📚 Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数
Fractions, decimals and percentages are three ways of expressing parts of a whole. In IGCSE Mathematics, you must be able to convert between them, perform calculations, and apply percentage change in real-life problems such as discounts, tax and compound interest. This article covers the key methods, common exam traps and worked-style examples to help you build confidence.
分数、小数和百分数是表示整体一部分的三种方式。在 IGCSE 数学中，你必须能够在它们之间相互转换、进行计算，并能在折扣、税收和复利等实际问题中应用百分数变化。本文涵盖关键方法、常见考试陷阱和例题式讲解，帮助你建立信心。
1. Understanding Fractions | 理解分数
A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator shows how many equal parts the whole is divided into; the numerator shows how many parts are taken. For example, 3/4 means 3 parts out of 4 equal parts.
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多少等份；分子表示取了多少份。例如，3/4 表示 4 等份中的 3 份。
Proper fractions have a numerator smaller than the denominator, such as 2/5. Improper fractions have a numerator larger than or equal to the denominator, such as 9/4 or 7/7. Mixed numbers combine a whole number and a proper fraction, such as 2 1/3.
真分数的分子小于分母，例如 2/5。假分数的分子大于或等于分母，例如 9/4 或 7/7。带分数由一个整数和一个真分数组成，例如 2 1/3。
All fractions can be placed on a number line. The distance between 0 and 1 is split into equal segments according to the denominator.
所有分数都可以标在数轴上。0 到 1 之间的距离根据分母分成等份。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions represent the same value even though they use different numbers. Multiply or divide both the numerator and denominator by the same non-zero number to create equivalent fractions. For example, 1/2 = 2/4 = 3/6 = 50/100.
等值分数表示相同的值，尽管使用的数字不同。将分子和分母同时乘或除以同一个非零数，可以得到等值分数。例如，1/2 = 2/4 = 3/6 = 50/100。
To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For 18/24, the HCF of 18 and 24 is 6, so 18/24 simplifies to 3/4.
化简分数时，将分子和分母同时除以它们的最大公因数（HCF）。对于 18/24，18 和 24 的最大公因数是 6，因此 18/24 化简为 3/4。
Always give fractions in their simplest form unless a question asks otherwise. In an exam, a fraction that is not simplified may lose accuracy marks.
除非题目另有要求，分数始终要化为最简形式。考试中，未化简的分数可能会丢失准确度分数。
3. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple (LCM) of the denominators. For 1/4 + 2/3, the LCM of 4 and 3 is 12, so 1/4 becomes 3/12 and 2/3 becomes 8/12; the sum is 11/12.
分数加减时，它们必须有相同的分母。求出分母的最小公倍数（LCM）。对于 1/4 + 2/3，4 和 3 的最小公倍数是 12，因此 1/4 变为 3/12，2/3 变为 8/12；和为 11/12。
For mixed numbers, add or subtract the whole number parts and fraction parts separately. If the fraction part of a subtraction is too small, borrow 1 from the whole number part and convert it into the required fraction.
对于带分数，整数部分和分数部分分别加减。如果减法中的分数部分不够减，就从整数部分借 1，并转换为所需分母的分数。
After adding or subtracting, simplify the result if possible. Also check whether an improper fraction should be converted back to a mixed number in the final answer.
加减后，如果可能应化简结果。还要检查最终答案中假分数是否应转换回带分数。
4. Multiplying and Dividing Fractions | 分数乘除法
To multiply fractions, multiply the numerators together and multiply the denominators together. For example, 2/3 × 4/5 = (2×4)/(3×5) = 8/15. Simplify before or after multiplying to make the numbers smaller.
分数乘法是将分子相乘、分母相乘。例如，2/3 × 4/5 = (2×4)/(3×5) = 8/15。可以在乘法前或乘法后化简，使数字更小。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a/b is b/a. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
除以一个分数等于乘以它的倒数。a/b 的倒数是 b/a。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
With mixed numbers, convert them to improper fractions first. For 1 1/2 × 2 1/3, convert to 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2.
带分数要先转换为假分数。对于 1 1/2 × 2 1/3，转换为 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2。
5. Decimals and Place Value | 小数与位值
A decimal uses a decimal point to separate the whole number part from the fractional part. The first digit after the point is tenths, the second is hundredths, and the third is thousandths. For example, 0.375 means 3/10 + 7/100 + 5/1000.
小数用小数点分隔整数部分和小数部分。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。例如，0.375 表示 3/10 + 7/100 + 5/1000。
To add or subtract decimals, align the decimal points vertically so that place values match. For 2.7 + 1.45, write 2.70 + 1.45 = 4.15.
小数加减时，小数点上下对齐，使数位匹配。对于 2.7 + 1.45，写作 2.70 + 1.45 = 4.15。
Multiplying decimals can be done by ignoring the points, multiplying as whole numbers, then putting the decimal point back so the answer has the same total number of decimal places as the original numbers combined. For example, 0.2 × 0.3 has 2 decimal places, so 2 × 3 = 6 becomes 0.06.
小数乘法可以先忽略小数点，按整数相乘，然后再加回小数点，使答案的小数位数等于原数字小数位数之和。例如，0.2 × 0.3 有两位小数，所以 2 × 3 = 6 变成 0.06。
6. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数互化
A percentage is a fraction with a denominator of 100. To convert a percentage to a decimal, divide by 100. For example, 35% = 0.35 and 7% = 0.07.
百分数是分母为 100 的分数。将百分数转换为小数时除以 100。例如，35% = 0.35，7% = 0.07。
To convert a decimal to a percentage, multiply by 100. To convert a fraction to a percentage, either change the denominator to 100 or convert to a decimal first, then multiply by 100. For example, 3/5 = 0.6 = 60%.
将小数转换为百分数时乘以 100。将分数转换为百分数时，可以将分母变为 100，也可以先转换为小数再乘以 100。例如，3/5 = 0.6 = 60%。
Common conversions should be memorised: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/3 ≈ 0.333 = 33.3%, 2/3 ≈ 0.667 = 66.7%, 1/8 = 0.125 = 12.5%.
常见互化应该牢记：1/2 = 0.5 = 50%，1/4 = 0.25 = 25%，3/4 = 0.75 = 75%，1/3 ≈ 0.333 = 33.3%，2/3 ≈ 0.667 = 66.7%，1/8 = 0.125 = 12.5%。
To compare quantities given in different forms, convert them all to the same form. For example, to decide whether 2/5, 0.45 or 43% is largest, convert: 2/5 = 0.40 = 40%, so 45% is largest.
比较不同形式的数量时，将它们全部转换为同一种形式。例如，判断 2/5、0.45 和 43% 哪个最大，转换后：2/5 = 0.40 = 40%，所以 45% 最大。
7. Percentages of Amounts | 求一个数的百分数
To find a percentage of an amount, multiply the amount by the percentage as a decimal. For example, 15% of 240 = 0.15 × 240 = 36.
求一个数的百分数，用该数乘以百分数对应的小数。例如，240 的 15% = 0.15 × 240 = 36。
Alternatively, use the fraction method: 15% = 15/100 = 3/20, so 15% of 240 = 240 × 3/20 = 36. This is useful for non-calculator questions.
也可以使用分数法：15% = 15/100 = 3/20，因此 240 的 15% = 240 × 3/20 = 36。这在非计算器题目中很有用。
For common percentages, use mental shortcuts: 50% is half, 25% is a quarter, 10% is one tenth, and 5% is half of 10%.
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