📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分比
Fractions, decimals and percentages are three equivalent ways of describing parts of a whole. In the Cambridge KS3 Mathematics course, you are expected to move fluently between these forms and use them confidently in calculations, comparisons and real-life problems.
分数、小数和百分比是描述整体与部分关系的三种等价方式。在剑桥 KS3 数学课程中，你需要在这三种形式之间灵活转换，并能在计算、比较和实际生活问题中自信地运用它们。
1. Equivalent Fractions | 等值分数
Equivalent fractions have the same value but are written with different numerators and denominators. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero whole number.
等值分数数值相同，但分子和分母不同。你可以将分子和分母同时乘或除以同一个非零整数，得到等值分数。
For example, 1/2, 2/4, 3/6 and 12/24 are all equivalent because they represent the same proportion of a whole.
例如，1/2、2/4、3/6 和 12/24 都是等值的，因为它们表示整体中相同的比例。
1/2 × 3/3 = 3/6
Always simplify a fraction to its lowest terms by dividing the numerator and denominator by their highest common factor (HCF).
化简分数时，要用分子和分母的最大公因数（HCF）同时去除，得到最简分数。
2. Converting Fractions to Decimals | 分数化为小数
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or long division, and the decimal may terminate or recur.
要把分数化为小数，用分子除以分母。你可以使用短除法或长除法，结果可能是有限小数或循环小数。
For example, 3/8 means 3 ÷ 8, so it becomes 0.375. The decimal terminates because the denominator 8 has only 2 as a prime factor.
例如，3/8 表示 3 ÷ 8，所以结果是 0.375。由于分母 8 的质因数只有 2，因此小数会终止。
3/8 = 3 ÷ 8 = 0.375
2 and 5, the decimal will recur. For instance, 1/3 = 0.333... which may be written as 0.3̇.
如果分母含有 2 和 5 以外的质因数，小数就会循环。例如 1/3 = 0.333...，可记作 0.3̇。
3. Converting Decimals to Fractions | 小数化为分数
To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000 or another power of 10, then simplify.
要把有限小数化为分数，先将小数写成分母为 10、100、1000 等 10 的幂的分数，再化简。
For example, 0.45 = 45/100 = 9/20 after dividing both top and bottom by 5.
例如，0.45 = 45/100，将分子分母同时除以 5 后得到 9/20。
0.45 = 45/100 = 9/20
KS3 work, but you should recognise simple cases such as 0.333... = 1/3 and 0.666... = 2/3.
对于循环小数，KS3 后期会引入用代数方法转换，但你应该先记住简单情况，如 0.333... = 1/3 和 0.666... = 2/3。
4. Understanding Percentages | 理解百分比
A percentage is a fraction with denominator 100. The symbol % means 'out of 100', so 45% means 45 out of every 100 equal parts.
百分比是分母为 100 的分数。符号 % 表示“每 100 份中”，因此 45% 表示每 100 等份中占 45 份。
To convert a percentage to a fraction, write it over 100 and simplify. For example, 25% = 25/100 = 1/4.
要把百分比化为分数，先写成分母为 100 的分数并化简。例如，25% = 25/100 = 1/4。
25% = 25/100 = 1/4 = 0.25
To convert a fraction to a percentage, first convert it to a decimal, then multiply by 100 and add the % sign.
要把分数化为百分比，先化为小数，再乘以 100 并加上百分号。
5. Ordering and Comparing Fractions, Decimals and Percentages | 比较与排序分数、小数和百分比
When comparing fractions, decimals and percentages, it is usually easiest to change all values to the same form. Decimals are often the fastest choice because they can be compared digit by digit.
比较分数、小数和百分比时，通常先将所有量转换成同一种形式。小数往往是最快的方式，因为可以逐位比较。
For example, to compare 3/5, 0.58 and 61%, convert them all to decimals: 3/5 = 0.6, 0.58 = 0.58, 61% = 0.61.
例如，比较 3/5、0.58 和 61%，把它们都化为小数：3/5 = 0.6，0.58 = 0.58，61% = 0.61。
3/5 = 0.60, 0.58, 61% = 0.61
Now you can see that 0.58 is the smallest and 61% is the largest. Write them in ascending order as 0.58, 3/5, 61%.
现在可以看出 0.58 最小，61% 最大。按从小到大排列为 0.58、3/5、61%。
6. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, they must have the same denominator. If they do not, first find the lowest common denominator (LCD) and rewrite each fraction as an equivalent fraction.
分数相加减时，分母必须相同。如果分母不同，先找出最小公分母（LCD），再把每个分数写成等值分数。
to calculate 1/4 + 2/5, use the LCD of 20: 1/4 = 5/20 and 2/5 = 8/20, so the sum is 13/20.
例如，计算 1/4 + 2/5，用最小公分母 20：1/4 = 5/20，2/5 = 8/20，所以和为 13/20。
1/4 + 2/5 = 5/20 + 8/20 = 13/20
For mixed numbers, add or subtract the whole number parts and fractional parts separately, regrouping when necessary.
对于带分数，分别加减整数部分和分数部分，必要时进行借位或进位。
7. Multiplying and Dividing Fractions | 分数的乘法与除法
To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify before or after multiplying if possible.
分数相乘时，分子乘分子，分母乘分母。可以在乘之前或之后进行约分。
/5 = 8/15. If you are multiplying mixed numbers, convert them to improper fractions first.
例如，2/3 × 4/5 = 8/15。如果乘的是带分数，要先化成假分数。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and denominator.
除以一个分数，等于乘以它的倒数。一个分数的倒数就是把分子和分母交换位置。
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Remember to simplify your final answer and, where appropriate, give it as a mixed number.
最后答案要化简，如果合适，写成带分数形式。
8. Finding Percentages of Quantities | 求一个数量的百分之几
To find a percentage of a quantity, write the percentage as a fraction out of 100 and multiply, or convert it to a decimal and multiply.
求一个数量的百分之几，可以把百分数写成分母为 100 的分数再相乘，或化成小数再相乘。
For example, 18% of 250 can be found by calculating 18/100 × 250 = 45, or 0.18 × 250 = 45.
例如，18% of 250 可以计算 18/100 × 250 = 45，或 0.18 × 250 = 45。
18% × 250 = 0.18 × 250 = 45
Common benchmark percentages can be used mentally: 10% is found by dividing by 10, 5% is half of 10%, and 1% is found by dividing by 100.
常见基准百分数可以心算：10% 是除以 10，5% 是 10% 的一半，1% 是除以 100。
9. Percentage Increase and Decrease | 百分比的增加与减少
A percentage increase or decrease is always calculated on the original amount. To increase an amount by a percentage, add that percentage of the amount to the original value.
百分比的增加或减少总是以原数为基础计算。要将一个数增加某个百分比，先求出该百分比对应的量，再加到原数上。
For example, increasing 80 by 15% gives 80 + 0.15 × 80 = 80 + 12 = 92.
例如，把 80 增加 15%，得到 80 + 0.15 × 80 = 80 + 12 = 92。
80 × 1.15 = 92
ntage, subtract that percentage of the amount from the original value. Decreasing 80 by 15% gives 80 - 12 = 68.
要将一个数减少某个百分比，先求出该百分比对应的量，再从原数中减去。把 80 减少 15%，得到 80 - 12 = 68。
10. Reverse Percentages | 逆推百分比问题
In a reverse percentage problem, you are given the final amount after a percentage change and must find the original amount. The key is to identify what percentage the final amount represents.
在逆推百分比问题中，已知百分比变化后的最终量，需要求原来的量。关键是确定最终量对应原数的百分之几。
For example, if a shirt is reduced by 20% and now costs £36, the sale price represents 80% of the original price.
例如，一件衬衫降价 20% 后卖 36 英镑，那么售价是原价的 80%。
80% → £36, so 1% → £36 ÷ 80 = £0.45, and 100% → £45
Divide the given amount by the percentage it represents, then multiply by 100 to find the original amount.
用已知量除以它所对应的百分数，再乘以 100，即可求出原数。
11. Common Misconceptions and Exam Tips | 常见误区与应试技巧
A common mistake is to add or subtract fractions by simply adding the numerators and denominators. Never do this: 1/2 + 1/3 is not 2/5.
一个常见错误是直接分子相加、分母相加。千万不要这样做：1/2 + 1/3 不等于 2/5。
Another misconception is thinking that 0.5 and 0.50 are different because they have a different number of digits. They are equal in value.
另一个误区是认为 0.5 和 0.50 因为小数位数不同而不同。实际上它们的值相等。
When converting between forms, always check that your answer is reasonable. For example, 90% must be less than the whole, so as a decimal it cannot be 9.0.
在形式之间转换时，要检查答案是否合理。例如 90% 必须小于整体，所以化成小数不可能是 9.0。
In exams, show clear steps, simplify final answers, and remember to include units such as £, kg or cm where the question uses them.
考试时要写出清晰步骤，化简最终答案，并在题目使用单位时写上 £、kg 或 cm 等单位。
12. Summary and Practice Challenge | 总结与练习挑战
s, practise converting between all three forms regularly. Use mental benchmarks such as 50% = 1/2 and 25% = 1/4 to speed up your work.
要掌握分数、小数和百分比，需要经常练习三者之间的转换。使用 50% = 1/2、25% = 1/4 等心算基准可以加快解题速度。
Try this challenge: convert 3/8 to a decimal and a percentage, then find 3/8 of 200. The answers are 0.375, 37.5% and 75.
试试这个挑战：把 3/8 化成小数和百分比，再求 200 的 3/8。答案分别是 0.375、37.5% 和 75。
3/8 = 0.375 = 37.5% and 3/8 × 200 = 75
With consistent practice, you will build the fluency needed for Cambridge Checkpoint tests and everyday mathematical reasoning.
通过持续练习，你将建立 Cambridge Checkpoint 考试以及日常数学推理所需的熟练度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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