📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three different ways of representing parts of a whole. They appear everywhere in IGCSE Mathematics, from basic arithmetic to probability and data interpretation. A solid understanding of how to convert between them and perform calculations is essential for success in the Edexcel IGCSE exam. This article will guide you through the key concepts, methods and common pitfalls, ensuring you can tackle any question with confidence.
分数、小数和百分数是表示整体的一部分的三种不同方式。它们在IGCSE数学中无处不在，从基本算术到概率和数据解释。扎实掌握它们之间的相互转换以及运算方法对于在爱德思IGCSE考试中取得成功至关重要。本文将引导你学习关键概念、方法和常见陷阱，确保你能充满信心地应对任何题目。
1. Understanding Fractions | 理解分数
A fraction represents a part of a whole or a division of quantities. It is written as a numerator over a denominator, separated by a vinculum. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have.
分数表示整体的一部分或数量的除法。它写作分子除以分母，中间用分数线隔开。分母告诉你整体被分成了多少等份，分子告诉你拥有其中的多少份。
Proper fractions have a numerator smaller than the denominator (e.g. ½, ¾). Improper fractions have a numerator larger than or equal to the denominator (e.g. ⁵⁄₃, ⁷⁄₄). Mixed numbers combine a whole number and a proper fraction (e.g. 1½, 2¾).
真分数的分子比分母小（例如½, ¾）。假分数的分子大于或等于分母（例如⁵⁄₃, ⁷⁄₄）。带分数由一个整数和一个真分数组合而成（例如1½, 2¾）。
In IGCSE, you must be comfortable converting between improper fractions and mixed numbers. To change a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the denominator.
在IGCSE中，你必须能熟练地在假分数和带分数之间转换。将带分数化为假分数：用整数乘以分母，加上分子，然后将结果放在分母上。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions look different but have the same value. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number. This is the basis of simplifying fractions – dividing the numerator and denominator by their highest common factor (HCF) until they cannot be reduced further. The result is a fraction in its simplest form.
等值分数看起来不同但数值相同。你可以通过将分子和分母同时乘以或除以同一个非零数来创造等值分数。这就是分数化简的基础——将分子和分母同时除以它们的最大公因数（HCF），直到不能再约分为止。所得结果即为最简分数。
For example, ¹²⁄₁₆ can be simplified by dividing both by 4, giving ¾. Always check if a fraction can be simplified before performing operations, as it makes calculations easier.
例如，¹²⁄₁₆可以通过分子分母同时除以4进行化简，得到¾。在进行运算前务必检查分数是否可以化简，因为这能让计算更简单。
- Multiply numerator and denominator by the same number to scale up: ½ = ²⁄₄ = ⁴⁄₈
- Divide by HCF to simplify: ⁸⁄₁₂ ÷ 4/4 = ²⁄₃
- 将分子分母乘以同一个数来扩展：½ = ²⁄₄ = ⁴⁄₈
- 除以最大公因数化简：⁸⁄₁₂ ÷ 4/4 = ²⁄₃
3. Adding and Subtracting Fractions | 分数的加法和减法
To add or subtract fractions, they must have the same denominator. If the denominators are different, find the lowest common denominator (LCD) – the lowest common multiple of the denominators. Convert each fraction to an equivalent fraction with the LCD, then add or subtract the numerators. Keep the denominator the same. Finally, simplify the result if possible.
要对分数进行加法或减法运算，它们必须具有相同的分母。如果分母不同，请找到最小公分母（LCD）——即分母的最小公倍数。将每个分数转换为以LCD为分母的等值分数，然后对分子进行加减运算。分母保持不变。最后，如可能则化简结果。
For mixed numbers, either change them to improper fractions first or add/subtract the whole numbers and fractions separately. Borrowing may be needed when subtracting mixed numbers.
对于带分数，要么先将它们化为假分数，要么分别对整数部分和分数部分进行加减。在做带分数减法时可能需要借位。
| Example | Solution |
| ¼ + ⅓ | LCD = 12 → ³⁄₁₂ + ⁴⁄₁₂ = ⁷⁄₁₂ |
| 2¼ − 1⅔ | Improper: ⁹⁄₄ − ⁵⁄₃ → LCD 12 → ²⁷⁄₁₂ − ²⁰⁄₁₂ = ⁷⁄₁₂ |
4. Multiplying and Dividing Fractions | 分数的乘法和除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. There is no need for a common denominator. Always cross-cancel common factors before multiplying to keep numbers small. For mixed numbers, convert them to improper fractions first.
分数乘法很直接：分子相乘，分母相乘。不需要公分母。在相乘前一定要先约去分子分母的公因数，让数字保持较小。对于带分数，先将其化为假分数。
Dividing fractions involves multiplying by the reciprocal. Write the first fraction, change the division sign to multiplication, and flip the second fraction (swap numerator and denominator). Then proceed as multiplication. This is often remembered as 'Keep, Change, Flip'.
分数除法涉及乘以倒数。写下第一个分数，将除号改为乘号，并将第二个分数翻转（交换分子和分母）。然后按照乘法进行。这通常被记作“保留、改变、翻转”。
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
5. Decimals and Place Value | 小数与位值
A decimal is another way of writing a fraction whose denominator is a power of 10. The digits after the decimal point represent tenths, hundredths, thousandths, and so on. The place value of each digit is crucial for ordering, rounding, and converting between fractions and decimals.
小数是分母为10的幂的分数的另一种写法。小数点后的数字分别表示十分位、百分位、千分位等等。每个数字的位值对于排序、四舍五入以及分数与小数之间的转换至关重要。
For example, 0.75 means 7 tenths and 5 hundredths, which is equivalent to ⁷⁵⁄₁₀₀ or simplified to ¾. Understanding decimal place value helps you read and write decimals accurately and perform arithmetic operations.
例如，0.75表示7个十分之一和5个百分之一，相当于⁷⁵⁄₁₀₀，化简后为¾。理解小数位值有助于你准确读写小数并进行算术运算。
6. Converting Fractions to Decimals | 分数化小数
hould recognise common fraction-decimal equivalences and know how to use division to find any decimal.
要将分数化为小数，用分子除以分母。你可以使用短除法或长除法。有些分数产生有限小数（如¼ = 0.25），而另一些则产生循环小数（如⅓ = 0.333...）。在IGCSE中，你要认识常见的分数-小数等价关系，并知道如何通过除法得出任何小数。
recurring decimals with a dot or line over the repeating digit(s). For instance, ⁵⁄₆ = 0.8̇3 or 0.83 with the 3 repeating.
如果分母含有2或5以外的质因数，小数将会循环。你可以在重复的数字上方加一个点或横线来表示循环小数。例如，⁵⁄₆ = 0.83̇，其中3循环。
7. Converting Decimals to Fractions | 小数化分数
to its lowest terms. Always count the number of decimal places to determine the power of 10 in the denominator.
要将有限小数化为分数，将小数写在其位值对应的分母上（例如0.7 = ⁷⁄₁₀，0.25 = ²⁵⁄₁₀₀ = ¼）。然后将分数化简到最简形式。务必数清小数位数以确定分母中10的幂。
For recurring decimals, a special algebraic method is needed. Set the decimal equal to x, multiply by a power of 10 to move the repeating block to the left of the decimal point, then subtract to eliminate the repeating part. Solve for x and simplify.
对于循环小数，需要一种特殊的代数方法。设该小数为x，乘以10的幂将循环节移到小数点左侧，然后通过减法消去循环部分。求解x并化简。
Example: 0.7̇ = 0.777... Let x = 0.777..., 10x = 7.777..., subtract: 9x = 7 → x = ⁷⁄₉.
例子：0.7̇ = 0.777... 设 x = 0.777...，10x = 7.777...，相减：9x = 7 → x = ⁷⁄₉。
8. Introduction to Percentages | 百分数简介
A percentage is a fraction with a denominator of 100. The symbol % means 'out of 100' or 'per hundred'. Percentages are widely used to compare proportions, express changes, and calculate discounts or interest. Understanding percentages as simply 'hundredths' makes conversions straightforward.
百分数是以100为分母的分数。符号%表示“每一百”或“百分之”。百分数广泛用于比较比例、表示变化以及计算折扣或利息。将百分数简单理解为“百分之一”使得转换变得简单直接。
For example, 45% means ⁴⁵⁄₁₀₀, which simplifies to ⁹⁄₂₀ as a fraction, and is 0.45 as a decimal. A percentage can be greater than 100% if a quantity increases to more than its original value.
例如，45%意指⁴⁵⁄₁₀₀，化简为分数是⁹⁄₂₀，写成小数是0.45。如果某个量增加到超过原值，百分数可以大于100%。
9. Converting Between Fractions, Decimals and Percentages | 三者间的相互转换
The three forms are interchangeable, and IGCSE questions often require you to switch between them fluently. To convert a percentage to a decimal, divide by 100 (move the decimal point two places left). To convert a decimal to a percentage, multiply by 100 (move two places right). To convert a fraction to a percentage, first convert to a decimal by division, then multiply by 100. Alternatively, find an equivalent fraction with denominator 100.
这三种形式可以互相转换，IGCSE题目常要求你熟练地进行切换。将百分数化为小数：除以100（将小数点向左移动两位）。将小数化为百分数：乘以100（向右移动两位）。将分数化为百分数：先通过除法化为小数，再乘以100。或者，找到一个分母为100的等值分数。
| Common Equivalents | Fraction | Decimal | Percentage |
| Half | ½ | 0.5 | 50% |
| Quarter | ¼ | 0.25 | 25% |
| Three quarters | ¾ | 0.75 | 75% |
| One fifth | ⅕ | 0.2 | 20% |
| One third | ⅓ | 0.3̇ | 33⅓% |
10. Finding a Percentage of a Quantity | 求一个量的百分之几
on and multiply by the quantity. Without a calculator, you can use mental methods: find 10% by dividing by 10, 1% by dividing by 100, and then build the required percentage. For example, 15% of 80 = (10% of 80) + (5% of 80) = 8 + 4 = 12.
求一个量的百分之几，将百分数转换成小数或分数，然后乘以这个量。在没有计算器的情况下，你可以使用心算方法：除以10得到10%，除以100得到1%，然后组合出所需的百分数。例如，求80的15%：80的10%为8，80的5%为4，相加得12。
When using a calculator, simply enter the percentage as a decimal: 15% h word problems involving percentage of amounts, especially when dealing with money or measurements – remember to include units in your final answer.
使用计算器时，只需将百分数作为小数输入：15% = 0.15，所以0.15 × 80 = 12。在涉及金额或测量的文字题中要小心，记住在最终答案中注明单位。
11. Percentage Increase and Decrease | 百分数的增加与减少
o its original value. The formula is: percentage change = (change ÷ original amount) × 100% e new value is greater than the original, it is an increase; if smaller, it is a decrease.
百分数增加或减少是将一个量的变化值与其原始值进行比较。公式为：百分数变化 = (变化值 ÷ 原始量) × 100%。如果新值大于原值，则为增加；如果小于原值，则为减少。
To calculate the new amount after a percentage change, you can use a multiplier. For an increase of R%, the multiplier is (1 + R/100). For a decrease of R%, it is (1 − R/100). Multiply the original amount by the multiplier to get the final amount. This method is efficient for compound changes, such as successive discounts or interest calculations.
要计算百分数变化后的新量，可以使用乘数。对于增加R%，乘数为(1 + R/100)。对于减少R%，乘数为(1 − R/100)。将原量乘以乘数即得最终量。这种方法对于复合变化非常高效，例如连续折扣或利息计算。
Example: Increase 200 by 12% → multiplier = 1.12 → 200 × 1.12 = 224.
例子：200增加12% → 乘数 = 1.12 → 200 × 1.12 = 224。
12. Reverse Percentages | 逆向百分数
Reverse percentage problems give you the final amount after a percentage increase or decrease and ask you to find the original amount. The key is to identify the multiplier that was applied and then divide the final amount by that multiplier. Do not simply subtract the percentage from the final amount – that is a common mistake.
逆向百分数问题给出百分数增减后的最终量，要求你求出原量。关键是找出所施加的乘数，然后用最终量除以该乘数。不要简单地从最终量中减去该百分数——这是一个常见错误。
For a price after a 20% increase, the final price represents 120% original, so the multiplier is 1.20. Original = Final ÷ 1.20. If given a price after a 15% discount, the final price is 85% of the original, multiplier 0.85 → Original = Final ÷ 0.85.
对于增长20%之后的价格，最终价格代表原价的120%，因此乘数是1.20。原价 = 最终价 ÷ 1.20。如果给出折扣15%之后的价格，最终价是原价的85%，乘数0.85 → 原价 = 最终价 ÷ 0.85。
Always check your answer by applying the percentage change to your calculated original amount – it should return the given final value.
务必通过将百分数变化施加在你计算出的原量上来进行检验——应当得到题目所给的最终值。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图