有用的记忆口诀：保留第一个分数，将除号改为乘号，翻转第二个分数。例如，3/5 除以 2/3 变为 3/5 x 3/2 = 9/10。对于带分数，先转换为假分数。1 1/2 除以 3/4 = 3/2 除以 3/4 = 3/2 x 4/3 = 12/6 = 2。记住：永远不要翻转第一个分数，只翻转你除以的那个分数。

Recurring Decimals | 循环小数

Some fractions, when converted to decimals, produce digits that repeat forever. These are called recurring decimals. For example, 1/3 = 0.333333…, which we write as 0.3 with a dot above the 3. The fraction 1/6 = 0.166666… is written as 0.16 with a dot above the 6. When a block of digits repeats, such as 1/7 = 0.142857142857…, we use dots above the first and last digits of the repeating block. Recurring decimals are rational numbers — they can always be expressed exactly as fractions. To convert a recurring decimal back to a fraction, use algebraic manipulation. For example, let x = 0.363636… Multiply by 100: 100x = 36.363636… Subtract: 100x – x = 36, so 99x = 36, and x = 36/99 = 4/11.

有些分数在转换为小数时会产生永远重复的数字，这些被称为循环小数。例如，1/3 = 0.333333…，我们写成 0.3 上面加一个点。分数 1/6 = 0.166666… 写成 0.16 上面在 6 上加一个点。当一组数字重复时，如 1/7 = 0.142857142857…，我们在重复块的首尾数字上加两个点。循环小数是有理数——它们总是可以精确地表示为分数。要将循环小数转回分数，使用代数操作。例如，设 x = 0.363636…，乘以 100：100x = 36.363636…。相减：100x – x = 36，所以 99x = 36，x = 36/99 = 4/11。

Working with Ratios and Proportions | 比和比例

Fractions naturally connect to ratios and proportions, another key KS3 topic. A ratio of 3:5 means that for every 3 parts of one thing, there are 5 parts of another — a total of 8 parts. The fraction representing the first quantity is 3/8 of the whole. If a recipe uses flour and sugar in the ratio 4:1, and you have 500g of flour, you can find how much sugar you need: 500g divided by 4 = 125g of sugar. This proportional reasoning is the bridge between fractions and the broader world of algebra, where you will work with unknown quantities represented by letters.

分数自然地与比和比例相联系，这是 KS3 另一个关键课题。比 3:5 意味着每 3 份一种东西对应 5 份另一种东西——总共 8 份。表示第一个量的分数是整体的 3/8。如果一个食谱使用面粉和糖的比例为 4:1，你有 500 克面粉，可以求出需要多少糖：500g 除以 4 = 125 克糖。这种比例推理是分数与更广泛的代数世界之间的桥梁，在代数中你将用字母表示未知量。

Converting Between Fractions, Decimals and Percentages: Complete Reference | 分数、小数与百分数转换：完整参考

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/3	0.333…	33.3%
2/3	0.666…	66.7%
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%
1/100	0.01	1%

This table is worth keeping as a reference. Memorising the most common conversions (halves, quarters, fifths, eighths, tenths) will speed up your mental arithmetic considerably. Notice the pattern: fifths go up by 0.2 each time, eighths by 0.125, tenths by 0.1 — these patterns make the relationships predictable once you spot them.

这张表格值得作为参考保存下来。记住最常见的转换（二分之一、四分之一、五分之一、八分之一、十分之一）将显著加快你的心算速度。注意规律：五分之一每次增加 0.2，八分之一每次增加 0.125，十分之一每次增加 0.1——一旦发现这些规律，它们的关系就变得可预测了。

Exam Tips for KS3 Cambridge Mathematics | KS3 剑桥数学考试技巧

Cambridge KS3 mathematics assessments test both your computational fluency and your problem-solving ability. Here are some specific strategies for the fractions-decimals-percentages topic. Show your working: in multi-step problems, write down each conversion clearly — marks are awarded for correct method even if the final answer has a small arithmetic error. Check reasonableness: after finding a percentage of an amount, does your answer make sense? 50% of something should be roughly half — if you get a number larger than the original, you have made a mistake. Use estimation: round decimals to one decimal place to quickly estimate answers before calculating precisely. Memorise key equivalents: knowing that 1/3 is approximately 33.3%, 2/3 is 66.7%, and that 0.1 = 10%, 0.01 = 1% will save you time. Read the question carefully: does it ask for the answer as a fraction, a decimal, or a percentage? Giving the right answer in the wrong form will lose marks.

剑桥 KS3 数学评估不仅测试你的计算熟练度，还测试你的问题解决能力。以下是针对分数-小数-百分数课题的一些具体策略。展示过程：在多步骤问题中，清晰地写下每次转换——即使最终答案有小算术错误，正确的方法也能得分。检查合理性：求出一个数的百分之几后，你的答案合理吗？某数的 50% 应该大约是它的一半——如果你得到的数比原数还大，你就犯了错误。使用估算：将小数四舍五入到一位小数，在精确计算之前快速估计答案。记住关键等值：知道 1/3 约等于 33.3%、2/3 约等于 66.7%、0.1 = 10%、0.01 = 1% 将为你节省时间。仔细读题：题目要求以分数、小数还是百分数给出答案？以错误形式给出正确答案将失分。

Summary | 总结

Fractions, decimals, and percentages are three interconnected ways of expressing parts of a whole. The key skills are: understanding what each form represents, converting fluently between them (fraction to decimal by division, decimal to percentage by multiplying by 100), and applying them to real-world problems including finding percentages of amounts and calculating percentage increases and decreases. The multiplier method for percentage change is particularly powerful and efficient. With practice, these skills become second nature and provide a strong foundation for all future mathematics learning at KS3 and beyond.

分数、小数和百分数是表示整体一部分的三种相互关联的方式。核心技能是：理解每种形式代表什么，熟练地在它们之间进行转换（分数转小数用除法，小数转百分数乘以 100），并将它们应用于实际问题，包括求一个数的百分之几以及计算百分数的增加和减少。百分数变化的乘法倍数法特别强大且高效。通过练习，这些技能将变成你的第二天性，为 KS3 及以后的所有数学学习奠定坚实基础。

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