IGCSE Maths: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数

📚 IGCSE Maths: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数

Fractions, decimals and percentages are three ways of representing parts of a whole. In the IGCSE Mathematics syllabus, these forms appear in arithmetic, algebra, data handling and problem-solving. This article reviews the key skills you need, from simplifying fractions to converting between forms and applying percentage change.

分数、小数和百分数是表示整体部分量的三种方式。在 IGCSE 数学大纲中，这些形式广泛出现在算术、代数、数据处理和问题解决中。本文复习你需要掌握的核心技能，包括约分、三种形式之间的转换以及百分数增减应用。

This article is part of the Year 6 theme series-900, bridging primary number work to IGCSE-level confidence.

本文属于 6 年级主题系列-900，帮助你把小学数感提升到 IGCSE 水平。

1. Understanding Fractions | 分数的基本概念

A fraction represents equal parts of a whole or a set. The numerator tells how many parts are selected, and the denominator tells into how many equal parts the whole is divided.

分数表示整体或一组物体中的等份。分子表示取了多少份，分母表示整体被分成多少等份。

Proper fractions have a numerator smaller than the denominator, such as 3/4. Improper fractions have a numerator larger than or equal to the denominator, such as 9/4. Mixed numbers combine an integer and a proper fraction, such as 2 1/4.

真分数的分子小于分母，例如 3/4。假分数的分子大于或等于分母，例如 9/4。带分数由一个整数和一个真分数组成，例如 2 1/4。

The denominator of a fraction must never be zero, because division by zero is undefined in mathematics.

分数的分母永远不能为零，因为除以零在数学中没有定义。

2. Equivalent Fractions and Simplifying | 等值分数与约分

Equivalent fractions have the same value but different numerators and denominators. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.

等值分数数值相同，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，从而得到等值分数。

For example, 2/3 = 4/6 = 10/15. To simplify a fraction, divide the numerator and denominator by their highest common factor, or HCF.

例如，2/3 = 4/6 = 10/15。要约分一个分数，需要用分子和分母的最大公因数 HCF 同时去除。

18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4

Always check whether your final fraction can be simplified further. A fraction is in simplest form when the numerator and denominator have no common factor other than 1.

始终检查最终分数是否还能继续约分。当分子和分母除了 1 以外没有其他公因数时，分数就是最简形式。

3. Decimals and Place Value | 小数与位值

Decimal numbers extend the base-10 place value system to parts smaller than one. The first decimal place is tenths, the second is hundredths, and the third is thousandths.

小数将十进制位值系统扩展到小于 1 的部分。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, 0.37 means 3 tenths plus 7 hundredths. It can be written as the fraction 37/100.

例如，0.37 表示 3 个十分之一加上 7 个百分之一。它可以写成分数 37/100。

Decimal	Fraction	Place value reading
0.4	4/10	4 tenths
0.07	7/100	7 hundredths
0.009	9/1000	9 thousandths

Understanding place value helps when converting decimals to fractions and when comparing decimal sizes.

理解位值有助于小数与分数的转换，也有助于比较小数的大小。

4. Converting Fractions to Decimals | 分数转换为小数

To convert a fraction to a decimal, divide the numerator by the denominator. The fraction bar is another way of writing division.

要将分数转换为小数，用分子除以分母。分数线实际上就是除号的另一种写法。

For example, 3/8 = 3 ÷ 8 = 0.375. Some fractions produce terminating decimals, while others produce recurring decimals such as 1/3 = 0.333…

例如，3/8 = 3 ÷ 8 = 0.375。有些分数会得到有限小数，而有些会得到循环小数，例如 1/3 = 0.333…

If the denominator is 10, 100 or 1000, you can write the decimal directly. For example, 7/10 = 0.7 and 23/100 = 0.23.

如果分母是 10、100 或 1000，可以直接写出小数。例如 7/10 = 0.7，23/100 = 0.23。

When a decimal recurs, use a dot or line above the repeating digit in IGCSE notation. For example, 2/11 = 0.1818… can be written with dots above 1 and 8.

当小数循环时，IGCSE 表示法是在循环数字上方加点或线。例如，2/11 = 0.1818… 可以在 1 和 8 上方加点。

5. Percentages and Their Meaning | 百分数及其含义

Percent means ‘out of 100’. The symbol % represents a denominator of 100, so 25% means 25 out of every 100.

百分数表示“每一百份中的多少份”。符号 % 代表分母为 100，因此 25% 表示每一百份中有 25 份。

Percentages are useful for comparing relative sizes because they use the same base of 100. For example, 25% of a class is easier to compare than 7 out of 28 students.

百分数适合比较相对大小，因为它们都以 100 为共同基准。例如，一个班的 25% 比 28 名学生中的 7 名更容易比较。

Common equivalents should be memorised: 50% = 1/2, 25% = 1/4, 75% = 3/4, 10% = 1/10, and 20% = 1/5.

常见等值关系需要记住：50% = 1/2，25% = 1/4，75% = 3/4，10% = 1/10，20% = 1/5。

6. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互相转换

IGCSE questions often ask you to move between all three forms. The key methods are straightforward if you remember the base of 100 for percentages.

IGCSE 题目经常要求你在三种形式之间转换。只要记住百分数以 100 为基准，关键方法就很简单。

To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.

将小数转换为百分数，乘以 100。将百分数转换为小数，除以 100。

To convert a percentage to a fraction, write it over 100 and simplify. To convert a fraction to a percentage, first convert the fraction to a decimal, then multiply by 100.

将百分数转换为分数，先写成分母为 100 的分数，再约分。将分数转换为百分数，先把分数转换为小数，再乘以 100。

Fraction	Decimal	Percentage
1/2	0.5	50%
3/5	0.6	60%
7/8	0.875	87.5%

Always simplify fractions in your final answer unless a question asks for a specific form.

除非题目要求特定形式，否则最终答案中的分数都要约分到最简。

7. Comparing and Ordering | 比较与排序

To compare fractions, decimals and percentages, convert them all into the same form. Decimals or percentages are usually the easiest forms for comparison.

要比较分数、小数和百分数，需要先把它们都转换成同一种形式。通常转换为小数或百分数最容易比较。

For example, order 2/5, 0.45 and 38% from smallest to largest. Convert: 2/5 = 0.4, 38% = 0.38, so 38% < 2/5 < 0.45.

例如，将 2/5、0.45 和 38% 从小到大排列。转换后：2/5 = 0.4，38% = 0.38，所以 38% < 2/5 < 0.45。

When ordering fractions without a calculator, find a common denominator and compare numerators. For example, 1/3 = 4/12 and 1/4 = 3/12, so 1/4 < 1/3.

在没有计算器的情况下给分数排序，可以先找公分母，再比较分子。例如 1/3 = 4/12，1/4 = 3/12，所以 1/4 < 1/3。

8. Adding and Subtracting Fractions | 分数的加法与减法

To add or subtract fractions, they must have the same denominator. Find a common denominator, rewrite each fraction, then add or subtract the numerators only.

分数相加或相减时，它们必须具有相同的分母。先找到公分母，重写每个分数，然后只对分子进行加减。

For example, 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The denominator stays as 12.

例如，1/3 + 1/4 = 4/12 + 3/12 = 7/12。分母保持为 12。

1/3 + 1/4 = 4/12 + 3/12 = 7/12

For mixed numbers, add or subtract the whole number parts and the fractional parts separately. If the fractional subtraction is negative, borrow 1 from the whole number part.

对于带分数，可以分别对整数部分和分数部分进行加减。如果分数部分相减不够，需要从整数部分借 1。

Example: 3 1/5 – 1 3/5 = 2 6/5 – 1 3/5 = 1 3/5.

例如：3 1/5 – 1 3/5 = 2 6/5 – 1 3/5 = 1 3/5。

9. Multiplying and Dividing Fractions | 分数的乘法与除法

Multiplying fractions is simpler than adding them. Multiply the numerators together and the denominators together, then simplify if possible.

分数相乘比相加更简单。分子与分子相乘，分母与分母相乘，然后尽可能约分。

2/3 × 4/5 = 8/15

To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction flips the numerator and denominator.

除以一个分数等于乘以它的倒数。一个分数的倒数就是把分子和分母交换位置。

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

For mixed numbers, convert to improper fractions before multiplying or dividing. Then simplify your final answer.

对于带分数，在乘法或除法之前要先转换为假分数，然后对最终答案进行约分。

10. Percentage Increase and Decrease | 百分数增减

Percentage change compares the actual change to the original amount. The formula is percentage change = change ÷ original × 100%.

百分数变化用实际变化量除以原始量。公式为：百分数变化 = 变化量 ÷ 原始量 × 100%。

For example, a price rises from $50 to $65. The change is $15, so the percentage increase is 15/50 × 100 = 30%.

例如，价格从 50 美元上涨到 65 美元。变化量是 15 美元，所以百分数增加为 15/50 × 100 = 30%。

Percentage increase = 15 ÷ 50 × 100% = 30%

百分数增加 = 15 ÷ 50 × 100% = 30%

To find a new amount after a percentage increase, multiply the original by 1 plus the percentage as a decimal. For a decrease, multiply by 1 minus the percentage as a decimal.

求百分数增加后的新量，用原量乘以 1 加上百分数对应的小数。求减少后的新量，用原量乘以 1 减去百分数对应的小数。

For example, increasing $80 by 15% gives 80 × 1.15 = $92. Decreasing $80 by 15% gives 80 × 0.85 = $68.

例如，80 美元增加 15% 得到 80 × 1.15 = 92 美元。80 美元减少 15% 得到 80 × 0.85 = 68 美元。

Reverse percentage questions ask you to find the original amount after a change. If a value after a 20% increase is $120, the original is 120 ÷ 1.20 = $100.

逆向百分数问题要求你根据变化后的量求原始量。若增加 20% 后为 120 美元，则原始量为 120 ÷ 1.20 = 100 美元。

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IGCSE Maths: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数