📚 Fractions, Decimals and Percentages: Converting and Comparing | 分数、小数与百分数：转换与比较
In KS3 mathematics, fractions, decimals and percentages are three different ways of expressing the same idea: a part of a whole. Being able to switch between them quickly is essential for solving problems in number work, algebra and data handling. This revision guide explains the key conversion rules, comparison strategies and common mistakes to avoid in Cambridge Checkpoint tests.
在 KS3 数学中，分数、小数和百分数是表达同一个概念的三种不同方式：整体的一部分。能够快速在它们之间进行切换对于解决数、代数和数据处理中的问题至关重要。本复习指南解释了关键转换规则、比较策略以及在剑桥 Checkpoint 考试中需要避免的常见错误。
1. The Three Forms of a Number | 数字的三种形式
A fraction such as 3/4 means 3 parts out of 4 equal parts. A decimal such as 0.75 uses the base-ten place value system to show the same amount. A percentage such as 75% means 75 parts out of 100. All three describe the same proportion of a whole.
分数（如 3/4）表示 4 等份中的 3 份。小数（如 0.75）使用十进制位值系统表示相同的量。百分数（如 75%）表示 100 份中的 75 份。这三者都描述整体中相同的比例。
The symbol % comes from the Latin phrase ‘per centum’, meaning ‘by the hundred’. That is why converting to a percentage often involves making the denominator 100 or multiplying by 100.
百分号 % 来自拉丁语短语 ‘per centum’，意思是 ‘每一百’。这就是为什么转换为百分数通常涉及将分母变为 100 或乘以 100。
You should be comfortable moving between these forms because mixed questions are very common. For example, a question may ask which is larger: 2/5, 0.45 or 41%. Without conversion, this is hard to judge.
你应该能够熟练地在这几种形式之间转换，因为混合形式的题目非常常见。例如，题目可能会问 2/5、0.45 和 41% 哪个更大。如果不进行转换，很难判断。
2. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number). For example, 3/8 becomes 3 ÷ 8 = 0.375. This works for proper fractions, improper fractions and mixed numbers once the mixed number is written as an improper fraction.
要将分数转换为小数，用分子（上面的数）除以分母（下面的数）。例如，3/8 变成 3 ÷ 8 = 0.375。这适用于真分数、假分数以及转化为假分数后的带分数。
3 ÷ 8 = 0.375
If the division does not terminate, you may need to round the decimal to a given number of decimal places or significant figures. Common recurring decimals include 1/3 = 0.333… and 2/7 = 0.285714285714…
如果除法不终止，你可能需要将小数四舍五入到指定的小数位或有效数字。常见的循环小数包括 1/3 = 0.333… 和 2/7 = 0.285714285714…
With mixed numbers, first convert to an improper fraction. For example, 1 3/4 = 7/4, and 7 ÷ 4 = 1.75. This avoids mistakes when using a calculator or doing long division.
对于带分数，首先将其转换为假分数。例如，1 3/4 = 7/4，然后 7 ÷ 4 = 1.75。这可以避免在使用计算器或进行长除法时出错。
3. Converting Decimals to Percentages | 小数转百分数
To convert a decimal to a percentage, multiply the decimal by 100 and add the % symbol. This moves the decimal point two places to the right. For example, 0.45 × 100 = 45%.
要将小数转换为百分数，将小数乘以 100 并加上 % 符号。这相当于将小数点向右移动两位。例如，0.45 × 100 = 45%。
0.45 × 100 = 45%
If the decimal is greater than 1, the percentage will be greater than 100. For instance, 1.2 = 1.2 × 100 = 120%. This is useful when describing increases or when a value is larger than the original whole.
如果小数大于 1，百分数将大于 100。例如，1.2 = 1.2 × 100 = 120%。这在描述增加量或数值大于原始整体时很有用。
Small decimals can be trickier. For example, 0.025 × 100 = 2.5%. Remember that 0.1 is 10%, 0.01 is 1% and 0.001 is 0.1%. Place value links directly to the percentage scale.
较小的小数可能更容易出错。例如，0.025 × 100 = 2.5%。请记住，0.1 是 10%，0.01 是 1%，0.001 是 0.1%。位值与百分数刻度直接相关。
4. Converting Percentages to Fractions | 百分数转分数
To convert a percentage to a fraction, write the percentage as a fraction with denominator 100, then simplify if possible. For example, 45% = 45/100 = 9/20.
要将百分数转换为分数，将百分数写成分母为 100 的分数，然后尽可能化简。例如，45% = 45/100 = 9/20。
45% = 45/100 = 9/20
If the percentage includes a decimal, such as 12.5%, first multiply numerator and denominator by 10 or 100 to remove the decimal: 12.5/100 = 125/1000 = 1/8.
如果百分数包含小数，例如 12.5%，首先将分子和分母乘以 10 或 100 以去掉小数：12.5/100 = 125/1000 = 1/8。
Percentages greater than 100 can be written as improper fractions or mixed numbers. For example, 140% = 140/100 = 7/5 = 1 2/5. Always simplify to the lowest terms unless the question says otherwise.
大于 100 的百分数可以写成假分数或带分数。例如，140% = 140/100 = 7/5 = 1 2/5。除非题目另有说明，否则始终化简到最简形式。
5. Converting Fractions to Percentages | 分数转百分数
To convert a fraction to a percentage, either convert the fraction to a decimal first and then multiply by 100, or find an equivalent fraction with denominator 100. For example, 3/5 = 3 ÷ 5 = 0.6 = 60%.
要将分数转换为百分数，可以先将分数转换为小数，然后乘以 100，或者找到一个分母为 100 的等价分数。例如，3/5 = 3 ÷ 5 = 0.6 = 60%。
3/5 = 60/100 = 60%
The equivalent-fraction method works well when the denominator is a factor of 100, such as 2, 4, 5, 10, 20, 25 or 50. For other denominators, division is usually easier and more reliable.
当分母是 100 的因数（如 2、4、5、10、20、25 或 50）时，等价分数法效果很好。对于其他分母，除法通常更容易且更可靠。
For 1/3, division gives 0.333…, which is 33.333…%. In many KS3 questions, you can write this as 33 1/3% or round to 33.3% if asked to give an answer to one decimal place.
对于 1/3，除法得到 0.333…，也就是 33.333…%。在许多 KS3 题目中，你可以将其写成 33 1/3%，或者如果要求保留一位小数，则四舍五入为 33.3%。
6. Comparing Fractions, Decimals and Percentages | 比较分数、小数和百分数
To compare two or more quantities written in different forms, convert all of them to the same form. Using decimals is often the fastest method because place value makes ordering straightforward.
要比较以不同形式表示的两个或多个数量，先将它们全部转换为同一种形式。使用小数通常是最快的方法，因为位值使排序变得简单。
For example, to compare 7/20, 0.36 and 35%, convert to decimals: 7/20 = 0.35, 0.36 stays 0.36, and 35% = 0.35. Now it is clear that 0.36 is the largest.
例如，要比较 7/20、0.36 和 35%，转换为小数：7/20 = 0.35，0.36 保持不变，35% = 0.35。现在很明显 0.36 最大。
| 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% |
Knowing these common equivalences by heart will save time in the test. They also help you estimate and check the reasonableness of your answers before writing them down.
熟记这些常见的等价关系可以在考试中节省时间。它们还能帮助你在写下答案之前进行估算并检查答案的合理性。
7. Ordering Mixed Quantities | 混合数量的排序
When a question asks you to order mixed fractions, decimals and percentages from smallest to largest, follow a clear method: convert every value to a decimal, record the original forms next to their decimals, then sort.
当题目要求你将分数、小数和百分数混合排序时，遵循明确的方法：将每个值转换为小数，在小数旁边记录原始形式，然后排序。
Example: Order 0.7, 3/8, 65% and 0.55 from smallest to largest. Converting gives 0.7, 0.375, 0.65, 0.55. The correct order is 3/8, 0.55, 65%, 0.7.
示例：将 0.7、3/8、65% 和 0.55 从小到大排序。转换后得到 0.7、0.375、0.65、0.55。正确顺序是 3/8、0.55、65%、0.7。
3/8 = 0.375, 65% = 0.65
Always write the original forms in your final ordered list, not the decimals you used for sorting. This shows the examiner that you know how to convert and compare accurately.
最终排序列表中始终写出原始形式，而不是你用于排序的小数。这向阅卷老师表明你能够准确地进行转换和比较。
If two values convert to the same decimal, such as 0.5 and 50%, write them in either order, but be ready to explain that they are equivalent.
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