📚 Fractions, Decimals and Percentages: Core Skills | 分数、小数与百分数：核心技能
This KS3 Cambridge mathematics revision guide covers the essential skills needed to move confidently between fractions, decimals and percentages. The topic appears regularly in Checkpoint tests and in everyday problem-solving such as discounts, data comparisons and probability.
本 KS3 剑桥数学复习指南涵盖分数、小数与百分数之间灵活转换所需的核心技能。该主题经常出现在 Checkpoint 测验中，也常用于折扣、数据比较和概率等日常生活问题。
1. Understanding the Three Forms | 理解三种形式
A fraction, a decimal and a percentage are three ways of writing the same proportion. The fraction 1/2, the decimal 0.5 and the percentage 50% all represent one half.
分数、小数和百分数是表示同一比例的三种写法。分数 1/2、小数 0.5 和百分数 50% 都表示一半。
Percent means 'out of 100', so 50% is exactly 50 out of 100. A decimal is based on place value, with the first decimal place showing tenths and the second showing hundredths.
百分数表示“每一百份中的几份”，因此 50% 就是 100 份中的 50 份。小数基于位值，小数点后第一位表示十分位，第二位表示百分位。
When you compare a fraction, a decimal and a percentage, you are comparing the same amount written in different notations. Choosing the most useful form makes calculation easier.
当你比较一个分数、一个小数和一个百分数时，你其实是在比较同一种数量的不同写法。选择最方便的形式会让计算更简单。
2. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. The fraction bar means division.
要把分数转换为小数，用分子除以分母。分数线就表示除法。
For example, 3/4 means 3 ÷ 4. Write 3 as 3.0 and divide: 3.0 ÷ 4 = 0.75, so 3/4 = 0.75.
例如，3/4 表示 3 ÷ 4。把 3 写成 3.0 再除：3.0 ÷ 4 = 0.75，因此 3/4 = 0.75。
3/4 = 3 ÷ 4 = 0.75
3/4 = 3 ÷ 4 = 0.75
nt test, you may need to write a recurring decimal using dots or by giving the first three decimal places.
有些分数会产生循环小数。例如，1/3 = 0.333...，2/11 = 0.181818...。在 Checkpoint 测验中，你可能需要用循环点表示循环小数，或者写出前三位小数。
Always check whether a fraction can be simplified before dividing. Simplifying first can make the division easier.
在计算前先检查分数是否能化简。先化简通常会让除法更简单。
3. Converting Decimals to Percentages | 小数转百分数
To convert a decimal to a percentage, multiply the decimal by 100 and write the percent symbol.
要把小数转换为百分数，将小数乘以 100，再加上百分号。
For example, 0.45 × 100 = 45, so 0.45 = 45%. The decimal point moves two places to the right.
例如，0.45 × 100 = 45，因此 0.45 = 45%。小数点向右移动两位。
0.45 = 45%
0.45 = 45%
For a decimal less than 1, the percentage is between 0% and 100%. For a decimal greater than 1, the percentage is greater than 100%. For example, 1.25 = 125%.
小于 1 的小数对应的百分数在 0% 到 100% 之间。大于 1 的小数对应的百分数大于 100%。例如，1.25 = 125%。
places, move the point two places and keep any remaining digits. For example, 0.375 = 37.5%.
如果小数超过两位，小数点向右移动两位后保留其余数字。例如，0.375 = 37.5%。
4. Converting Percentages to Fractions | 百分数转分数
To convert a percentage to a fraction, write the percentage over 100 and simplify if possible.
要把百分数转换为分数，先写成分母为 100 的分数，再尽可能化简。
For example, 25% = 25/100. Divide the numerator and denominator by 25 to get 1/4.
例如，25% = 25/100。分子和分母同时除以 25，得到 1/4。
25% = 25/100 = 1/4
25% = 25/100 = 1/4
If the percentage includes a decimal, such as 12.5%, write 12.5/100 and multiply both numerator and denominator by 10 to remove the decimal. This gives 125/1000, which simplifies to 1/8.
如果百分数含有小数，例如 12.5%，先写成 12.5/100，再把分子和分母同时乘以 10 去掉小数点。得到 125/1000，化简为 1/8。
If the percentage is greater than 100%, the fraction is an improper fraction or a mixed number. For example, 150% = 150/100 = 3/2 = 1 1/2.
如果百分数大于 100%，得到的分数是假分数或带分数。例如，150% = 150/100 = 3/2 = 1 1/2。
5. Ordering and Comparing | 排序与比较
To compare fractions, decimals and percentages, convert all three forms into the same form. Decimals are often the easiest for comparison.
要比较分数、小数和百分数，先把三种形式都转换成同一种形式。小数通常最容易比较。
For example, arrange 0.4, 3/10 and 35% in ascending order. Convert all to decimals: 0.4, 0.3 and 0.35. The order is 0.3, 0.35, 0.4, so the original order is 3/10, 35%, 0.4.
例如，将 0.4、3/10 和 35% 按从小到大排列。全部转换成小数：0.4、0.3 和 0.35。排列为 0.3、0.35、0.4，因此原顺序为 3/10、35%、0.4。
Another method is to change all values to percentages. This is useful when the question asks for percentages or when fractions have different denominators.
另一种方法是把所有数值都转换为百分数。当题目要求百分数或分数的分母不同时，这种方法很有用。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
6. Percentages of Amounts | 求一个数的百分数
To find a percentage of an amount, write the percentage as a fraction over 100 and multiply by the amount.
要求一个数的百分数，先把百分数写成分母为 100 的分数，再与该数相乘。
For example, find 20% of 45. Write 20% as 20/100 = 1/5, then 1/5 × 45 = 9.
例如，求 45 的 20%。将 20% 写成 20/100 = 1/5，然后 1/5 × 45 = 9。
20% of 45 = 0.20 × 45 = 9
45 的 20% = 0.20 × 45 = 9
For a non-simple percentage such as 17%, multiply by 0.17. For example, 17% of 60 = 0.17 × 60 = 10.2.
对于 17% 这样的百分数，可以乘以 0.17。例如，60 的 17% = 0.17 × 60 = 10.2。
Use the 10% method for mental calculation: find 10% first, then scale up or down. For example, 30% of 80 = 3 × 8 = 24 because 10% of 80 is 8.
心算时可以使用“先找 10%”的方法：先求出 10%，再按倍数推算。例如，80 的 30% = 3 × 8 = 24，因为 80 的 10% 是 8。
7. Percentage Increase and Decrease | 百分数增减
To increase an amount by a percentage, find the percentage of the amount and add it to the original amount.
要将一个数增加某个百分数，先求出该百分数对应的量，再加到原数上。
For example, increase 120 by 15%. First find 15% of 120 = 0.15 × 120 = 18. Then 120 + 18 = 138.
例如，将 120 增加 15%。先求 120 的 15% = 0.15 × 120 = 18。然后 120 + 18 = 138。
To decrease an amount by a percentage, find the percentage of the amount and subtract it from the original amount.
要将一个数减少某个百分数，先求出该百分数对应的量，再从原数中减去。
For example, decrease 80 by 25%. First find 25% of 80 = 20. Then 80 - 20 = 60.
例如，将 80 减少 25%。先求 80 的 25% = 20。然后 80 - 20 = 60。
A quicker method uses a multiplier. For an increase of 15%, multiply by 1.15. For a decrease of 25%, multiply by 0.75.
更快的方法是使用乘数。增加 15% 就乘以 1.15；减少 25% 就乘以 0.75。
Increase by 15%: × 1.15
减少 25%：× 0.75
8. Fraction of an Amount | 求一个数的几分之几
To find a fraction of an amount, divide the amount by the denominator and then multiply by the numerator.
要求一个数的几分之几，先除以分母，再乘以分子。
For example, find 3/5 of 40. First divide 40 by 5 to get 8, then multiply by 3 to get 24.
例如，求 40 的 3/5。先将 40 除以 5 得到 8，再乘以 3 得到 24。
3/5 of 40 = 40 ÷ 5 × 3 = 24
40 的 3/5 = 40 ÷ 5 × 3 = 24
1 1/4, first convert it to an improper fraction. For example, 1 1/4 of 24 = 5/4 × 24 = 30.
对于 1 1/4 这样的带分数，先转换成假分数。例如，24 的 1 1/4 = 5/4 × 24 = 30。
9. Reverse Percentage Problems | 逆百分数问题
In a reverse percentage problem, you know the final amount after a percentage change and need to find the original amount.
在逆百分数问题中，你知道百分数变化后的最终量，需要求出原来的量。
To solve this, write an equation using the multiplier. If a price has been increased by 20%, the final amount is 1.20 times the original.
解题时使用乘数建立方程。例如，价格增加了 20%，最终量就是原来量的 1.20 倍。
For example, a jacket costs 96 pounds after a 20% increase. Let the original price be x. Then 1.20x = 96, so x = 96 ÷ 1.20 = 80 pounds.
例如，一件夹克涨价 20% 后售价为 96 英镑。设原价为 x。则 1.20x = 96，所以 x = 96 ÷ 1.20 = 80 英镑。
Original price = 96 ÷ 1.20 = 80
原价 = 96 ÷ 1.20 = 80
Do not simply subtract 20% of 96, because 20% applies to the original amount, not the final amount.
不要简单地减去 96 的 20%，因为 20% 是针对原价计算的，不是最终价格。
10. Common Misconceptions | 常见错误
One common error is adding fractions by adding numerators and denominators. For example, 1/2 + 1/3 is not 2/5. The correct answer is 5/6 after using a common denominator.
一个常见错误是分数相加时分子加分子、分母加分母。例如，1/2 + 1/3 不等于 2/5。正确做法是先通分，得到 5/6。
Another error is confusing 0.5 with 0.05. 0.5 is 50%, while 0.05 is 5%.
另一个常见错误是混淆 0.5 和 0.05。0.5 是 50%，而 0.05 是 5%。
- 1/2 = 0.5 = 50%
- 1/20 = 0.05 = 5%
Some students move the decimal point the wrong way when converting to a percentage. Remember: decimal to percentage moves the point two places to the right.
有些学生在把小数转换为百分数时会把小数点移错方向。记住：小数转百分数时小数点向右移动两位。
When simplifying a fraction, always divide both the numerator and denominator by the same number.
化简分数时，一定要用同一个数同时除以分子和分母。
11. Exam-Style Practice | 考试风格练习
Try these questions to test your understanding. Write your answers as fractions, decimals or percentages as requested.
尝试以下题目来检测你的理解。按照要求用分数、小数或百分数写出答案。
- Convert 7/8 to a decimal and a percentage. | 将 7/8 转换为小数和百分数。
- Write 0.45 as a fraction in its simplest form. | 将 0.45 写成最简分数。
- Increase 240 by 15%. | 将 240 增加 15%。
- Decrease 180 by 30%. | 将 180 减少 30%。
- Find 3/8 of 56. | 求 56 的 3/8。
- A price is reduced by 10% to 72 pounds. Find the original price. | 某价格减少 10% 后为 72 英镑。求原价。
Show all steps clearly. In the Checkpoint test, method marks are often awarded even if the final answer is incorrect.
请清晰地写出所有步骤。在 Checkpoint 测验中，即使最终答案错误，过程和步骤通常也能得分。
12. Summary Checklist | 总结清单
Make sure you can do the following before your test:
考试前请确保你能完成以下内容：
- Convert between fractions, decimals and percentages. | 在分数、小数和百分数之间转换。
- Order a mixture of fractions, decimals and percentages. | 对分数、小数和百分数混合排序。
- Find percentage increases and decreases using multipliers. | 使用乘数求百分数增减。
- Solve reverse percentage problems. | 解决逆百分数问题。
- Explain common mistakes and correct them. | 解释常见错误并加以改正。
Regular practice with mixed examples will build speed and confidence for the Cambridge Checkpoint Mathematics test.
经常练习混合题型有助于提升速度，并增强参加 Cambridge Checkpoint 数学测验的信心。
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