📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three connected ways of describing parts of a whole. In IGCSE Mathematics, you must be able to convert between them, compare their sizes and use them in real calculations. This article covers the key ideas and skills needed for Year 7 and early IGCSE work.
分数、小数和百分数是描述整体的一部分的三种相互关联的方式。在 IGCSE 数学中，你必须能够在它们之间相互转换、比较大小，并在实际计算中运用它们。本文涵盖七年级及 IGCSE 早期阶段所需的关键概念和技能。
1. What Are Fractions? | 分数是什么？
A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts are being considered.
分数写作 a/b，其中 a 是分子，b 是分母。分母表示整体被分成了多少等份，分子表示你正在考虑其中的几份。
For example, 3/4 means that a whole is divided into 4 equal parts and we take 3 of them. Fractions can be proper (numerator smaller than denominator) or improper (numerator larger than or equal to denominator).
例如，3/4 表示一个整体被分成 4 等份，我们取其中的 3 份。分数可以是真分数（分子小于分母），也可以是假分数（分子大于或等于分母）。
A mixed number combines an integer and a proper fraction, such as 2 1/3. An improper fraction can be converted to a mixed number by dividing the numerator by the denominator.
带分数由一个整数和一个真分数组成，例如 2 1/3。假分数可以通过分子除以分母转换为带分数。
Fractions are also used as operators. Saying '3/4 of 20' means you multiply 20 by 3/4, which gives 15. This operator meaning is important in word problems.
分数也可以作为运算符号使用。说“20 的 3/4”意味着用 20 乘以 3/4，得到 15。这种运算含义在应用题中非常重要。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value even though they look different. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.
等值分数虽然看起来不同，但数值相同。你可以将分子和分母同时乘以或除以同一个非零数，从而得到一个等值分数。
1/2 = 2/4 = 3/6 = 4/8
The fraction 1/2 is in simplest form because 1 and 2 have no common factor other than 1. To simplify, divide the numerator and denominator by their highest common factor (HCF).
1/2 是最简形式，因为 1 和 2 除了 1 以外没有公因数。化简时，将分子和分母同时除以它们的最大公因数（HCF）。
12 and 18 is 6, so 12/18 = 2/3. Always check whether the result can be simplified further.
例如，12/18：12 和 18 的最大公因数是 6，所以 12/18 = 2/3。始终检查结果是否还能继续化简。
Simplifying fractions makes them easier to compare and calculate with. It is also expected in final answers unless the question asks for a decimal or percentage.
化简分数使它们更容易比较和计算。除非题目要求小数或百分数，否则最终答案通常应写成最简分数。
3. Converting Between Fractions and Decimals | 分数与小数互化
.75. Some fractions produce terminating decimals, while others produce recurring decimals.
将分数转换为小数，用分子除以分母。例如，3/4 = 3 ÷ 4 = 0.75。有些分数得到有限小数，有些则得到循环小数。
A terminating decimal has a finite number of decimal places, such as 0.25 or 0.375. A recurring decimal has a repeating pattern, such as 0.333... or 0.142857142857...
有限小数的小数位数是有限的，例如 0.25 或 0.375。循环小数有重复模式，例如 0.333... 或 0.142857142857...。
ver a power of 10 and simplify. For example, 0.6 = 6/10 = 3/5, and 0.125 = 125/1000 = 1/8.
将小数转换为分数，把小数写成分母是 10 的幂的分数，然后化简。例如，0.6 = 6/10 = 3/5，0.125 = 125/1000 = 1/8。
The place value of the last digit tells you the denominator: tenths for 1 decimal place, hundredths for 2 decimal places, thousandths for 3 decimal places, and so on.
最后一位数字的位值决定分母：1 位小数对应十分位，2 位小数对应百分位，3 位小数对应千分位，依此类推。
4. Converting Between Fractions and Percentages | 分数与百分数互化
Percent means 'per hundred', so a percentage is a fraction with denominator 100. To convert a fraction to a percentage, multiply the fraction by 100%.
百分数表示“每一百”，因此百分数就是分母为 100 的分数。将分数转换为百分数，用分数乘以 100%。
For example, 3/5 × 100% = 60%. To convert a percentage to a fraction, write it over 100 and simplify: 75% = 75/100 = 3/4.
例如，3/5 × 100% = 60%。将百分数转换为分数，写成除以 100 的形式并化简：75% = 75/100 = 3/4。
Common conversions should be memorised: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%, 1/8 = 12.5%.
常见转换需要记住：1/2 = 50%，1/4 = 25%，3/4 = 75%，1/5 = 20%，1/10 = 10%，1/8 = 12.5%。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
This table summarises the most frequently tested conversions. Learning them saves time in the exam.
这个表格总结了最常考的转换。记住它们可以在考试中节省时间。
5. Converting Between Decimals and Percentages | 小数与百分数互化
To convert a decimal to a percentage, multiply by 100 and add the % sign. For example, 0.45 × 100 = 45%.
将小数转换为百分数，乘以 100 并加上 % 符号。例如，0.45 × 100 = 45%。
To convert a percentage to a decimal, divide by 100. For example, 7% = 0.07 and 125% = 1.25.
将百分数转换为小数，除以 100。例如，7% = 0.07，125% = 1.25。
Be careful with place value: 0.5 = 50%, 0.05 = 5%, and 0.005 = 0.5%. Moving the decimal point two places is a quick method.
注意位值：0.5 = 50%，0.05 = 5%，0.005 = 0.5%。小数点移动两位是一种快捷方法。
When a percentage is greater than 100%, the decimal is greater than 1. When a percentage is less than 1%, the decimal is less than 0.01.
当百分数大于 100% 时，对应小数大于 1。当百分数小于 1% 时，对应小数小于 0.01。
6. Comparing and Ordering Fractions, Decimals and Percentages | 比较与排序分数、小数和百分数
To compare fractions, decimals and percentages, convert them all to the same form, usually decimals or percentages. Then arrange them in order.
比较分数、小数和百分数时，将它们全部转换成相同的形式，通常转换成小数或百分数。然后按顺序排列。
For example, compare 3/5, 0.62 and 60%. Convert: 3/5 = 0.6, 0.62 = 0.62, 60% = 0.6. So 0.62 is the largest, while 3/5 and 60% are equal.
例如，比较 3/5、0.62 和 60%。转换：3/5 = 0.6，0.62 = 0.62，60% = 0.6。因此 0.62 最大，而 3/5 和 60% 相等。
When comparing fractions with different denominators, finding a common denominator is another reliable method. For example, 2/3 and 3/5: common denominator 15 gives 10/15 and 9/15, so 2/3 > 3/5.
比较不同分母的分数时，找公分母是另一种可靠方法。例如，2/3 和 3/5：公分母 15 得到 10/15 和 9/15，所以 2/3 > 3/5。
- Convert all values to decimals for a quick comparison. | 将所有值转换为小数以便快速比较。
- Use a common denominator when comparing several fractions. | 比较多个分数时使用公分母。
- Remember that 0.1 = 10% and 0.01 = 1%. | 记住 0.1 = 10%，0.01 = 1%。
7. Fractions of Amounts | 求一个量的几分之几
multiply by the numerator. For example, to find 3/4 of 80: 80 ÷ 4 = 20, then 20 × 3 = 60.
求一个量的几分之几，
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