📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比
In KS3 Cambridge Mathematics, fractions, decimals and percentages are not separate topics but three ways of writing the same idea: a part of a whole. Mastering the links between them will help you solve problems in number, algebra, data handling and everyday life.
在 KS3 剑桥数学中，分数、小数和百分比并不是三个独立的主题，而是同一个概念——整体的一部分——的三种表达方式。掌握它们之间的联系将帮助你在数、代数、数据处理和日常生活中解决问题。
1. The Fraction-Decimal-Percent Triangle | 分数、小数和百分比的三角关系
Every fraction, terminating decimal and percentage can be written in the other two forms. The key idea is that they all compare a part to a whole, but use different notation for the same ratio.
每个分数、有限小数和百分比都可以写成另外两种形式。核心思想是它们都在比较部分与整体，只是用不同的符号表示同一个比值。
fraction ↔ decimal ↔ percentage
This triangle is the foundation for all conversions you will meet in KS3 and in Cambridge Checkpoint tests. Once you can move confidently between the three forms, percentage problems and fraction calculations become much easier.
这个三角关系是你在 KS3 和剑桥 Checkpoint 考试中会遇到的所有转换的基础。一旦你能自信地在三种形式之间切换，百分比问题和分数计算就会变得简单得多。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value but different numerators and denominators. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.
等值分数数值相同，但分子和分母不同。你可以通过将分子和分母同时乘或除以同一个非零数来得到等值分数。
2/3 = 4/6 = 8/12 = 20/30
To simplify a fraction to its lowest terms, divide the numerator and denominator by their highest common factor (HCF). For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6.
要把分数化简到最简形式，用分子和分母的最大公因数（HCF）去除它们。例如，18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6。
3. Converting Fractions to Decimals | 分数转换为小数
To convert a fraction to a decimal, divide the numerator by the denominator. This works for both proper and improper fractions, and the result may be terminating or recurring.
要把分数转换为小数，用分子除以分母。这适用于真分数和假分数，结果可能是有限小数或循环小数。
decimal = numerator ÷ denominator
cimal. However, 1/3 = 0.333... is a recurring decimal because the digit 3 repeats forever.
例如，3/8 = 3 ÷ 8 = 0.375，这是一个有限小数。但是 1/3 = 0.333... 是一个循环小数，因为数字 3 永远重复。
4. Converting Decimals to Percentages | 小数转换为百分比
To convert a decimal to a percentage, multiply the decimal by 100 and add the percent sign. Equivalently, move the decimal point two places to the right.
要把小数转换为百分比，将小数乘以 100 并加上百分号。等价地，把小数点向右移动两位。
0.47 × 100 = 47%
For decimals smaller than 0.01, such as 0.006, the conversion is 0.006 × 100 = 0.6%, which is less than 1%. You may also need to convert decimals greater than 1, for example 1.25 = 125%.
对于小于 0.01 的小数，例如 0.006，转换结果是 0.006 × 100 = 0.6%，小于 1%。你可能还需要转换大于 1 的小数，例如 1.25 = 125%。
5. Converting Percentages to Fractions | 百分比转换为分数
To convert a percentage to a fraction, write the percentage over 100 and then simplify if possible. This method works because 'percent' literally means 'per hundred'.
要把百分比转换为分数，将百分比写在 100 之上，然后尽可能化简。这个方法有效是因为“percent”的字面意思就是“每百”。
65% = 65/100 = 13/20
If the percentage includes a decimal, such as 12.5% thin a fraction, multiply both numerator and denominator by 10: 12.5/100 = 125/1000 = 1/8.
如果百分比包含小数，例如 12.5%，写成 12.5/100。为了避免分数中出现小数，把分子和分母同时乘以 10：12.5/100 = 125/1000 = 1/8。
6. Recurring Decimals and Rounding | 循环小数与四舍五入
Some fractions such as 2/3, 5/11 and 4/9 produce recurring decimals when converted. A recurring decimal repeats a digit or a block of digits forever, and we often round it to a given number of decimal places.
一些分数，如 2/3、5/11 和 4/9，在转换时会产生循环小数。循环小数会无限重复一个数字或一组数字，我们通常将其四舍五入到指定的小数位数。
unds to 0.67 to two decimal places. To round correctly, look at the next digit: if it is 5 or more, round up; if it is 4 or less, leave the last digit unchanged.
例如，2/3 = 0.666...，保留两位小数时四舍五入为 0.67。要正确四舍五入，看下一位数字：如果是 5 或以上则进位；如果是 4 或以下则保持最后一位不变。
7. Adding and Subtracting Fractions | 分数的加减法
Before adding or subtracting fractions, they must have the same denominator. Find the lowest common multiple (LCM) of the denominators, rewrite each fraction with that common denominator, then add or subtract the numerators only.
在加减分数之前，它们必须有相同的分母。找出分母的最小公倍数（LCM），把每个分数改写为以这个公倍数为分母的等值分数，然后只加减分子。
2/3 + 1/4 = 8/12 + 3/12 = 11/12
For mixed numbers, convert them to improper fractions first, perform the addition or subtraction, then change the answer back to a mixed number if required.
对于带分数，先把它们转换为假分数，进行加减运算，然后再根据需要把答案转换回带分数。
8. Multiplying and Dividing Fractions | 分数的乘除法
To multiply fractions, multiply the numerators together and multiply the denominators together. You can simplify before multiplying by cancelling any common factor between a numerator and a denominator.
要乘分数，将分子相乘，分母相乘。你可以在乘法之前先约分，消去分子和分母之间的任何公因数。
3/4 × 8/9 = 24/36 = 2/3
on, multiply by its reciprocal. For example, 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 or 1 1/4.
要除以一个分数，乘以它的倒数。例如，5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4，即 1 1/4。
9. Percentage of an Amount | 求一个数的百分比
rt the percentage to a decimal or a fraction, then multiply by the amount. For example, 15% of 240 can be found as 0.15 × 240 = 36, or as 15/100 × 240 = 36.
要求一个数的百分比，先将百分比转换为小数或分数，再乘以这个数。例如，240 的 15% 可以计算为 0.15 × 240 = 36，或者 15/100 × 240 = 36。
percentage of amount = percentage as decimal × amount
Common benchmark percentages are useful in mental calculations: 50% is half, 25% is one quarter, 10% is one tenth, and 5% is half of 10%.
常见的基准百分比在心算中很有用：50% 是一半，25% 是四分之一，10% 是十分之一，5% 是 10% 的一半。
10. Percentage Increase and Decrease | 百分比增减
To increase an amount by a percentage, add the percentage to 100%, then multiply. To decrease an amount by a percentage, subtract the percentage from 100%, then multiply.
要将一个数增加某个百分比，将百分比加到 100%，再乘以这个数。要将一个数减少某个百分比，从 100% 中减去该百分比，再乘以这个数。
increase by 20%: multiply by 1.20
decrease by 15%: multiply by 0.85
For example, a £60 jacket with a 20% increase costs £60 × 1.20 = £72. A £60 jacket with a 15% decrease costs £60 × 0.85 = £51.
例如，一件 60 英镑的夹克涨价 20% 后价格为 60 × 1.20 = 72 英镑。一件 60 英镑的夹克降价 15% 后价格为 60 × 0.85 = 51 英镑。
11. Comparing Quantities Using Percentages | 用百分比比较数量
Percentages allow you to compare quantities fairly when the totals are different. For example, a test score of 17 out of 20 is 85%, while 21 out of 25 is 84%, so the first result is slightly better relative to the total marks.
百分比使你能在总数不同的情况下公平地比较数量。例如，20 分中得到 17 分是 85%，而 25 分中得到 21 分是 84%，因此相对于总分而言，第一个成绩稍好。
A percentage change compares the actual change to the original amount using the formula:
百分比变化用以下公式将实际变化与原数量进行比较：
percentage change = (change ÷ original) × 100
If a price rises from £40 to £50, the change is £10 and the percentage increase is (10 ÷ 40) × 100 = 25%.
如果价格从 40 英镑上升到 50 英镑，变化量是 10 英镑，百分比增加为 (10 ÷ 40) × 100 = 25%。
12. Exam-style Strategies and Common Mistakes | 考试策略与常见错误
In Cambridge Checkpoint questions, always show your conversion steps clearly. Write the decimal or percentage form before multiplying, and simplify fractions at the end unless the question asks for a specific form.
在剑桥 Checkpoint 考题中，始终清晰地展示转换步骤。先写出小数或百分比形式再相乘，除非题目要求特定形式，否则最后要化简分数。
- Do not add or subtract fractions without a common denominator.
- 分数没有公分母时不能直接加减。
- Do not forget to add the percent sign after multiplying by 100.
- 乘以 100 后不要忘记加上百分号。
- Check that your final answer makes sense: 70% of a number should be less than the original number.
- 检查最终答案是否合理：一个数的 70% 应该小于原数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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