📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Mastering the links between fractions, decimals and percentages is essential for IGCSE Mathematics. This article walks you through the key skills, common pitfalls and exam-ready methods.
掌握分数、小数和百分数之间的联系是 IGCSE 数学的核心基本功。本文将带你梳理关键技能、常见错误和应试方法。
1. Understanding Fractions | 理解分数
A fraction shows equal parts of a whole. The top number is the numerator, and the bottom number is the denominator.
分数表示整体的等份。上面的数叫分子，下面的数叫分母。
The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts are being considered.
分母表示整体被分成多少等份，分子表示取了多少份。
For example, 3/4 means 3 out of 4 equal parts. You can picture a pizza cut into 4 slices and taking 3 slices.
例如，3/4 表示 4 等份中的 3 份。你可以想象一个披萨被切成 4 块，取走 3 块。
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 the denominator by the same non-zero number.
等值分数虽然形式不同，但数值相同。你可以把分子和分母同时乘以或除以同一个非零数，得到等值分数。
To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF).
化简分数时，用分子和分母的最大公因数（HCF）同时除它们。
ple, 6/8 simplifies to 3/4 because the HCF of 6 and 8 is 2, and 6 ÷ 2 = 3 while 8 ÷ 2 = 4.
例如，6/8 化简为 3/4，因为 6 和 8 的最大公因数是 2，6 ÷ 2 = 3，8 ÷ 2 = 4。
3. Mixed Numbers and Improper Fractions | 带分数与假分数
A mixed number contains a whole number part and a proper fraction part, such as 2 1/3.
带分数包含整数部分和真分数部分，例如 2 1/3。
An improper fraction has a numerator that is greater than or equal to its denominator, such as 7/3.
假分数的分子大于或等于分母，例如 7/3。
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 original denominator.
把带分数化为假分数：用整数乘分母，加上分子，把结果放在原分母上方。
× 3 + 1)/3 = 7/3. To change back, divide 7 by 3 to get 2 with remainder 1, so 7/3 = 2 1/3.
例如，2 1/3 = (2 × 3 + 1)/3 = 7/3。反过来，7 除以 3 得商 2 余 1，所以 7/3 = 2 1/3。
4. Adding and Subtracting Fractions | 分数的加减法
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common denominator (LCD) first.
分数相加减时，它们必须有相同的分母。如果分母不同，要先找最小公分母（LCD）。
D of 4 and 6 is 12. Convert: 1/4 = 3/12 and 1/6 = 2/12, so 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
例如，计算 1/4 + 1/6，4 和 6 的最小公分母是 12。转换：1/4 = 3/12，1/6 = 2/12，所以 1/4 + 1/6 = 3/12 + 2/12 = 5/12。
When adding mixed numbers, add the whole number parts and the fraction parts separately. If the fraction part becomes improper, convert it and carry over to the whole number.
带分数相加时，整数部分和分数部分分别相加。如果分数部分变成假分数，要进位到整数部分。
For example, 2 3/4 + 1 2/4 = 3 5/4 = 4 1/4 because 5/4 = 1 1/4.
例如，2 3/4 + 1 2/4 = 3 5/4 = 4 1/4，因为 5/4 = 1 1/4。
5. Multiplying and Dividing Fractions | 分数的乘除法
To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the result if possible.
分数相乘时，分子乘分子，分母乘分母，能化简就化简。
For example, 2/5 × 3/4 = (2 × 3)/(5 × 4) = 6/20 = 3/10.
例如，2/5 × 3/4 = (2 × 3)/(5 × 4) = 6/20 = 3/10。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a/b is b/a.
除以一个分数等于乘以它的倒数。a/b 的倒数是 b/a。
For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which can also be written as 1 7/8.
例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8，也可以写成 1 7/8。
6. Introduction to Decimals | 小数入门
A decimal uses place value to show parts of a whole. The first digit after the decimal point is tenths, the second is hundredths, and the third is thousandths.
小数通过位值表示整体的一部分。小数点后第一位是十分位，第二位是百分位，第三位是千分位。
For example, in 0.375, the 3 is in the tenths place, the 7 is in the hundredths place, and the 5 is in the thousandths place.
例如，在 0.375 中，3 在十分位，7 在百分位，5 在千分位。
Decimals can be terminating, such as 0.25, or recurring, such as 0.333..., where the digit 3 repeats forever.
小数可以是有限小数，如 0.25，也可以是循环小数，如 0.333...，其中数字 3 无限循环。
7. Converting Fractions to Decimals | 分数转小数
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator.
把分数化为小数，用分子除以分母。可以用短除法或计算器。
For example, 3/8 = 3 ÷ 8 = 0.375. This is a terminating decimal.
例如，3/8 = 3 ÷ 8 = 0.375，这是一个有限小数。
fractions produce recurring decimals. For example, 1/3 = 0.333..., and 2/11 = 0.181818...
有些分数会得到循环小数。例如，1/3 = 0.333...，2/11 = 0.181818...
am, you should recognise common conversions such as 1/4 = 0.25, 1/5 = 0.2, and 3/10 = 0.3.
考试中要熟悉常见转换，如 1/4 = 0.25，1/5 = 0.2，3/10 = 0.3。
8. Converting Decimals to Fractions | 小数转分数
To convert a terminating decimal to a fraction, write the decimal digits as the numerator over a power of 10, then simplify.
把有限小数化为分数：将小数数字作为分子，分母为对应的 10 的幂，再化简。
For example, 0.45 = 45/100 = 9/20 because 45 and 100 both divide by 5.
例如，0.45 = 45/100 = 9/20，因为 45 和 100 都能被 5 整除。
For recurring decimals, use an algebraic method: let x equal the recurring decimal, multiply by a power of 10 to align the repeating part, and subtract.
对于循环小数，可用代数方法：设 x 等于该循环小数，乘以 10 的幂使循环部分对齐，然后相减。
example, let x = 0.777... Then 10x = 7.777..., so 10x - x = 7, giving 9x = 7 and x = 7/9.
例如，设 x = 0.777...，则 10x = 7.777...，所以 10x - x = 7，得 9x = 7，x = 7/9。
9. Understanding Percentages | 理解百分数
A percentage means 'per hundred'. The symbol % represents dividing by 100.
百分数表示每一百份中的多少。符号 % 表示除以 100。
To convert a percentage to a fraction, write it over 100 and simplify. For example, 25% = 25/100 = 1/4.
把百分数化为分数：写成百分数除以 100，再化简。例如，25% = 25/100 = 1/4。
decimal, divide by 100 by moving the decimal point two places to the left. For example, 8% = 0.08.
把百分数化为小数：除以 100，即将小数点向左移动两位。例如，8% = 0.08。
Common conversions to memorise include 50% = 1/2 = 0.5, 25% = 1/4 = 0.25, and 10% = 1/10 = 0.1.
需要记住的常见转换有 50% = 1/2 = 0.5，25% = 1/4 = 0.25，10% = 1/10 = 0.1。
10. Converting Between Fractions, Decimals and Percentages | 三者之间的转换
The three forms are interchangeable. Use the methods from the previous sections to move between them in any direction.
这三种形式可以互相转换。使用前面小节的方法可以在任意两者之间进行转换。
A useful summary table is shown below.
下面是一个实用的汇总表。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
| 7/10 | 0.7 | 70% |
change it to a decimal by dividing, then multiply by 100. For example, 3/8 = 0.375 = 37.5%.
把分数化为百分数时，先通过除法化为小数，再乘以 100。例如，3/8 = 0.375 = 37.5%。
ng a percentage to a fraction, write the percentage over 100 and simplify. For example, 65% = 65/100 = 13/20.
把百分数化为分数时，将百分数写在 100 上方并化简。例如，65% = 65/100 = 13/20。
11. Percentage Calculations and Common Errors | 百分数计算与常见错误
rcentage of an amount, multiply the amount by the percentage as a decimal. For example, 15% of 200 = 0.15 × 200 = 30.
求一个数的百分之几，用这个数乘以百分数对应的小数。例如，200 的 15% = 0.15 × 200 = 30。
rease an amount by a percentage, multiply by (1 + p/100). For example, increasing 80 by 25% gives 80 × 1.25 = 100.
把一个数增加百分之 p，乘以 (1 + p/100)。例如，80 增加 25% 得 80 × 1.25 = 100。
rease an amount by a percentage, multiply by (1 - p/100). For example, decreasing 80 by 25% gives 80 × 0.75 = 60.
把一个数减少百分之 p，乘以 (1 - p/100)。例如，80 减少 25% 得 80 × 0.75 = 60。
Common errors include adding denominators when adding fractions, forgetting to find a common denominator, or misplacing the decimal point when converting percentages.
常见错误包括：分数相加时误把分母相加，忘记找公分母，或转换百分数时点错小数点。
Always check whether your answer makes sense. A percentage greater than 100 means an increase beyond the original amount, while a percentage less than 100 means a part of the original.
始终检查答案是否合理。大于 100% 表示超过原始数量，小于 100% 表示原始数量的一部分。
12. Exam-Style Practice and Revision Summary | 考试风格练习与复习总结
Try these quick questions: Convert 7/20 to a decimal and a percentage. Add 2/3 and 1/4. Find 12% of 250. Decrease 60 by 15%.
尝试以下快速练习：把 7/20 化为小数和百分数。计算 2/3 + 1/4。求 250 的 12%。把 60 减少 15%。
Check your answers: 7/20 = 0.35 = 35%; 2/3 + 1/4 = 8/12 + 3/12 = 11/12; 12% of 250 = 0.12 × 250 = 30; 60 decreased by 15% = 60 × 0.85 = 51.
核对答案：7/20 = 0.35 = 35%；2/3 + 1/4 = 8/12 + 3/12 = 11/12；250 的 12% = 0.12 × 250 = 30；60 减少 15% = 60 × 0.85 = 51。
For revision, focus on memorising common conversions, practising mixed number operations, and showing clear workings in your answers.
复习时，重点记忆常见转换，练习带分数的运算，并在答案中展示清晰的步骤。
Remember that fractions, decimals and percentages are different ways of representing the same quantity. Fluency in switching between them will save time and reduce errors in the IGCSE exam.
记住分数、小数和百分数是表示同一数量的不同方式。熟练地在它们之间转换将帮助你节省时间并减少 IGCSE 考试中的错误。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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