📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分比
In IGCSE Mathematics, fractions, decimals and percentages are not three separate topics. They are three different ways of describing the same proportion. You will see them in number calculations, estimation, probability, ratio and percentage problems. This revision guide covers the conversions, arithmetic rules and percentage techniques that examiners expect you to use accurately and quickly.
在 IGCSE 数学中，分数、小数和百分比并不是三个独立的主题。它们是描述同一个比例的三种不同方式。你会看到它们出现在数值计算、估算、概率、比和百分比问题中。本复习指南涵盖转换方法、运算规则和百分比技巧，这些正是考官希望你准确、快速掌握的内容。
1. The Core Idea: Three Forms of One Quantity | 核心概念：同一个量的三种形式
Every percentage, decimal and fraction is just a number expressed in a different notation. One half can be written as 1/2, 0.5 or 50%, but all three refer to exactly the same amount. The key skill is not memorising three isolated topics; it is seeing how they switch between forms.
每个百分比、小数和分数都只是同一个数用不同记号表示。二分之一可以写成 1/2、0.5 或 50%，但三者表示的数量完全相同。关键技能不是把三个主题割裂记忆，而是理解它们如何相互转换。
Use this equivalence as a mental anchor: 1/2 = 0.5 = 50%. Once you can turn any one form into the other two, percentage and fraction questions become far simpler. In many IGCSE questions, changing a percentage into a decimal makes multiplication easier, while changing a decimal into a fraction helps you simplify an exact answer.
请把这个等式当作心理锚点：1/2 = 0.5 = 50%。一旦你能把任意一种形式转换成另外两种，百分比和分数问题就会简单得多。在许多 IGCSE 题目中，把百分比转换为小数会让乘法更简单，而把小数转换为分数有助于化简精确答案。
Choose the form that fits the job. A percentage is best for comparing sizes, describing growth and reading statistics. A decimal is best when you are multiplying repeatedly or using a calculator. A fraction is best when the answer must be exact, or when you need to cancel and simplify.
要选择适合任务的形式。百分比最适合比较大小、描述增长和读取统计数据；小数最适合反复相乘或使用计算器的场合；分数最适合要求答案精确、或需要约分化简的场合。
2. Converting Fractions to Decimals and Percentages | 将分数转换为小数和百分比
onvert that decimal to a percentage, multiply by 100 and add the percent sign: 0.375 × 100% = 37.5%.
要把分数转换为小数，用分子除以分母。例如 3/8 表示 3 ÷ 8，得到 0.375。要把该小数转换为百分比，乘以 100 并加上百分号：0.375 × 100% = 37.5%。
decimal = numerator ÷ denominator
percentage = (numerator ÷ denominator) × 100%
art first and then add the whole number: 3/4 = 0.75, so 1¾ = 1 + 0.75 = 1.75, which is 175% is that a proper fraction gives a decimal smaller than 1 and a percentage smaller than 100% while a top-heavy fraction gives a decimal larger than 1 and a percentage larger than 100%. This single check catches most careless errors in an exam.
遇到 1¾ 这样的带分数，先转换小数部分，再加上整数部分：3/4 = 0.75，所以 1¾ = 1 + 0.75 = 1.75，也就是 175%。一个快速检验是：真分数得到小于 1 的小数和小于 100% 的百分比，而假分数得到大于 1 的小数和大于 100% 的百分比。这一条检验就能抓出考场上大多数粗心错误。
- on does not terminate, keep the fraction for later work: 1/3 = 0.333..., so write 33.33...% or a rounded 33.3% only if the question asks for a rounded value.
如果除法除不尽，后续计算请保留分数形式：1/3 = 0.333...，只有在题目要求取近似值时才写成 33.33...% 或四舍五入的 33.3%。 - Long division is often faster than reaching for a calculator when the denominator is 8, 20 or 25.
当分母是 8、20 或 25 时，长除法往往比按计算器更快。
3. Converting Decimals to Fractions and Percentages | 将小数转换为分数和百分比
To convert a decimal to a percentage, multiply by 100 and add the percent sign: 0.62 × 100% = 62% es, so 0.375 = 375/1000 = 3/8. Similarly, 0.36 = 36/100 = 9/25 and 0.125 = 125/1000 = 1/8.
要把小数转换为百分比，乘以 100 并加上百分号：0.62 × 100% = 62%。要把小数转换为分数，看最后一位数字的位值。0.375 有三位小数，所以 0.375 = 375/1000 = 3/8。同理，0.36 = 36/100 = 9/25，0.125 = 125/1000 = 1/8。
percentage = decimal × 100%
decimal = fraction of tenths / hundredths / thousandths, then cancel
and 0.0833... = 1/12. Remember that once a decimal has been rounded, it is no longer exact, so keep the fraction if the final answer must be precise.
循环小数有值得记住的分数搭档：0.333... = 1/3，0.666... = 2/3，0.1666... = 1/6，0.111... = 1/9，0.0833... = 1/12。要记住，小数一旦经过四舍五入就不再精确，所以如果最终答案必须精确，就保留分数形式。
- Step 1: count the decimal places and write the digits over 10, 100 or 1000.
第一步：数清小数位数，把数字写在 10、100 或 1000 上面作分子。 - Step 2: cancel the fraction down to its simplest form using common factors.
第二步：用公因数把分数约到最简形式。
4. Converting Percentages to Fractions and Decimals | 将百分比转换为分数和小数
To convert a percentage to a decimal, divide by 100 and drop the percent sign: 37.5% = 37.5 ÷ 100 = 0.375, and 7% .07. To convert a percentage to a fraction, place the percentage over 100 and simplify: 50% = 50/100 = 1/2, 20% = 20/100 = 1/5, and 37.5% = 37.5/100 = 375/1000 = 3/8.
要把百分比转换为小数，除以 100 并去掉百分号：37.5% = 37.5 ÷ 100 = 0.375，7% = 0.07。要把百分比转换为分数，把百分数放在 100 上面再化简：50% = 50/100 = 1/2，20% = 20/100 = 1/5，37.5% = 37.5/100 = 375/1000 = 3/8。
by 10 repeatedly until the numerator is a whole number, then cancel. Percentages above 100% give improper fractions: 125% = 125/100 = 5/4 = 1.25. Percentages below 1% give very small decimals: 0.5% = 0.005 = 1/200.
如果百分比中含有小数，把分子和分母反复乘以 10，直到分子变成整数，然后再约分。大于 100% 的百分比得到假分数：125% = 125/100 = 5/4 = 1.25。小于 1% 的百分比得到很小的小
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