📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分数
This revision guide covers the core KS3 Cambridge Mathematics skills for working confidently with fractions, decimals and percentages. You will learn how to convert between the three forms, compare values, perform calculations and apply percentage change in real problems. Each section pairs a clear explanation in English with a Chinese translation.
本复习指南涵盖剑桥 KS3 数学中分数、小数和百分数的核心技能。你将学习如何在三种形式之间转换、比较大小、进行计算，并在实际问题中应用百分比变化。每一节都配有清晰的英文解释和中文对照。
1. Equivalent Forms of Numbers | 数的等价形式
Fractions, decimals and percentages are three ways of representing the same proportion or part of a whole. For example, one half can be written as the fraction 1/2, the decimal 0.5 and the percentage 50%. Understanding that these forms are equivalent is the foundation for all later work.
分数、小数和百分数是表示同一个比例或整体一部分的三种方式。例如，一半可以写作分数 1/2、小数 0.5 和百分数 50%。理解这些形式是等价的，是后续所有学习的基础。
A fraction consists of a numerator and a denominator: in 3/4, 3 is the numerator and 4 is the denominator. A decimal uses place value after the decimal point. A percentage is a fraction with denominator 100, so 7% means 7 out of 100.
分数由分子和分母组成：在 3/4 中，3 是分子，4 是分母。小数使用小数点后的位值。百分数是分母为 100 的分数，因此 7% 表示 100 份中的 7 份。
When you see 0.25, 25% and 1/4, you should recognise them immediately as the same value. Quick recognition saves time in calculations and helps when ordering mixed sets of numbers.
当你看到 0.25、25% 和 1/4 时，应该立刻认出它们是同一个值。快速识别可以节省计算时间，也有助于对混合数字进行排序。
2. Converting Fractions to Decimals | 分数化为小数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 means 3 ÷ 8, which gives 0.375. You can use short division or a calculator for this conversion.
要将分数化为小数，用分子除以分母。例如，3/8 表示 3 ÷ 8，结果是 0.375。你可以使用短除法或计算器完成这一转换。
Some fractions produce terminating decimals, such as 1/4 = 0.25 and 7/20 = 0.35. Others produce recurring decimals, such as 1/3 = 0.333… and 2/11 = 0.181818… . A recurring decimal can be written with a dot or bar over the repeating digit or digits.
有些分数会得到有限小数，例如 1/4 = 0.25 和 7/20 = 0.35。另一些分数会得到循环小数，例如 1/3 = 0.333… 和 2/11 = 0.181818… 。循环小数可以在重复的数字上方加点或横线表示。
If the denominator is a power of 10, the conversion is immediate: 9/10 = 0.9, 47/100 = 0.47 and 231/1000 = 0.231. Knowing these place-value relationships is useful for mental work.
如果分母是 10 的幂，转换可以直接写出：9/10 = 0.9，47/100 = 0.47，231/1000 = 0.231。掌握这些位值关系对心算很有帮助。
3. Converting Decimals and Percentages | 小数与百分数互化
To convert a decimal to a percentage, multiply the decimal by 100. This is the same as moving the decimal point two places to the right. For example, 0.45 becomes 45%, and 0.075 becomes 7.5%.
要将小数化为百分数，把小数乘以 100。这相当于把小数点向右移动两位。例如，0.45 变为 45%，0.075 变为 7.5%。
To convert a percentage to a decimal, divide by 100, which moves the decimal point two places to the left. Thus 32% becomes 0.32 and 4.5% becomes 0.045.
要将百分数化为小数，除以 100，也就是把小数点向左移动两位。因此 32% 变为 0.32，4.5% 变为 0.045。
When converting a percentage to a fraction, write it over 100 and simplify if possible. For example, 40% = 40/100 = 2/5, and 12.5% = 12.5/100 = 1/8 after multiplying numerator and denominator by 2.
将百分数化为分数时，先写成分母为 100 的分数，再尽可能化简。例如，40% = 40/100 = 2/5，而 12.5% = 12.5/100，分子分母同乘 2 后得到 1/8。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
Memorising the common conversions in the table will speed up your work in tests and in word problems.
记住表格中的常见转换可以加快你在考试和文字题中的解题速度。
4. Ordering Mixed Sets of Numbers | 混合数字的大小排序
When ordering fractions, decimals and percentages, change all values into the same form first. Decimals are usually the easiest form for comparison because you can compare place values digit by digit.
在给分数、小数和百分数排序时，应先把所有数值化为同一种形式。小数通常是最容易比较的形式，因为可以逐位比较位值。
For example, arrange 0.4, 3/8, 35% and 2/5 in ascending order. Convert: 0.4 = 0.40, 3/8 = 0.375, 35% = 0.35, 2/5 = 0.40. The ascending order is 35%, 3/8, 0.4, 2/5.
例如，将 0.4、3/8、35% 和 2/5 按从小到大排列。转换后：0.4 = 0.40，3/8 = 0.375，35% = 0.35，2/5 = 0.40。从小到大为 35%、3/8、0.4、2/5。
Be careful with decimals that have different numbers of digits: 0.3 is larger than 0.299 because at the second decimal place 0 is larger than 9 in 0.29? Actually compare place by place: 0.3 = 0.300 and 0.299, so 0.300 > 0.299. Adding zeros to the right of a decimal does not change its value.
注意小数位数不同时的比较：0.3 大于 0.299，因为 0.3 = 0.300，而 0.300 > 0.299。在小数右侧补零不会改变它的大小。
A useful strategy is to write all decimals to the same number of decimal places before comparing them. This avoids mistakes with negative numbers and values less than 1.
一个有用的策略是在比较前把所有小数写成相同的小数位数。这样可以避免负数和小于 1 的数值出现错误。
5. Adding and Subtracting Fractions | 分数的加减法
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common denominator first. For example, 1/4 + 1/6 has a common denominator of 12, so write 3/12 + 2/12 = 5/12.
分数相加减时，必须具有相同的分母。如果分母不同，先找出最小公分母。例如，1/4 + 1/6 的公分母是 12，因此写成 3/12 + 2/12 = 5/12。
The lowest common denominator is the lowest common multiple of the denominators. For 1/6 and 3/8, the LCM of 6 and 8 is 24, so 1/6 = 4/24 and 3/8 = 9/24, giving 13/24 when added.
最小公分母就是各分母的最小公倍数。对于 1/6 和 3/8，6 和 8 的最小公倍数是 24，所以 1/6 = 4/24，3/8 = 9/24，相加得到 13/24。
When subtracting mixed numbers, convert them to improper fractions first. Example: 2 1/3 − 1 5/6 = 7/3 − 11/6 = 14/6 − 11/6 = 3/6 = 1/2.
进行带分数减法时，先将它们化为假分数。例如：2 1/3 − 1 5/6 = 7/3 − 11/6 = 14/6 − 11/6 = 3/6 = 1/2。
Always simplify your answer and, where appropriate, convert an improper fraction back to a mixed number. In the example 14/6 − 11/6 = 3/6, the simplified answer is 1/2.
答案一定要化简，并且在合适的情况下把假分数化回带分数。在 14/6 − 11/6 = 3/6 的例子中，化简后的答案是 1/2。
6. Multiplying and Dividing Fractions | 分数的乘除法
To multiply fractions, multiply the numerators together and multiply the denominators together. For example, 2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2. You can simplify before multiplying by cancelling common factors.
分数相乘时，分子相乘作为新分子，分母相乘作为新分母。例如，2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2。你可以在相乘前通过约去公因数来化简。
To divide by a fraction, multiply by its reciprocal. The reciprocal of 2/5 is 5/2. So 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
除以一个分数等于乘以它的倒数。2/5 的倒数是 5/2。因此 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
When multiplying mixed numbers, convert them to improper fractions first. For example, 1 1/2 × 2 1/3 = 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2.
带分数相乘时，先化为假分数。例如，1 1/2 × 2 1/3 = 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2。
The word ‘of’ in problems often means multiply. For example, 2/3 of 90 means 2/3 × 90 = 60. In context questions, look for ‘of’ to decide the operation.
题目中的 “of” 通常表示乘法。例如，90 的 2/3 表示 2/3 × 90 = 60。在情境题中，看到 “of” 就可以判断应该用乘法。
7. Calculating Percentages of Amounts | 求一个数的百分之几
To find a percentage of an amount, multiply the amount by the percentage expressed as a decimal. For example, 15% of 200 = 0.15 × 200 = 30. This method works for any percentage, including decimal percentages.
要求一个数的百分之几，用这个数乘以百分数对应的小数。例如，200 的 15% = 0.15 × 200 = 30。这个方法适用于任何百分数，包括带小数的百分数。
You can also use a fraction method: 25% of 80 = 1/4 × 80 = 20, and 10% of 80 = 8, so 35% of 80 = 20 + 8 = 28. Splitting a percentage into easier parts is useful for mental calculation.
你也可以使用分数方法：80 的 25% = 1/4 × 80 = 20，80 的 10% = 8，所以 80 的 35% = 20 + 8 = 28。把百分数拆成容易计算的部分有助于心算。
For non-calculator questions, first find 10% by dividing by 10, then scale up or down. For example, 5% of 320 = 32 ÷ 2 = 16, and 15% of 320 = 32 + 16 = 48.
在非计算器题目中，先除以 10 求出 10%，再进行放大或缩小。例如，320 的 5% = 32 ÷ 2 = 16，320 的 15% = 32 + 16 = 48。
Always check that your answer makes sense: 100% of a number is the number itself, and 50% is half of it. If your answer is larger than the original amount for a percentage less than 100%, you have made an error.
一定要检查答案是否合理：一个数的 100% 就是它本身，50% 是它的一半。如果百分数小于 100%，答案却比原数大，就说明计算出错了。
8. Percentage Increase and Decrease | 百分数增减
A percentage increase adds a percentage of the original value to the original amount. For example, increasing 150 by 20% gives 20% of 150 = 30, so the new amount is 150 + 30 = 180.
百分数增加是在原数的基础上加上原数的一定百分比。例如，把 150 增加 20%，先求 150 的 20% = 30，所以新数量是 150 + 30 = 180。
A percentage decrease subtracts a percentage of the original value. Decreasing 80 by 15% gives 15% of 80 = 12, so the new amount is 80 − 12 = 68.
百分数减少是在原数的基础上减去原数的一定百分比。把 80 减少 15%，先求 80 的 15% = 12，所以新数量是 80 − 12 = 68。
Using a multiplier is faster: for an increase of 20%, multiply the original by 1.2; for a decrease of 15%, multiply by 0.85. Thus 150 × 1.2 = 180 and 80 × 0.85 = 68.
使用乘数更快：增加 20% 时，用原数乘以 1.2；减少 15% 时，乘以 0.85。因此 150 × 1.2 = 180，80 × 0.85 = 68。
To find the percentage increase or decrease, use the formula: percentage change = change ÷ original × 100%. If a price rises from 40 to 50, the change is 10, so the percentage increase is 10/40 × 100% = 25%.
求增长或减少的百分比，使用公式：百分比变化 = 变化量 ÷ 原数 × 100%。如果价格从 40 涨到 50，变化量是 10，所以增长百分比为 10/40 × 100% = 25%。
Percentage change = (New value − Original value) ÷ Original value × 100%
百分比变化 = (新值 − 原值) ÷ 原值 × 100%
9. Reverse Percentages | 逆向百分数
Reverse percentage problems ask you to find the original amount before a percentage change. If a price after a 20% increase is 240, the multiplier used was 1.2, so the original price was 240 ÷ 1.2 = 200.
逆向百分数问题要求你求出百分数变化之前的原数。如果价格增加 20% 后是 240，那么所用的乘数是 1.2，所以原价是 240 ÷ 1.2 = 200。
If a price after a 15% decrease is 68, the multiplier was 0.85, so the original price was 68 ÷ 0.85 = 80. To find the original, divide by the multiplier, not multiply.
如果价格减少 15% 后是 68，那么乘数是 0.85，所以原价是 68 ÷ 0.85 = 80。求原数时应除以乘数，而不是乘以乘数。
Be careful with wording: ‘after a 20% increase’ means the final value is 120% of the original, so the multiplier is 1.2. ‘After a 20% decrease’ means the final value is 80% of the original, so the multiplier is 0.8.
注意题意：”增加 20% 后” 表示最终值是原值的 120%，所以乘数是 1.2。”减少 20% 后” 表示最终值是原值的 80%，所以乘数是 0.8。
A common mistake is to take 20% of the new amount and subtract or add it. This does not recover the original because percentages are always based on the original value unless stated otherwise.
常见错误是求出新数量的 20% 然后再减去或加上。这样做无法还原原数，因为除非特别说明，百分数始终以原值为基准。
10. Fractions, Decimals and Percentages in Word Problems | 文字题中的分数、小数和百分数
Word problems often combine fractions, decimals and percentages in one situation. For example, 3/5 of the students in a school walk to school, 0.25 come by bus, and the rest come by car. Convert 3/5 = 0.6 and 0.25 = 1/4 to compare easily.
文字题常常在一个情境中结合分数、小数和百分数。例如，一所学校 3/5 的学生步行上学，0.25 的学生乘公共汽车，其余乘小汽车。将 3/5 化为 0.6，0.25 化为 1/4，就容易比较。
The remaining fraction who come by car is 1 − (3/5 + 1/4) = 1 − (12/20 + 5/20) = 1 − 17/20 = 3/20. Changing all values to twentieths makes the calculation clear.
乘小汽车的学生所占的剩余分数是 1 − (3/5 + 1/4) = 1 − (12/20 + 5/20) = 1 − 17/20 = 3/20。把所有数化为二十分之几后计算就很清楚。
In percentage problems, read the question twice and underline the original amount, the change and the final amount. This helps you decide whether to add, subtract, multiply or divide.
在百分数问题中，读题两遍，在原数、变化量和最终量下面画线。这有助于你判断该加减还是乘除。
When comparing offers, use percentages: a shop offers 30% off 80 or 1/3 off 80. Since 30% = 0.3 and 1/3 ≈ 0.333, the 1/3 offer saves slightly more. Converting both to decimals makes the comparison reliable.
比较优惠时使用百分数：一家商店提供 80 减 30% 或 80 减 1/3。由于 30% = 0.3，而 1/3 ≈ 0.333，所以减 1/3 的优惠节省得略多。把两者都化为小数可以让比较更可靠。
11. Common Errors and How to Avoid Them | 常见错误及避免方法
One common error is adding fractions without finding a common denominator: 1/2 + 1/3 is not 2/5. The correct method gives 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
一个常见错误是分数相加时不找公分母：1/2 + 1/3 不等于 2/5。正确的方法是 1/2 + 1/3 = 3/6 + 2/6 = 5/6。
Another error is moving the decimal point the wrong way when converting percentages. 0.05 is 5%, not 50%; 0.5 is 50%. Moving two places to the right must be done carefully.
另一个错误是百分数转换时小数点移动方向弄错。0.05 是 5%，不是 50%；0.5 是 50%。向右移动两位时必须仔细。
When finding the whole from a part, students sometimes forget to divide. If 25% of a number is 30, the number is not 30 × 0.25; instead solve 0.25 × n = 30, so n = 30 ÷ 0.25 = 120.
由部分求整体时，学生有时忘记用除法。如果某数的 25% 是 30，这个数不是 30 × 0.25；而应解 0.25 × n = 30，所以 n = 30 ÷ 0.25 = 120。
Percentage increase followed by the same percentage decrease does not return the original amount. Increasing 100 by 10% gives 110, then decreasing 110 by 10% gives 99. This shows why you must identify the correct base each time.
先增加百分之几再减少相同的百分之几不会回到原数。100 增加 10% 得到 110，再减少 10% 得到 99。这说明每次都必须找准基数。
12. Summary and Key Facts | 总结与关键事实
Always remember that 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20% and 1/10 = 0.1 = 10%. These basic conversions are the building blocks for harder questions.
始终记住 1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/5 = 0.2 = 20%，以及 1/10 = 0.1 = 10%。这些基本转换是解决难题的基础。
For increase, multiply by (1 + percentage as decimal); for decrease, multiply by (1 − percentage as decimal). For example, a 12% increase uses 1.12, and a 12% decrease uses 0.88.
增加时，乘以 (1 + 百分数对应的小数)；减少时，乘以 (1 − 百分数对应的小数)。例如，增加 12% 用 1.12，减少 12% 用 0.88。
When a question gives a fraction, decimal and percentage together, convert them all to the same form before doing anything else. This removes confusion and makes comparison or calculation straightforward.
当题目同时出现分数、小数和百分数时，先全部化为同一种形式再做其他运算。这样可以消除混淆，使比较或计算更直接。
Practice by writing your own conversion table and testing yourself with mixed sets. The more automatic your conversions are, the more working memory you free up for the problem itself.
通过自制转换表并混合自测来练习。你的转换越熟练，解决问题时就能腾出越多的工作记忆。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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