📚 Working with Fractions, Decimals and Percentages | 分数、小数与百分数的转换与运算
In the Cambridge Key Stage 3 mathematics curriculum, fractions, decimals and percentages form a connected family of number representations. Understanding how to convert between them and carry out calculations accurately is essential for algebra, geometry, data handling and everyday problem solving. This article revises the key methods, common exam question types and the most frequent mistakes students make.
在剑桥 Key Stage 3 数学课程中，分数、小数和百分数构成了一组相互关联的数的表示方式。理解它们之间的转换并能准确进行计算，是学习代数、几何、数据处理以及解决日常生活中问题的基础。本文将系统复习关键方法、常见考试题型以及学生最常犯的错误。
1. Understanding the Core Equivalence | 理解核心等价关系
All three forms represent parts of a whole. A fraction such as 1/4, a decimal 0.25 and a percentage 25% have exactly the same value. To convert between them, start from the idea that 'percent' means 'out of 100'.
三种形式都表示整体的一部分。分数 1/4、小数 0.25 和百分数 25% 具有完全相同的值。在它们之间转换时，首先要记住 “percent” 的意思是 “每 100”。
1/4 = 0.25 = 25%
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/10 | 0.1 | 10% |
| 3/4 | 0.75 | 75% |
2. Converting Fractions to Decimals | 分数转小数
0.375. Some fractions terminate, such as 1/4 = 0.25; others recur, such as 1/3 = 0.333...
要将分数转换为小数，用分子除以分母。例如 3/8 表示 3 ÷ 8，得到 0.375。有些分数是有限小数，如 1/4 = 0.25；有些是循环小数，如 1/3 = 0.333...
3/8 = 3 ÷ 8 = 0.375
If you do not have a calculator, use long division. Write the numerator inside the division bracket and the denominator outside, then add decimal zeros as needed. Recognising common fraction-decimal equivalents also speeds up your work.
如果没有计算器，可以使用长除法。把分子放在除号内，分母放在除号外，然后根据需要补零。熟记常见的分数与小数对应关系也能加快计算速度。
3. Converting Decimals to Percentages | 小数转百分数
uivalent to moving the decimal point two places to the right. For example, 0.85 × 100 = 85%, and 0.025 × 100 = 2.5%. A decimal greater than 1, such as 1.25, becomes 125%.
要将小数转换为百分数，乘以 100。这相当于将小数点向右移动两位。例如 0.85 × 100 = 85%，0.025 × 100 = 2.5%。大于 1 的小数，如 1.25，变成 125%。
0.85 × 100 = 85%
Be careful with very small decimals. For instance, 0.004 is equal to 0.4%, not 4%, because the decimal point moves two places to the right and the zero place value must be preserved.
注意非常小的小数。例如 0.004 等于 0.4%，而不是 4%，因为小数点只向右移动两位，必须保留零的位值。
4. Converting Percentages to Fractions | 百分数转分数
ge to a fraction, write the percentage over 100 and simplify if possible. For instance, 40% = 40/100 = 2/5. If the percentage contains a decimal, such as 12.5%, multiply both numerator and denominator by 10 or 100 to clear the decimal first.
要将百分数转换为分数，先把百分数写在 100 之上，然后尽可能约分。例如 40% = 40/100 = 2/5。如果百分数含有小数，如 12.5%，先将分子和分母同时乘以 10 或 100 以去掉小数点，再约分。
40% = 40/100 = 2/5
12.5% = 12.5/100 = 125/1000 = 1/8
Always check whether your final fraction can be simplified. Use the highest common factor of the numerator and denominator to reduce it in one step.
最后务必检查分数是否可以约分。用分子和分母的最大公因数进行一次性约分最为高效。
5. Ordering and Comparing | 排序与比较
t. Decimals are usually the easiest for comparison. For example, to order 2/5, 0.45 and 38%, write 2/5 = 0.40, 0.45 = 0.45 and 38% = 0.38. The ascending order is 38%, 2/5, 0.45.
当需要对分数、小数和百分数的混合形式进行排序时，先将它们都转换为同一种形式。通常转换为小数最容易比较。例如，要排序 2/5、0.45 和 38%，可写为 2/5 = 0.40、0.45 = 0.45、38% = 0.38。升序排列为 38%、2/5、0.45。
Use the same number of decimal places when comparing. Writing 0.40 and 0.45 makes it clear that both are larger than 0.38, and that 0.45 is larger than 0.40.
比较时使用相同的小数位数。写成 0.40 和 0.45 后，可以清楚地看出两者都大于 0.38，且 0.45 大于 0.40。
6. Calculating Percentages of Amounts | 求一个数的百分数
onvert the percentage to a decimal or fraction and multiply by the amount. For example, 15% 0 is 0.15 × 80 = 12. Using fractions, 15/100 × 80 = 12 as well. Mental benchmarks help: 50% is a half, 25% is a quarter, and 10% is a tenth.
要求一个数的百分数，先把百分数转换为小数或分数，再乘以这个数。例如，80 的 15% 是 0.15 × 80 = 12。使用分数计算也一样：15/100 × 80 = 12。心算时可以利用基准值：50% 是一半，25% 是四分之一，10% 是十分之一。
15% of 80 = 0.15 × 80 = 12
For non-calculator questions, break the percentage into easy parts. To find 35% of 80, calculate 10% of 80 = 8, 5% of 80 = 4, then combine three lots of 10% plus one lot of 5%: 24 + 4 = 28.
对于不使用计算器的题目，可以把百分数拆成容易计算的部分。要求 80 的 35%，先算 80 的 10% = 8，80 的 5% = 4，然后用三个 10% 加一个 5%：24 + 4 = 28。
7. Percentage Increase and Decrease | 百分数增减
To increase an amount by p%, multiply by (1 + p/100). To decrease an amount by p%, multiply by (1 - p/100). For example, increasing 60 by 20% gives 60 × 1.20 = 72, while decreasing 80 by 15% gives 80 × 0.85 = 68. Percentage change is calculated by (change ÷ original) × 100.
要将一个数增加 p%，乘以 (1 + p/100)；减少 p%，乘以 (1 - p/100)。例如，60 增加 20% 得 60 × 1.20 = 72；80 减少 15% 得 80 × 0.85 = 68。百分比变化用（变化量 ÷ 原值）× 100 计算。
Increase: 60 × 1.20 = 72
Decrease: 80 × 0.85 = 68
When a price increases then decreases by the same percentage, the final value is not the original. A £100 item increased by 10% becomes £110; a 10% decrease then gives £99, because the 10% is taken from the new, larger amount.
当一个价格先增加再减少相同的百分数时，最终值并不是原值。100 英镑增加 10% 变为 110 英镑；再减少 10% 得到 99 英镑，因为减少的 10% 是从新的、更大的数值中扣除的。
8. Fraction Arithmetic: Addition and Subtraction | 分数加减法
rking with mixed numbers, convert them to improper fractions first or combine whole-number parts and fraction parts separately.
要进行分数加减法，首先找到公分母。例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12。对于带分数，可以先转换为假分数，也可以分别处理整数部分和分数部分。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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