📚 Topic 500: Fractions, Decimals and Percentages | 主题500：分数、小数与百分数
In IGCSE Mathematics, fractions, decimals and percentages are three different ways of representing the same idea: a part of a whole. Topic 500 brings together the key skills you need to convert between these forms and use them confidently in calculations, from simplifying fractions to solving percentage increase problems. Many exam questions test your ability to switch quickly between all three forms, so a strong grasp of this topic is essential.
在IGCSE数学中，分数、小数和百分数是表示同一个概念——整体的一部分——的三种不同方式。主题500汇总了你需要掌握的关键技能，包括在这三种形式之间进行转换，以及自信地运用它们进行计算，从化简分数到解决百分数增减问题。许多考试题目会考察你在三种形式之间快速转换的能力，因此扎实掌握本主题非常重要。
1. Understanding Fractions | 理解分数
A fraction has a numerator (top) and a denominator (bottom). The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many of those parts you have.
分数有分子（上方）和分母（下方）。分母表示整体被分成多少等份；分子表示你取了多少份。
For example, ¾ means 3 parts out of 4 equal parts. If a pizza is cut into 4 slices and you eat 3, you have eaten ¾ of the pizza.
例如，¾ 表示 4 等份中的 3 份。如果一个披萨被切成 4 片，你吃了 3 片，那么你就吃了这个披萨的 ¾。
Fractions can be proper (numerator less than denominator, such as ⅝), improper (numerator greater than or equal to denominator, such as 9/4), or mixed numbers (a whole number with a proper fraction, such as 2 ¼).
分数可以是真分数（分子小于分母，如 ⅝）、假分数（分子大于或等于分母，如 9/4）或带分数（整数与真分数组合，如 2 ¼）。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value but different numerators and denominators. You can create equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.
等值分数具有相同的数值，但分子和分母不同。你可以将分子和分母同时乘以或除以同一个非零数来得到等值分数。
eir highest common factor (HCF). For example, 8/12 = 2/3 because the HCF of 8 and 12 is 4.
化简分数时，将分子和分母同时除以它们的最大公因数（HCF）。例如，8/12 = 2/3，因为 8 和 12 的最大公因数是 4。
Another example: 25/100 simplifies to ¼ by dividing both 25 and 100 by 25. Always check whether your final answer can be reduced further.
另一个例子：25/100 通过将 25 和 100 同时除以 25 化简为 ¼。始终检查最终答案是否可以继续约分。
When comparing fractions, it is easier to rewrite them with a common denominator. For example, to compare 5/6 and 7/9, the common denominator 18 gives 15/18 and 14/18, so 5/6 > 7/9.
比较分数时，先把它们改写为公分母会更容易。例如，比较 5/6 和 7/9，公分母 18 得到 15/18 和 14/18，因此 5/6 > 7/9。
3. Mixed Numbers and Improper Fractions | 带分数与假分数
An improper fraction has a numerator that is larger than or equal to its denominator, such as 7/4. A mixed number combines a whole number with a proper fraction, such as 1 ¾.
假分数的分子大于或等于分母，例如 7/4。带分数由整数和真分数组成，例如 1 ¾。
and the denominator stays the same. For example, 7/4 = 1 ¾ because 7 ÷ 4 = 1 remainder 3.
将假分数转换为带分数时，用分子除以分母。商是整数部分，余数成为新的分子，分母保持不变。例如，7/4 = 1 ¾，因为 7 ÷ 4 = 1 余 3。
tor, add the numerator, and place the result over the original denominator. For example, 2 ⅗ = (2 × 5 + 3)/5 = 13/5.
将带分数转换为假分数时，将整数乘以分母，加上分子，再把结果放在原分母之上。例如，2 ⅗ = (2 × 5 + 3)/5 = 13/5。
Before multiplying or dividing mixed numbers, always convert them to improper fractions first. This avoids mistakes and makes the calculation easier.
在乘或除带分数之前，始终先将它们转换为假分数。这样可以避免错误，并使计算更简单。
4. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common denominator (LCD) and rewrite each fraction with that denominator.
分数相加或相减时，它们必须有相同的分母。如果分母不同，找出最小公分母（LCD），并将每个分数改写为该分母。
Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. Here the LCD of 3 and 4 is 12.
示例：1/3 + 1/4 = 4/12 + 3/12 = 7/12。这里 3 和 4 的最小公分母是 12。
For subtraction, the same rule applies: 5/6 - 1/4 = 10/12 - 3/12 = 7/12.
减法同样适用：5/6 - 1/4 = 10/12 - 3/12 = 7/12。
When adding or subtracting mixed numbers, handle the whole number parts and fraction parts separately. If the fraction part becomes negative, borrow 1 from the whole number part.
加减带分数时，分别处理整数部分和分数部分。如果分数部分出现负数，从整数部分借 1。
5. Multiplying and Dividing Fractions | 分数乘除法
To multiply fractions, multiply the numerators together and the denominators together. Simplify before or after multiplying if possible.
分数相乘时，分子相乘，分母相乘。如有可能，在相乘前或后化简。
= 6/15 = 2/5. You can also cancel the common factor 3 before multiplying: 2/1 × 1/5 = 2/5.
示例：2/3 × 3/5 = 6/15 = 2/5。你也可以在相乘前约去公因数 3：2/1 × 1/5 = 2/5。
ts reciprocal (flip the second fraction). For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 ⅞.
分数相除时，乘以倒数（将第二个分数分子分母颠倒）。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 ⅞。
, first convert them to improper fractions. For example, 1 ½ × 2 ⅔ = 3/2 × 8/3 = 24/6 = 4.
对于乘除法中的带分数，先将它们转换为假分数。例如，1 ½ × 2 ⅔ = 3/2 × 8/3 = 24/6 = 4。
6. Decimals and Place Value | 小数与位值
A decimal uses a decimal point to separate whole numbers from fractional parts. Each position after the point represents tenths, hundredths, thousandths, and so on.
小数用小数点把整数部分和小数部分分开。小数点后的每一位分别表示十分之一、百分之一、千分之一等。
0 + 5/1000, which equals 375/1000. In expanded form, it is 3 × 0.1 + 7 × 0.01 + 5 × 0.001.
例如，0.375 表示 3/10 + 7/100 + 5/1000，等于 375/1000。展开形式为 3 × 0.1 + 7 × 0.01 + 5 × 0.001。
When comparing decimals, write them with the same number of decimal places. For example, 0.7 is the same as 0.70, and 0.65 < 0.7 because 0.65 < 0.70.
比较小数时，将它们写成相同的小数位数。例如，0.7 等于 0.70，且 0.65 < 0.7，因为 0.65 < 0.70。
Adding trailing zeros to the right of a decimal does not change its value. This is useful when lining up addition and subtraction calculations.
在小数右侧添加末尾的零不会改变其数值。这在列竖式进行加减法时非常有用。
7. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. Some fractions give terminating decimals; others give recurring decimals.
将分数转换为小数时，用分子除以分母。有些分数得到有限小数；有些得到循环小数。
1/8 = 0.125 are terminating decimals. By contrast, 1/3 = 0.333... is a recurring decimal.
示例：3/8 = 0.375 和 1/8 = 0.125 是有限小数。相比之下，1/3 = 0.333... 是循环小数。
l as a fraction over 10, 100, 1000, etc., and simplify. For example, 0.45 = 45/100 = 9/20.
将有限小数转换为分数时，先把小数写成以 10、100、1000 等为分母的分数，再化简。例如，0.45 = 45/100 = 9/20。
For recurring decimals, you can use a dot or bar notation, but at IGCSE level you are often asked to convert only terminating decimals to fractions.
对于循环小数，可以使用点或横线记号，但在 IGCSE 阶段通常只要求将有限小数转换为分数。
8. Percentages Basics | 百分数基础
A percentage is a fraction with denominator 100. The symbol % means 'per hundred'. For example, 45% = 45/100 = 9/20.
百分数是分母为 100 的分数。符号 % 表示“每一百”。例如，45% = 45/100 = 9/20。
ntage of a quantity, multiply the quantity by the percentage as a decimal. For example, 20% of 60 = 0.20 × 60 = 12.
求一个量的百分数时，将百分数化为小数后与该量相乘。例如，60 的 20% = 0.20 × 60 = 12。
Published by TutorHao | IGCSE 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