📚 Mastering Fractions, Decimals & Percentages | 掌握分数、小数与百分比
Numbers can be expressed in many forms, and three of the most common are fractions, decimals, and percentages. In KS3 Cambridge Mathematics, you are expected to move confidently between these representations, compare them, and use them in calculations. This revision guide covers all the essential concepts, conversion methods, and problem-solving skills you need to master the topic, with clear bilingual explanations to strengthen your understanding.
数字可以用多种形式表示，其中最常见的是分数、小数和百分比。在KS3剑桥数学中，你需要熟练地在这些表示形式之间转换、比较它们，并用于计算。本复习指南涵盖所有基本概念、转换方法和解题技巧，并以清晰的中英双语解释巩固你的理解。
1. What Are Fractions, Decimals & Percentages? | 分数、小数与百分比是什么？
A fraction represents a part of a whole, written as a numerator over a denominator, such as 3/4. It shows how many equal parts are taken from a total number of equal parts.
分数表示整体的一部分，写作分子在上、分母在下的形式，如 3/4。它表示从相等的总份数中取出的份数。
A decimal is another way of writing a fraction, using a decimal point to separate the whole number part from the fractional part. For example, 0.75 is the decimal equivalent of 3/4.
小数是分数的另一种写法，使用小数点将整数部分与小数部分分开。例如，0.75 就是 3/4 的小数表示。
A percentage is a number expressed as a fraction of 100. The symbol % means 'per hundred', so 75% directly means 75 out of 100, or 75/100.
百分比是以100为分母的分数表示法。符号 % 表示“每一百”，所以 75% 直接表示 100 份中的 75 份，即 75/100。
Together, these three forms are deeply connected. Every fraction can be written as a decimal or a percentage, and vice versa. Mastering the links between them unlocks many mathematical shortcuts.
这三种形式紧密相连。每个分数都能写成小数或百分比，反之亦然。掌握它们之间的联系能开启许多数学捷径。
2. Converting Fractions to Decimals | 将分数转换为小数
numerator by the denominator. For a simple fraction like 1/4, you calculate 1 ÷ 4 = 0.25.
要将分数转换为小数，用分子除以分母。以简单的分数 1/4 为例，计算 1 ÷ 4 = 0.25。
. For instance, 7/10 = 0.7, 23/100 = 0.23, and 9/1000 = 0.009. Be careful with place value – the number of zeros in the denominator tells you how many decimal places you need.
如果分母是10的幂（10、100、1000），转换立即可得。例如，7/10 = 0.7，23/100 = 0.23，9/1000 = 0.009。注意位值——分母中零的个数告诉你需要几位小数。
bove the repeating digit. Similarly, 2/11 = 0.1818... These are called recurring decimals.
对于像 1/3 这样的分数，除法会得到循环小数：1 ÷ 3 = 0.333... 可以在循环数字上方加点表示。类似地，2/11 = 0.1818... 这些被称为循环小数。
t practice mental conversions for common fractions like 1/2 = 0.5, 1/5 = 0.2, 3/8 = 0.375.
记住只有在允许时才使用计算器做除法，但要练习常见分数的心算转换，如 1/2 = 0.5，1/5 = 0.2，3/8 = 0.375。
3. Converting Decimals to Fractions | 将小数转换为分数
Write the decimal as a fraction with a denominator of 10, 100, 1000, or a suitable power of 10, then simplify. For 0.6, it becomes 6/10, which simplifies to 3/5 by dividing numerator and denominator by 2.
将小数写成分母为10、100、1000或适当10的幂的分数，然后化简。对于 0.6，变成 6/10，分子分母同时除以2，化简为 3/5。
For 0.75, the last digit is in the hundredths place, so write 75/100. Divide by 25 to get 3/4. Always remove common factors to express the fraction in its simplest form.
对于 0.75，最后一位是百分位，所以写作 75/100。除以25得到 3/4。总是消去公因数，以最简形式表示分数。
o the answer is 2 2/5. You can also convert to an improper fraction: (2 × 5 + 2)/5 = 12/5.
当遇到像 2.4 这样的带小数时，分开处理：整数2保留，0.4 变成 4/10 = 2/5，所以答案是 2 2/5。你也可以转换为假分数：(2 × 5 + 2)/5 = 12/5。
thods beyond KS3, but you may be asked to recognise that 0.333... = 1/3 or 0.666... = 2/3.
循环小数需要特殊的代数方法，超出了KS3的范围，但你可能需要认出 0.333... = 1/3 或 0.666... = 2/3。
4. Converting Fractions to Percentages | 将分数转换为百分比
age, first convert the fraction to a decimal by dividing, then multiply by 100 and add the % symbol. For example, 3/5 = 0.6, then 0.6 × 100 = 60%.
要将分数转换为百分比，先用除法将分数转换为小数，然后乘以100并添加百分号。例如，3/5 = 0.6，然后 0.6 × 100 = 60%。
/20, multiply denominator by 5 to get 100, so multiply numerator by 5 as well: 35/100 = 35%.
如果分母是100的因数，一种简单的方法是将分母乘以某个数变成100，同时分子也乘以相同的数。对于 7/20，分母乘以5得100，分子也乘以5：35/100 = 35%。
inator that easily becomes 100, use division: 5/8 = 5 ÷ 8 = 0.625, then 0.625 × 100 = 62.5%. Watch out for percentages that include a decimal or fraction.
当分数的分母不容易变成100时，使用除法：5/8 = 5 ÷ 8 = 0.625，然后 0.625 × 100 = 62.5%。留意那些包含小数或分数的百分比。
It is useful to memorise common equivalents: 1/4 = 25%, 1/2 = 50%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%, 1/3 ≈ 33.3%, 2/3 ≈ 66.7%.
记住常见的等价形式很有帮助：1/4 = 25%，1/2 = 50%，3/4 = 75%，1/5 = 20%，1/10 = 10%，1/3 ≈ 33.3%，2/3 ≈ 66.7%。
5. Converting Percentages to Fractions | 将百分比转换为分数
Write the percentage as a fraction with denominator 100, then simplify. 45% means 45/100. Divide numerator and denominator by 5 to get 9/20. Always check if you can reduce further.
将百分比写成分母为100的分数，然后化简。45% 表示 45/100。分子分母同时除以5得到 9/20。始终检查能否进一步约分。
For percentages with a decimal, such as 12.5%, first write as 12.5/100. Multiply numerator and denominator by 10 to remove the decimal: 125/1000, then simplify by dividing by 125 to get 1/8.
对于带小数的百分比，如 12.5%，先写成 12.5/100。分子分母同乘以10去掉小数：125/1000，然后除以125化简得到 1/8。
For mixed number percentages like 33 1/3% 33 1/3 = 100/3, so (100/3)/100 = 100/300 = 1/3. This confirms the well-known equivalence.
对于带分数百分比如 33 1/3%，先转化为假分数：33 1/3 = 100/3，所以 (100/3)/100 = 100/300 = 1/3。这证实了众所周知的等价关系。
Percentages greater than 100, like 150% ions greater than 1: 150/100 = 3/2 or 1 1/2. This is important for representing increases.
大于100的百分比，如150%，会变成大于1的分数：150/100 = 3/2 或 1 1/2。这对表示增长很重要。
6. Converting Between Decimals and Percentages | 小数与百分比的互相转换
To convert a decimal to a percentage, multiply by 100 and add the % sign. This is the same as moving the decimal point two places to the right. 0.34 becomes 34%, 0.07 becomes 7%, and 1.2 becomes 120%.
要将小数转换为百分比，乘以100并添加百分号。这相当于把小数点向右移动两位。0.34 变成 34%，0.07 变成 7%，1.2 变成 120%。
rcentage to a decimal, divide by 100, or move the decimal point two places to the left. 85% = 0.85, 5% = 0.05, 0.1% = 0.001. Be careful with single-digit percentages – you need a zero in the tenths place.
要将百分比转换为小数，除以100，或将小数点向左移动两位。85% = 0.85，5% = 0.05，0.1% = 0.001。注意个位数百分比——十分位需要一个零。
This two-way conversion is often the quickest way to compare a mixture of numbers. Line up values as decimals to see which is largest or smallest.
这种双向转换通常是比较混合数字的最快方法。将数值按小数排列，就能看出哪个最大或最小。
Example question: Which is larger, 0.4, 39%, or 2/5? Convert all to decimals: 0.4, 39% = 0.39, 2/5 = 0.4, so 0.4 and 2/5 are tied and both are larger than 39%.
例题：0.4、39% 和 2/5 哪个最大？全部转换为小数：0.4，39% = 0.39，2/5 = 0.4，所以 0.4 和 2/5 相等，且都大于 39%。
7. Ordering and Comparing FDP | 分数、小数与百分比的排序与比较
When you face a list of fractions, decimals, and percentages, pick one form (usually decimals) and convert everything. Then place them on a number line or simply order by size.
当你面对一列分数、小数和百分比时，选择一种形式（通常是小数）并全部转换。然后将它们放在数轴上，或简单地按大小排序。
For example, order 0.65, 3/5, 58%, 2/3 from smallest to largest. Convert: 0.65, 3/5 = 0.6, 58% = 0.58, 2/3 ≈ 0.666... So the order is 0.58 (58%), 0.6 (3/5), 0.65, 0.666... (2/3).
例如，将 0.65、3/5、58%、2/3 从小到大排列。转换：0.65，3/5 = 0.6，58% = 0.58，2/3 ≈ 0.666... 所以顺序是 0.58（58%），0.6（3/5），0.65，0.666...（2/3）。
are 5/6 and 7/9, use 18 as common denominator: 5/6 = 15/18, 7/9 = 14/18, so 5/6 is larger.
当直接比较分数时，可以找到公分母。比较 5/6 和 7/9，用18作公分母：5/6 = 15/18，7/9 = 14/18，所以 5/6 更大。
Using a calculator is allowed for conversions, but you must know the underlying method. In non-calculator exams, stick to common denominators or simple decimal equivalents.
可以使用计算器进行转换，但必须了解其底层方法。在非计算器考试中，坚持用公分母或简单的小数等值。
8. Calculating Percentages of Quantities | 计算一个数量的百分比
To find a percentage of a number, convert the percentage to a decimal and multiply. 30% of 80 = 0.30 × 80 = 24. Alternatively, find 10% first and scale up. 10% of 80 is 8, so 30% is 3 × 8 = 24.
求一个数的百分之几，将百分比转换为小数再相乘。80 的 30% = 0.30 × 80 = 24。或者先求 10% 再放大。80 的 10% 是 8，所以 30% 就是 3 × 8 = 24。
This building-up method is efficient in non-calculator papers. For example, to find 17.5% of 200: 10% = 20, 5% = 10, 2.5% = half of 5% = 5, so total = 20 + 10 + 5 = 35.
这种逐步构建的方法在非计算器试卷中很高效。例如，求 200 的 17.5%：10% = 20，5% = 10，2.5% = 5% 的一半 = 5，所以总和 = 20 + 10 + 5 = 35。
When a percentage is greater than 100, the result is larger than the original. 120% of 50 = 1.2 × 50 = 60. This occurs in problems about increase or profit.
当百分比大于100时，结果会比原数大。50 的 120% = 1.2 × 50 = 60。这出现在关于增长或利润的问题中。
Word problems often involve finding discounts, tax, or interest. A 15% discount on a £40 item means you save 0.15 × 40 = £6, so the sale price is £34.
应用题通常涉及求折扣、税费或利息。一件40英镑的商品打15%的折扣，意味着你节省 0.15 × 40 = 6英镑，所以售价为34英镑。
9. Percentage Increase and Decrease | 百分比增减
by a percentage, find the percentage of the amount and add it to the original. Increasing £50 by 8%: 8% of £50 = £4, new amount = £54.
要将一个数量增加百分之几，先求该数量的百分比，然后加到原数上。将50英镑增加8%：50英镑的8% = 4英镑，新金额 = 54英镑。
A faster method uses a multiplier. For an increase of 8%, the multiplier is 1 + 0.08 = 1.08. £50 × 1.08 = £54. For a decrease of 12%, the multiplier is 1 - 0.12 = 0.88.
更快的方法是使用乘数。对于增加8%，乘数是 1 + 0.08 = 1.08。50英镑 × 1.08 = 54英镑。对于减少12%，乘数是 1 - 0.12 = 0.88。
riginal amount before a percentage change, divide by the multiplier. If a price after a 20% increase is £72, the original price = £72 ÷ 1.20 = £60.
要找到百分比变化前的原始数量，除以乘数。如果一个价格增加了20%后是72英镑，则原价 = 72 ÷ 1.20 = 60英镑。
Decrease by a percentage works similarly: a shirt reduced by 25% now costs £15. The multiplier is 0.75, so original price = £15 ÷ 0.75 = £20.
按百分比减少同理：一件衬衫降价25%后售价15英镑。乘数为0.75，所以原价 = 15 ÷ 0.75 = 20英镑。
10. Fractions in Real-Life Contexts | 现实情境中的分数
needs 3/4 cup of sugar and you want to make half the recipe, you need 3/4 × 1/2 = 3/8 cup.
分数出现在食谱、测量和分享问题中。如果一份食谱需要 3/4 杯糖，而你想做一半的量，就需要 3/4 × 1/2 = 3/8 杯。
e. To combine 1/3 of a pizza with 1/4, find a common denominator (12): 4/12 + 3/12 = 7/12.
分数加减时，分母必须相同。要合并一个披萨的 1/3 和 1/4，找到公分母（12）：4/12 + 3/12 = 7/12。
s (2+1=3) and fractions separately: 1/2 + 2/3 = 3/6 + 4/6 = 7/6 = 1 1/6, so total = 4 1/6.
带分数需要小心处理。2 1/2 + 1 2/3 可以通过整数部分（2+1=3）和分数部分分别相加：1/2 + 2/3 = 3/6 + 4/6 = 7/6 = 1 1/6，所以总和为 4 1/6。
ultiply the denominators. 2/3 × 4/5 = (2×4)/(3×5) = 8/15. No common denominator is needed.
分数乘法很简单：分子乘分子，分母乘分母。2/3 × 4/5 = (2×4)/(3×5) = 8/15。不需要公分母。
the same as multiplying by its reciprocal. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. Remember 'Keep, Change, Flip'.
除以一个分数等于乘以其倒数。3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。记住“保留、变号、翻转”。
11. Decimals in Measurement and Money | 测量与货币中的小数
arly, 2.3 kg = 2 kg and 300 g. Understanding decimal place value avoids conversion errors.
小数在公制单位中至关重要。1.5 公里是 1 公里 500 米，因为 0.5 公里 = 500 米。类似地，2.3 千克 = 2 千克 300 克。理解小数位值可避免换算错误。
In money, amounts are always written to 2 decimal places: £3.50 not £3.5. When adding or subtracting, align the decimal points vertically.
在货币中，金额总是写到两位小数：3.50英镑，而不是3.5英镑。加减时要将小数点垂直对齐。
rs. 0.3 × 0.7: 3×7=21, one decimal place in each factor, so total 2 decimal places → 0.21.
小数乘法：忽略小数点，按整数相乘，然后在积中点上小数点，使小数位数等于因数中小数位数的总和。0.3 × 0.7：3×7=21，每个因数有一位小数，所以总共两位小数 → 0.21。
make the divisor a whole number. 4.5 ÷ 0.15 = (4.5 × 100) ÷ (0.15 × 100) = 450 ÷ 15 = 30.
小数除法：将两个数同时乘以10的幂，使除数变为整数。4.5 ÷ 0.15 = (4.5 × 100) ÷ (0.15 × 100) = 450 ÷ 15 = 30。
12. Problem-Solving Strategy and Common Pitfalls | 解题策略与常见陷阱
Always read the question carefully to identify what form the answer should be in. Many marks are lost by giving a decimal when a fraction is required, or by not simplifying fully.
始终仔细阅读题目，确定答案应该以何种形式呈现。很多分数是因为题目要求分数却给出了小数，或者没有完全化简而丢失。
When converting, double-check your decimal places. Remember that 0.5% is not the same as 0.5 (which is 50%). 0.5% = 0.005 as a decimal, and 1/200 as a fraction.
转换时，反复检查小数位数。记住 0.5% 与 0.5（即50%）不同。0.5% 作为小数是 0.005，作为分数是 1/200。
In multi-step problems, write down each conversion step clearly. For instance, '35% of 240' is clearer if you show 240 × 0.35 = 84, rather than doing it in one mental leap.
在多步问题中，清晰地写下每一步转换。例如，“240 的 35%”，展示 240 × 0.35 = 84，比心算一步到位更清晰。
Practice linking percentages to fractions: knowing that 12.5% = 1/8, 33 1/3% = 1/3, and 66 2/3% = 2/3 helps to tackle problems without a calculator.
练习将百分比与分数联系起来：知道 12.5% = 1/8，33 1/3% = 1/3，66 2/3% = 2/3，有助于不借助计算器解决问题。
Finally, check your answer with an alternative method whenever possible. If you used division, verify by multiplication. This will catch silly mistakes and boost your confidence.
最后，尽可能用另一种方法检查答案。如果你用了除法，用乘法验证。这能发现粗心错误，增强你的信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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