📚 Fractions, Decimals and Percentages | 分数、小数和百分数
Fractions, decimals and percentages are three connected ways of describing parts of a whole. In KS3 Cambridge Mathematics, you need to convert fluently between all three forms and apply them to real problems such as discounts, proportions and probability.
分数、小数和百分数是描述整体一部分的三种相互关联的方式。在剑桥 KS3 数学中，你需要灵活地在三种形式之间转换，并运用它们解决折扣、比例和概率等实际问题。
1. Understanding the Three Forms | 理解三种形式
A fraction such as 3/5 means 3 parts out of 5 equal parts. A decimal such as 0.6 places the part over a power of ten. A percentage such as 60% means 60 parts out of 100.
分数（如 3/5）表示 5 等份中的 3 份。小数（如 0.6）把部分表示为十的幂次方。百分数（如 60%）表示 100 份中的 60 份。
3/5 = 0.6 = 60%
2. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 means 3 ÷ 8 = 0.375. If the denominator is 10, 100 or 1000, you can use place value: 7/100 = 0.07.
要将分数转换为小数，用分子除以分母。例如 3/8 表示 3 ÷ 8 = 0.375。如果分母是 10、100 或 1000，可以直接用位值转换：7/100 = 0.07。
5/8 = 5 ÷ 8 = 0.625
Some fractions are recurring decimals. For example, 1/3 = 0.333… and 2/3 = 0.666… . In Checkpoint answers, three dots show that the decimal continues forever.
有些分数是循环小数。例如 1/3 = 0.333…，2/3 = 0.666…。在 Checkpoint 答案中，三个点表示小数无限循环。
3. Converting Decimals to Percentages | 小数转百分数
To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, 0.35 × 100 = 35%. To convert a percentage to a decimal, divide by 100.
要将小数转换为百分数，乘以 100 并加上 % 符号。例如 0.35 × 100 = 35%。要将百分数转换为小数，则除以 100。
0.125 = 0.125 × 100% = 12.5%
Reverse conversion is just as important: 7% = 7 ÷ 100 = 0.07.
反向转换同样重要：7% = 7 ÷ 100 = 0.07。
4. Converting Percentages to Fractions | 百分数转分数
Write the percentage as a fraction over 100 and simplify if possible. For example, 40% = 40/100 = 2/5. If the percentage contains a decimal, multiply numerator and denominator to make both integers.
将百分数写成分母为 100 的分数，并尽可能化简。例如 40% = 40/100 = 2/5。如果百分数含有小数，可以将分子和分母同时放大为整数。
12.5% = 12.5/100 = 125/1000 = 1/8
The table below shows some common conversions that are worth memorising.
下表列出一些值得记住的常见转换。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
5. Ordering and Comparing Values | 比较与排序
To compare fractions, decimals and percentages, convert all values to the same form. Decimals are often the easiest choice. For example, compare 2/5, 0.45 and 38%. Convert 2/5 = 0.4 and 38% = 0.38.
比较分数、小数和百分数时，把所有值转换为同一种形式。小数通常最方便。例如比较 2/5、0.45 和 38%。先转换：2/5 = 0.4，38% = 0.38。
2/5 = 0.40, 38% = 0.38, 0.45 = 0.45
In ascending order, the values are 38%, 2/5, 0.45. Always write the final answer using the original forms given in the question.
按升序排列为 38%、2/5、0.45。最终答案必须使用题目中给出的原始形式。
6. Finding a Percentage of an Amount | 求一个数量的百分数
To find a percentage of an amount, convert the percentage to a decimal and multiply. For example, 15% of 80 = 0.15 × 80 = 12. You can also build up from 10%. Since 10% of 80 = 8, then 15% = 8 + 4 = 12.
求一个数量的百分数，先把百分数转换为小数再相乘。例如 15% of 80 = 0.15 × 80 = 12。也可以先求 10%。因为 80 的 10% = 8，所以 15% = 8 + 4 = 12。
20% of 45 = 0.20 × 45 = 9
This method is useful for mental calculation, especially when the percentage is a multiple of 10 or 5.
这种方法对心算很有用，尤其是当百分数是 10 或 5 的倍数时。
7. Percentage Increase and Decrease | 百分数增减
To increase an amount by a percentage, multiply by 1 plus the percentage as a decimal. For example, a £60 item increased by 25% becomes 60 × 1.25 = £75.
将一个数量增加某个百分数，用 1 加上该百分数化为小数去乘。例如 £60 增加 25% 后变为 60 × 1.25 = £75。
New amount = original × (1 + p)
To decrease an amount, multiply by 1 minus the percentage as a decimal. A 25% discount on £60 is 60 × 0.75 = £45.
减少一个数量，则用 1 减去该百分数化为小数去乘。£60 减少 25% 为 60 × 0.75 = £45。
New amount = original × (1 –
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