Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分数

📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分数

Fractions, decimals and percentages are three different ways to describe parts of a whole. They appear throughout the KS3 Cambridge Mathematics curriculum and are essential for solving problems in algebra, geometry and data handling. This article explains the key ideas, conversion methods and exam techniques you need to master.

分数、小数和百分数是描述整体部分的三种不同方式。它们贯穿 KS3 剑桥数学课程，对解决代数、几何和数据处理问题至关重要。本文将解释你需要掌握的核心概念、转换方法和考试技巧。

1. Understanding Fractions | 理解分数

A fraction shows a part of a whole. It is written with a numerator above a denominator. The numerator tells how many equal parts are taken; the denominator tells how many equal parts make one whole.

分数表示整体的一部分。它写作分子在上、分母在下。分子表示取了多少等份；分母表示一个整体被分成多少等份。

For example, in 3/4, the denominator 4 means the whole is divided into 4 equal parts, and the numerator 3 means we take 3 of those parts. Fractions can represent parts of shapes, amounts or measurements.

例如，在 3/4 中，分母 4 表示整体被分成 4 等份，分子 3 表示取其中的 3 份。分数可以表示图形的一部分、数量的一部分或测量值的一部分。

3/4 = 3 parts out of 4 equal parts

2. Equivalent Fractions and Simplifying | 等值分数与化简

Equivalent fractions have the same value even though they look different. To create equivalent fractions, multiply or divide 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。要得到等值分数，可以将分子和分母同时乘以或除以同一个不为零的数。

For example, 1/2 is equivalent to 2/4 and 4/8. To simplify a fraction, divide the numerator and denominator by their greatest common factor. For 6/8, divide both by 2 to get 3/4.

例如，1/2 等于 2/4 和 4/8。要化简分数，用分子和分母的最大公因数同时去除。对于 6/8，分子分母都除以 2 得到 3/4。

• Always multiply or divide both terms by the same number.
• 始终用同一个数去乘或除分子和分母。
• Simplify fractions fully in final answers unless told otherwise.
• 除非题目另有要求，最终答案都要化为最简分数。

3. Decimals and Place Value | 小数与位值

Decimals are another way to represent parts of a whole. The digits after the decimal point have place values of tenths, hundredths, thousandths, and so on.

小数是表示整体部分的另一种方法。小数点后的数字位值分别为十分位、百分位、千分位等。

For example, 0.37 has 3 tenths and 7 hundredths, meaning 37/100. The digit 5 in 0.5 means 5 tenths, which is the same as 1/2.

例如，0.37 有 3 个十分之一和 7 个百分之一，即 37/100。0.5 中的数字 5 表示 5 个十分之一，等于 1/2。

0.37 = 3/10 + 7/100 = 37/100

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

To convert a fraction to a decimal, divide the numerator by the denominator. Some fractions convert to terminating decimals, while others produce recurring decimals.

要将分数转换为小数，用分子除以分母。有些分数会转换为有限小数，有些则会产生循环小数。

For example, 3/4 = 3 ÷ 4 = 0.75 and 1/8 = 1 ÷ 8 = 0.125. However, 1/3 = 0.333… which is a recurring decimal often written as 0.3 with a dot above the 3.

例如，3/4 = 3 ÷ 4 = 0.75，1/8 = 1 ÷ 8 = 0.125。然而，1/3 = 0.333… 是循环小数，通常记作 0.3 并在 3 上方加点。

• Use short division for simple conversions.
• 简单转换可使用短除法。
• Check whether the decimal terminates or recurs.
• 检查小数是有限小数还是循环小数。

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

Percent means ‘per hundred’. A percentage represents a number out of 100. Percentages are useful for comparing ratios because they all use the same base of 100.

百分数表示“每一百”。百分数表示一个数占 100 的比例。百分数便于比较比例，因为它们都使用 100 这个相同的基准。

For example, 45% means 45 out of 100, which can be written as 45/100 or 0.45. Similarly, 5% means 5 out of 100, or 1/20 after simplifying.

例如，45% 表示 100 份中的 45 份，可以写成 45

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