📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数
In KS3 Mathematics, understanding how fractions, decimals and percentages relate to each other is a core skill. This article will guide you through their conversions, comparisons and calculations, with clear examples and practical tips. Whether you are tackling a test or building confidence for IGCSE, these fundamentals are essential.
在 KS3 数学中，理解分数、小数和百分数之间的关系是一项核心技能。本文将带你掌握它们之间的转换、比较和计算，提供清晰的例子和实用技巧。无论你是在准备考试还是为 IGCSE 打基础，这些基础知识都至关重要。
1. Understanding the Three Forms | 理解三种表示形式
A fraction shows a part of a whole using a numerator and a denominator, like ½ or ¾. A decimal uses a decimal point to show the same value, for example 0.5 or 0.75. A percentage means ‘out of 100’ and is written with the % symbol, such as 50% or 75%.
分数用分子和分母表示整体的一部分，如 ½ 或 ¾。小数使用小数点表示相同的值，例如 0.5 或 0.75。百分数意为“每一百份中的份数”，并用 % 符号书写，如 50% 或 75%。
All three forms can represent exactly the same amount. For instance, one-half is ½ as a fraction, 0.5 as a decimal and 50% as a percentage. Recognising this equivalence is the first step towards fluency.
这三种形式都可以表示完全相同的数量。例如，一半可以是分数 ½、小数 0.5 以及百分数 50%。认识到这种等价关系是达到熟练的第一步。
2. Converting Fractions to Decimals | 分数转换为小数
To change a fraction to a decimal, divide the numerator by the denominator. For example, to write ¾ as a decimal, divide 3 ÷ 4 = 0.75. If the division does not terminate, the decimal will be recurring; for example, ⅓ becomes 0.333…, often written as 0.3̅.
要将分数转换为小数，用分子除以分母即可。例如，把 ¾ 写成小数，计算 3 ÷ 4 = 0.75。如果除法不能终止，小数就会循环；例如 ⅓ 变成 0.333…，通常记作 0.3̅。
A straightforward fraction like ⅛ gives 0.125. Knowing common fraction-to-decimal equivalents by heart, such as ½ = 0.5, ¼ = 0.25, ⅕ = 0.2, can speed up mental calculations.
像 ⅛ 这样简单的分数得到 0.125。熟记常见的分数对应的小数，如 ½ = 0.5、¼ = 0.25、⅕ = 0.2，可以加快心算速度。
3. Converting Decimals to Fractions | 小数转换为分数
To convert a terminating decimal to a fraction, write the digits after the decimal point as the numerator over a power of ten. The denominator depends on the number of decimal places: one place uses 10, two places use 100, three places use 1000, and so on. Then simplify the fraction if possible.
要将有限小数转换为分数，把小数点后的数字作为分子，分母为 10 的幂。分母取决于小数位数：一位小数分母为 10，两位为 100，三位为 1000，以此类推。如果可能，再将分数化简。
For example, 0.6 = 6/10 = ⅗. And 0.125 = 125/1000 = ⅛. With recurring decimals like 0.3̅, we use algebraic methods for conversion, which will be covered later in KS3.
例如，0.6 = 6/10 = ⅗。0.125 = 125/1000 = ⅛。对于循环小数如 0.3̅，我们使用代数方法转换，这会在 KS3 后续课程中学习。
4. Converting Fractions to Percentages | 分数转换为百分数
Because percentage means ‘out of 100’, we can convert a fraction by finding an equivalent fraction with a denominator of 100. If that is not convenient, multiply the fraction by 100%. For instance, ⅗ = (3 ÷ 5) × 100% = 0.6 × 100% = 60%.
因为百分数意为“每一百中的份数”，我们可以通过寻找一个分母为 100 的等价分数来转换。若不方便，则将分数乘以 100%。例如，⅗ = (3 ÷ 5) × 100% = 0.6 × 100% = 60%。
Another method: first change the fraction to a decimal, then multiply by 100. So ⅞ = 0.875, and 0.875 × 100 = 87.5%, which is perfectly acceptable as a percentage.
另一种方法：先将分数化为小数，再乘以 100。所以 ⅞ = 0.875，0.875 × 100 = 87.5%，这完全可以用作百分数。
5. Converting Percentages to Fractions | 百分数转换为分数
To write a percentage as a fraction, put the percentage number over 100 and simplify. For example, 45% = 45/100 = 9/20. If the percentage contains a decimal, like 12.5%, write it as 12.5/100, multiply numerator and denominator by 10 to eliminate the decimal, then simplify: 125/1000 = ⅛.
要把百分数写成分数，将百分号前的数字放在 100 之上，然后化简。例如，45% = 45/100 = 9/20。如果百分数包含小数，如 12.5%，写成 12.5/100，分子分母同乘 10 以去掉小数点，然后化简：125/1000 = ⅛。
Always reduce the fraction to its simplest form by dividing the numerator and denominator by their highest common factor. This neatens your work and makes comparisons easier.
一定要用分子和分母的最大公因数约分，把分数化成最简形式。这会让你的解答更整洁，也比较更容易。
6. Converting Decimals to Percentages and Back | 小数与百分数的相互转换
To change a decimal to a percentage, multiply by 100 and add the % sign. Thus 0.47 becomes 47%, and 1.2 becomes 120%. To convert a percentage to a decimal, divide by 100. So 85% = 0.85 and 7% = 0.07.
要将小数转换为百分数，乘以 100 并加上 % 号。因此 0.47 变成 47%，1.2 变成 120%。要把百分数转换为小数，除以 100。所以 85% = 0.85，7% = 0.07。
Remember: moving the decimal point two places to the right converts a decimal to a percentage; moving it two places to the left converts a percentage to a decimal. This works for any number.
记住：小数点向右移动两位可将小数转为百分数；向左移动两位可将百分数转为小数。这对任何数字都适用。
7. Comparing Fractions, Decimals and Percentages | 比较分数、小数和百分数
To compare quantities given in different forms, convert them all to the same form – usually decimals or percentages. For example, to decide which is largest among ⅖, 0.35 and 30%, convert: ⅖ = 0.4, 30% = 0.3. Clearly 0.4 > 0.35 > 0.3, so ⅖ is the greatest.
要比较用不同形式给出的数量，先把它们都转换成同一种形式——通常是小数或百分数。例如，要判断 ⅖、0.35 和 30% 中哪个最大，转换：⅖ = 0.4，30% = 0.3。显然 0.4 > 0.35 > 0.3，所以 ⅖ 最大。
Ordering numbers is straightforward once they share the same format. Practising this skill helps with data interpretation and probability questions throughout KS3.
一旦数字采用相同的形式，排序就很简单。练习这项技能有助于在整个 KS3 阶段处理数据分析和概率问题。
8. Calculations Involving Percentages of Amounts | 涉及一个数的百分数的计算
To find a percentage of an amount, change the percentage to a decimal or fraction and multiply. For instance, 20% of 150 = 0.2 × 150 = 30. Alternatively, find 10% first (15) and double it. This method is useful for mental maths.
要求一个数的百分数，先把百分数化成小数或分数，再相乘。例如，150 的 20% = 0.2 × 150 = 30。或者，先求 10%（15）再乘以 2。这种方法对心算很有用。
When working with harder percentages like 17.5%, change to a decimal (0.175) and multiply, or break it down: 10% + 5% + 2.5%. Both approaches give the same answer.
遇到较难的百分数如 17.5% 时，化成小数（0.175）再相乘，或者拆分为 10% + 5% + 2.5%。两种方法得到的结果相同。
9. Adding and Subtracting Fractions | 分数的加减
To add or subtract fractions, they must share the same denominator. Find a common denominator – usually the least common multiple of the denominators. For ⅓ + ¼, the LCM of 3 and 4 is 12. Convert: ⅓ = 4/12, ¼ = 3/12, then add to get 7/12.
要加减分数，它们必须有相同的分母。找一个公分母——通常是分母的最小公倍数。对于 ⅓ + ¼，3 和 4 的最小公倍数是 12。转换：⅓ = 4/12，¼ = 3/12，然后相加得到 7/12。
With mixed numbers, deal with whole parts first or convert to improper fractions. For example, 1¼ + 2½ = 5/4 + 5/2 = 5/4 + 10/4 = 15/4 = 3¾.
对于带分数，可以先处理整数部分，或者化为假分数。例如，1¼ + 2½ = 5/4 + 5/2 = 5/4 + 10/4 = 15/4 = 3¾。
10. Multiplying and Dividing Fractions | 分数的乘除
Multiplying fractions is straightforward: multiply the numerators together and the denominators together. For ⅔ × ⅘, we get (2×4)/(3×5) = 8/15. Cancel any common factors before multiplying to keep numbers small.
分数乘法很简单：分子乘分子，分母乘分母。对于 ⅔ × ⅘，得到 (2×4)/(3×5) = 8/15。在乘之前先约去公因数，可以让数字保持较小。
To divide by a fraction, multiply by its reciprocal (flip the second fraction). So ⅔ ÷ ⅘ becomes ⅔ × 5/4 = 10/12, which simplifies to ⅚.
除以一个分数，乘以它的倒数（将第二个分数翻转）。所以 ⅔ ÷ ⅘ 变成 ⅔ × 5/4 = 10/12，化简为 ⅚。
For mixed numbers, convert them to improper fractions before multiplying or dividing. This ensures accurate results every time.
对于带分数，在乘除之前先把它们化成假分数。这每次都能保证结果准确。
11. Common Mistakes to Avoid | 常见错误要避免
A typical error is adding denominators when adding fractions. Remember: only add numerators if denominators are the same. Another mistake is confusing 0.5% with 50% – 0.5% equals 0.005 as a decimal, while 50% equals 0.5. Pay close attention to the % symbol.
一个典型错误是在分数加法时将分母相加。记住：只有当分母相同时，才只加分子。另一个错误是把 0.5% 与 50% 混淆——0.5% 等于小数 0.005，而 50% 等于 0.5。要密切注意 % 符号。
When converting recurring decimals to fractions, do not round early; use the exact algebraic method to capture the infinite repeating part. With percentages greater than 100%, remember that 150% = 1.5, not 0.15.
在将循环小数转换成分数时，不要过早四舍五入；要使用精确的代数方法来处理无限循环部分。遇超过 100% 的百分数时，记住 150% = 1.5，而不是 0.15。
12. Practice Techniques for Fluency | 提升熟练度的练习方法
Build speed by memorising a set of key equivalences: ½ = 0.5 = 50%, ¼ = 0.25 = 25%, ⅓ ≈ 33.3%, ⅔ ≈ 66.7%, ⅕ = 0.2 = 20%, ⅛ = 0.125 = 12.5%. These appear again and again in word problems and real-life contexts.
通过记忆一组关键的等价关系来提高速度：½ = 0.5 = 50%，¼ = 0.25 = 25%，⅓ ≈ 33.3%，⅔ ≈ 66.7%，⅕ = 0.2 = 20%，⅛ = 0.125 = 12.5%。这些在应用题和现实情境中会反复出现。
Use flashcards or quick-fire quizzes to test yourself. Draw a three-column table with headings Fraction, Decimal, Percentage and fill in missing values. This deepens understanding and reveals relationships instantly.
使用抽认卡或快速提问来测试自己。画一个三列表格，标题为分数、小数、百分数，然后填入空缺值。这能加深理解，并让你迅速发现关联。
Apply your skills to real problems, such as working out discounts (25% off a £40 item) or recipe quantities (¾ of a cup of flour). Linking maths to everyday situations makes learning stick.
将你的技能应用于实际问题，如计算折扣（40英镑商品的 25% 折扣）或食谱分量（¾ 杯面粉）。将数学与日常情境联系起来，能让学习内容记得更牢。
Published by TutorHao | Cambridge KS3 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导