📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three different ways of representing parts of a whole. In the Cambridge KS3 mathematics curriculum, you must be able to convert between these forms, compare them, and use them accurately in calculations and real-life problems. This revision guide explains the key methods step by step, with worked examples and exam tips.
分数、小数和百分数是表示整体中某一部分的三种不同方式。在剑桥 KS3 数学课程中，你必须能够在这些形式之间进行转换、比较大小，并在计算和实际问题中准确使用它们。本复习指南将逐步讲解关键方法，并提供例题和考试技巧。
1. Equivalent Fractions and Simplifying | 等值分数与约分
A fraction has a numerator, the top number, and a denominator, the bottom number. Equivalent fractions have the same value but use different numerators and denominators. To simplify a fraction, divide both the numerator and the denominator by their highest common factor, or HCF.
分数有分子，即上面的数，和分母，即下面的数。等值分数具有相同的值，但使用不同的分子和分母。要化简分数，需要把分子和分母同时除以它们的最大公因数，即 HCF。
For example, to simplify 18/24, the highest common factor of 18 and 24 is 6. Divide both terms by 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4. Therefore, 18/24 simplifies to 3/4. You should always give answers in their simplest form unless a question says otherwise.
例如，要化简 18/24，18 和 24 的最大公因数是 6。将两项都除以 6：18 ÷ 6 = 3，24 ÷ 6 = 4。因此，18/24 化简为 3/4。除非题目另有要求，答案应始终写成最简形式。
18/24 = 3/4
When simplifying, dividing by the HCF in one step is faster than repeated division by small factors. If you cannot see the HCF immediately, list the factors of the numerator and denominator first.
化简时，一步除以最大公因数比重复除以小因数更快。如果无法立即看出最大公因数，可以先列出分子和分母的因数。
2. Mixed Numbers and Improper Fractions | 带分数与假分数
An improper fraction has a numerator that is equal to or larger than its denominator, such as 11/4. A mixed number combines a whole number with a proper fraction, such as 2 3/4. Cambridge KS3 questions often ask you to convert between these two forms.
假分数的分子等于或大于分母，例如 11/4。带分数由整数和真分数组成，例如 2 3/4。剑桥 KS3 题目经常要求你在这两种形式之间进行转换。
To change a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For 2 3/4: 2 × 4 = 8, then 8 + 3 = 11, so the improper fraction is 11/4.
要把带分数化为假分数，先将整数乘以分母，再加上分子，然后把结果写在原分母之上。对于 2 3/4：2 × 4 = 8，然后 8 + 3 = 11，所以假分数是 11/4。
2 3/4 = (2 × 4 + 3)/4 = 11/4
To change an improper fraction into a mixed number, divide the numerator by the denominator. The quotient is the whole number part, and the remainder is the new numerator. For 11/4, 11 ÷ 4 gives 2 remainder 3, so the mixed number is 2 3/4.
要把假分数化为带分数，用分子除以分母。商是整数部分，余数是新的分子。对于 11/4，11 ÷ 4 得到 2 余 3，所以带分数是 2 3/4。
3. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple, or LCM, of the denominators, then convert each fraction so that both have this common denominator. Once the denominators are the same, add or subtract the numerators only.
要对分数进行加减运算，它们必须有相同的分母。先找出分母的最小公倍数，即 LCM，然后将每个分数转换为以这个公分母为分母的分数。当分母相同后，只对分子进行加减。
For 1/4 + 2/5, the LCM of 4 and 5 is 20. Convert: 1/4 = 5/20 and 2/5 = 8/20. Now add the numerators: 5 + 8 = 13, so the answer is 13/20. Do not add the denominators.
对于 1/4 + 2/5，4 和 5 的最小公倍数是 20。转换：1/4 = 5/20，2/5 = 8/20。现在将分子相加：5 + 8 = 13，所以答案是 13/20。不要把分母相加。
1/4 + 2/5 = 5/20 + 8/20 = 13/20
For mixed numbers, you can convert them to improper fractions first, or add the whole number parts and fractional parts separately. After calculating, simplify the result and convert back to a mixed number if needed.
对于带分数，可以先转换为假分数，也可以将整数部分和分数部分分别相加。计算后，化简结果，如有需要再转换回带分数。
4. Multiplying and Dividing Fractions | 分数乘除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. You can simplify before multiplying by cancelling any common factor across the numerator and denominator, which often makes the calculation easier.
分数乘法很直接：分子相乘，分母相乘。在相乘前，可以通过约去分子和分母之间的公因数来化简，这通常会让计算更简单。
For 3/8 × 4/9, multiply: (3 × 4) / (8 × 9) = 12/72. Simplify by dividing both terms by 12 to get 1/6. Alternatively, cancel first: 3 and 9 share a factor of 3, and 4 and 8 share a factor of 4, leaving 1/2 × 1/3 = 1/6.
对于 3/8 × 4/9，相乘：(3 × 4) / (8 × 9) = 12/72。将两项都除以 12 化简，得到 1/6。也可以先约分：3 和 9 有公因数 3，4 和 8 有公因数 4，剩下 1/2 × 1/3 = 1/6。
3/8 × 4/9 = 12/72 = 1/6
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is found by swapping the numerator and denominator. For 2/3 ÷ 4/5, rewrite as 2/3 × 5/4, giving 10/12, which simplifies to 5/6.
要除以一个分数，乘以它的倒数。一个分数的倒数可以通过交换分子和分母得到。对于 2/3 ÷ 4/5，改写为 2/3 × 5/4，得到 10/12，化简为 5/6。
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
5. Converting Fractions to Decimals | 分数化为小数
A fraction can be converted to a decimal by dividing the numerator by the denominator. Some fractions give terminating decimals, which stop after a finite number of digits, while others give recurring decimals, which repeat a digit or pattern forever.
分数可以通过用分子除以分母转换为小数。有些分数得到有限小数，即在小数点后有限位数结束；有些分数得到循环小数，即某个数字或模式无限重复。
For example, 3/8 means 3 ÷ 8, which is 0.375. This is a terminating decimal. In contrast, 1/3 means 1 ÷ 3, which is 0.333…, a recurring decimal often written as 0.3 with a dot over the 3.
例如，3/8 表示 3 ÷ 8，结果是 0.375。这是一个有限小数。相反，1/3 表示 1 ÷ 3，结果是 0.333…，这是一个循环小数，通常写作在 3 上方加点的形式。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
You should memorise the common conversions in the table because they appear frequently in KS3 tests and help you compare values quickly.
你应该记住表格中的常见转换，因为这些转换在 KS3 考试中经常出现，能帮助你快速比较数值。
6. Converting Decimals to Percentages and Back | 小数与百分数互化
A percentage means ‘out of 100’. To convert a decimal to a percentage, multiply it by 100, which is the same as moving the decimal point two places to the right. To convert a percentage back to a decimal, divide by 100, which moves the decimal point two places to the left.
百分数表示 ‘每一百中有多少’。要把小数转换为百分数，将其乘以 100，这相当于把小数点向右移动两位。要把百分数转换回小数，除以 100，这相当于把小数点向左移动两位。
For example, 0.65 × 100 = 65%, so 0.65 is equal to 65%. Similarly, 7% ÷ 100 = 0.07. Be careful with percentages less than 1%, such as 0.5% = 0.005 as a decimal.
例如，0.65 × 100 = 65%，所以 0.65 等于 65%。同样，7% ÷ 100 = 0.07。注意小于 1% 的百分数，例如 0.5% 写成小数是 0.005。
0.65 = 65% and 7% = 0.07
To convert a fraction to a percentage, first convert the fraction to a decimal by dividing, then multiply by 100. For example, 3/5 = 0.6 = 60%. Alternatively, make an equivalent fraction with denominator 100, so 3/5 = 60/100 = 60%.
要把分数转换为百分数，先通过除法把分数转换为小数，再乘以 100。例如，3/5 = 0.6 = 60%。也可以先化为分母为 100 的等值分数，所以 3/5 = 60/100 = 60%。
7. Ordering Fractions, Decimals and Percentages | 分数、小数和百分数排序
When comparing fractions, decimals and percentages, convert them all into the same form. Decimals are usually the easiest form for comparison because you can compare place values directly. Once converted, arrange the decimals from smallest to largest or largest to smallest as required.
比较分数、小数和百分数时，先把它们都转换成同一种形式。小数通常是最容易比较的形式，因为可以直接比较数位。转换后，按照要求从小到大或从大到小排列。
For example, order 3/5, 0.58, 56% and 2/3 from smallest to largest. Convert to decimals: 3/5 = 0.6, 0.58 stays the same, 56% = 0.56, and 2/3 = 0.666… . The correct order is 0.56, 0.58, 0.6, 0.666…, so 56% < 0.58 < 3/5 < 2/3.
例如，将 3/5、0.58、56% 和 2/3 从小到大排列。转换为小数：3/5 = 0.6，0.58 不变，56% = 0.56，2/3 = 0.666…。正确顺序是 0.56、0.58、0.6、0.666…，所以 56% < 0.58 < 3/5 < 2/3。
56% < 0.58 < 3/5 < 2/3
Using a number line can help you see the positions clearly. Mark 0 and 1, then place each converted decimal. This method reduces mistakes when several different forms are mixed together.
使用数轴可以帮助你清楚地看到各数的位置。标出 0 和 1，然后放置每个转换后的小数。当多种不同形式混合在一起时，这种方法能减少错误。
8. Finding a Percentage of an Amount | 求一个数的百分之几
To find a percentage of an amount, convert the percentage to a decimal or fraction and multiply by the amount. For calculator questions, change the percentage to a decimal: for example, 15% becomes 0.15. Then multiply.
要求一个数的百分之几，先把百分数转换为小数或分数，再乘以这个数。对于可使用计算器的题目，把百分数化为小数：例如 15% 变成 0.15，然后相乘。
For 15% of 240, calculate 0.15 × 240 = 36. For non-calculator questions, use a mental method: find 10% first by dividing by 10, then scale up or down. For 35% of 80, 10% is 8, 5% is 4, so 35% = 8 + 8 + 8 + 4 = 28.
对于 240 的 15%，计算 0.15 × 240 = 36。对于不可使用计算器的题目，使用心算方法：先除以 10 得到 10%，再进行放大或缩小。对于 80 的 35%，10% 是 8，5% 是 4，所以 35% = 8 + 8 + 8 + 4 = 28。
15% of 240 = 0.15 × 240 = 36
This skill is essential for problems involving discounts, deposits, tax, interest and data interpretation. Always check whether the final answer needs to be rounded to a sensible degree of accuracy, especially when money is involved.
这项技能对于涉及折扣、定金、税费、利息和数据解释的问题至关重要。始终检查最终答案是否需要四舍五入到合理的精度，尤其是在涉及金钱时。
9. Percentage Increase and Decrease | 百分数增减
To increase an amount by a percentage, multiply by 1 plus the percentage as a decimal. For example, to increase a value by 12%, multiply by 1.12. To decrease by a percentage, multiply by 1 minus the percentage as a decimal, so a decrease of 15% means multiply by 0.85.
要将一个数增加某个百分数，乘以 1 加上该百分数对应的小数。例如，增加 12%，乘以 1.12。要减少某个百分数，乘以 1 减去该百分数对应的小数，所以减少 15% 表示乘以 0.85。
Increase: new = original × (1 + p/100) and Decrease: new = original × (1 − p/100)
For example, increase £45 by 12%: 45 × 1.12 = £50.40. Decrease 80 kg by 15%: 80 × 0.85 = 68 kg. The multiplier method is more efficient than finding the change and then adding or subtracting, especially in multi-step problems.
例如，将 £45 增加 12%：45 × 1.12 = £50.40。将 80 kg 减少 15%：80 × 0.85 = 68 kg。与先求出变化量再加减相比，使用乘数法更高效，尤其是在多步问题中。
To find the percentage change itself, use the formula: percentage change = (new value − original value) / original value × 100%. A positive result means an increase; a negative result means a decrease.
要求百分数变化本身，使用公式：百分数变化 = (新值 − 原值) / 原值 × 100%。正值表示增加，负值表示减少。
10. Fraction of an Amount and Real-Life Problems | 求一个数的几分之几与实际应用题
To find a fraction of an amount, divide the amount by the denominator, then multiply the result by the numerator. For 3/5 of 200, first calculate 200 ÷ 5 = 40, then multiply 40 × 3 = 120. Therefore, 3/5 of 200 is 120.
要求一个数的几分之几，先用这个数除以分母，再将结果乘以分子。对于 200 的 3/5，先计算 200 ÷ 5 = 40，再计算 40 × 3 = 120。因此，200 的 3/5 是 120。
3/5 of 200 = (200 ÷ 5) × 3 = 120
Real-life problems often combine fractions, decimals and percentages. For example, a jacket originally costs £60 and is reduced by 25%. The saving is 25% of £60 = £15, so the new price is £60 − £15 = £45. You can also say the customer pays 75% of the original price: 0.75 × 60 = £45.
实际问题经常综合运用分数、小数和百分数。例如，一件夹克原价为 £60，降价 25%。节省的金额是 £60 的 25% = £15，所以新价格是 £60 − £15 = £45。也可以说顾客支付原价的 75%：0.75 × 60 = £45。
11. Common Misconceptions and Exam Tips | 常见错误与考试技巧
One common mistake is adding or subtracting fractions without finding a common denominator first. For example, 1/2 + 1/3 is not 2/5. The correct method gives 3/6 + 2/6 = 5/6. Always check that the denominators are the same before adding or subtracting.
一个常见错误是在没有先找到公分母的情况下进行分数加减。例如，1/2 + 1/3 不等于 2/5。正确的方法是 3/6 + 2/6 = 5/6。在加减之前，始终检查分母是否相同。
Another mistake is confusing a percentage increase with a percentage point change. If a rate rises from 20% to 25%, that is a 5 percentage point increase, but the percentage increase is (25 − 20) / 20 × 100% = 25%. Also, when dividing fractions, remember to multiply by the reciprocal of the divisor, not the first fraction.
另一个错误是混淆百分数增加与百分点变化。如果比例从 20% 上升到 25%，这是增加了 5 个百分点，但百分数增加是 (25 − 20) / 20 × 100% = 25%。另外，在分数除法时，记住乘以除数的倒数，而不是第一个分数的倒数。
In exams, show all working clearly, simplify final answers, and write money using two decimal places where appropriate. Use the multiplier method for percentage increase and decrease to reduce the number of separate steps and avoid arithmetic errors.
在考试中，要清晰地展示所有步骤，化简最终答案，并在合适时将金额写成两位小数。使用乘数法进行百分数增减可以减少单独步骤的数量，并避免计算错误。
12. Quick Reference and Practice | 速查与练习
Use this quick reference to check your working: fraction to decimal means divide the numerator by the denominator; decimal to percentage means multiply by 100; fraction of an amount means divide by the denominator and multiply by the numerator; percentage increase uses the multiplier 1 + p/100; percentage decrease uses the multiplier 1 − p/100.
使用这份速查表检查你的步骤：分数化小数表示用分子除以分母；小数化百分数表示乘以 100；求一个数的几分之几表示除以分母再乘以分子；百分数增加使用乘数 1 + p/100；百分数减少使用乘数 1 − p/100。
Try these practice questions: simplify 24/36; convert 2 1/5 to an improper fraction; calculate 2/3 + 5/6; calculate 1 1/2 ÷ 3/8
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