Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数

📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数

In KS3 Cambridge Mathematics, fractions, decimals and percentages are three ways of expressing the same value. Understanding how to convert between them is a core skill for proportional reasoning, probability and many real-life problems.

在 KS3 剑桥数学中，分数、小数与百分数是表示同一数值的三种形式。掌握它们之间的互化是比例推理、概率以及许多实际问题中的核心技能。

1. Understanding Equivalent Forms | 理解等价形式

A fraction, a decimal and a percentage can all represent the same part of a whole. For example, 1/2 = 0.5 = 50%. The symbol % means ‘out of 100’, so 50% is 50 out of 100.

分数、小数和百分数都可以表示同一个整体的一部分。例如，1/2 = 0.5 = 50%。符号 % 表示 “每一百”，所以 50% 就是 100 份中的 50 份。

Equivalent fractions can be made by multiplying or dividing the numerator and denominator by the same number. This is important when comparing different forms.

等价分数可以通过将分子和分母同时乘以或除以同一个数得到。这在比较不同形式时非常重要。

1/2 = 2/4 = 3/6 = 50/100 = 0.5 = 50%

The top number in a fraction is called the numerator and the bottom number is called the denominator. The denominator tells you how many equal parts the whole has been divided into.

分数中上面的数叫分子，下面的数叫分母。分母表示整体被平均分成了多少份。

2. Converting Between Fractions and Decimals | 分数与小数互化

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 means 3 ÷ 4 = 0.75.

要将分数化为小数，用分子除以分母。例如，3/4 表示 3 ÷ 4 = 0.75。

To convert a terminating decimal to a fraction, write the decimal over its place value and simplify. For example, 0.25 = 25/100 = 1/4.

要将有限小数化成分数，把小数写在对应位值的分母上并约分。例如，0.25 = 25/100 = 1/4。

Some decimals recur, such as 1/3 = 0.333…, which is written as 0.3 with a dot over the 3. Similarly, 2/9 = 0.222… = 0.2 with a dot over the 2.

有些小数会循环，如 1/3 = 0.333…，记作在 3 上方加点。类似地，2/9 = 0.222…，记作在 2 上方加点。

1/8 = 1 ÷ 8 = 0.125

Practice with simple fractions first: 1/4 = 0.25, 3/10 = 0.3, and 1/5 = 0.2. These are common in KS3 exam questions.

先练习简单的分数：1/4 = 0.25、3/10 = 0.3、1/5 = 0.2。这些在 KS3 考试中很常见。

3. Converting Decimals to Percentages | 小数转百分数

To change a decimal to a percentage, multiply by 100 and add the % sign. This is the same as moving the decimal point two places to the right.

要把小数化为百分数，将其乘以 100 并加上 % 符号。这相当于把小数点向右移动两位。

0.37 × 100 = 37%

For a decimal greater than 1, such as 1.2, the percentage is 120%. This means the quantity is larger than the original whole.

对于大于 1 的小数，如 1.2，百分数为 120%。这表示数量大于原来的整体。

To convert a percentage back to a decimal, divide by 100, which moves the decimal point two places to the left. For example, 65% = 0.65.

要把百分数转回小数，除以 100，也就是把小数点向左移动两位。例如，65% = 0.65。

8% = 0.08 and 120% = 1.2

4. Converting Fractions to Percentages | 分数转百分数

There are two common methods. First, convert the fraction to a decimal by dividing, then multiply by 100. For example, 2/5 = 0.4 = 40%.

有两种常用方法。先把分数化为小数，再乘以 100。例如，2/5 = 0.4 = 40%。

Alternatively, find an equivalent fraction with denominator 100. For example, 7/20 = 35/100 = 35%.

另一种方法是找到分母为 100 的等价分数。例如，7/20 = 35/100 = 35%。

If the denominator does not divide 100 neatly, use division. For example, 5/8 = 0.625 = 62.5%.

如果分母不能整除 100，则使用除法。例如，5/8 = 0.625 = 62.5%。

Percentages can also be written as mixed numbers when needed, such as 125% = 1 1/4. This often appears in percentage increase problems.

百分数在需要时也可以写成带分数，例如 125% = 1 1/4。这在百分数增加问题中经常出现。

5. Ordering and Comparing | 排序与比较

To compare fractions, decimals and percentages, convert all values to the same form. Decimals are usually easiest for ordering.

要比较分数、小数和百分数，先把所有值化为同一种形式。通常小数最容易排序。

Fraction	Decimal	Percentage
3/10	0.3	30%
2/5	0.4	40%
1/4	0.25	25%

In ascending order: 1/4 (0.25), 3/10 (0.3), then 2/5 (0.4). Always double-check the direction required by the question.

按升序排列：1/4 (0.25)、3/10 (0.3)、2/5 (0.4)。务必再次确认题目要求的顺序方向。

When comparing percentages, remember that 100% is the whole. Percentages below 100% are less than the whole, while percentages above 100% are greater than the whole.

比较百分数时，记住 100% 代表整体。低于 100% 表示小于整体，高于 100% 表示大于整体。

6. Finding a Percentage of a Quantity | 求一个数量的百分数

To find a percentage of a quantity, convert the percentage to a decimal or fraction and multiply. For example, 20% of 60 = 0.2 × 60 = 12.

要求一个数量的百分数，先将百分数化为小数或分数再相乘。例如，60 的 20% = 0.2 × 60 = 12。

For non-calculator work, use benchmark percentages: 50%, 25%, 10%, 1%. For example, 15% of 80 = 10% (8) + 5% (4) = 12.

在非计算器题目中，可以使用基准百分数：50%、25%、10%、1%。例如，80 的 15% = 10% (8) + 5% (4) = 12。

You can also use fractions: 25% is 1/4, 50% is 1/2, and 75% is 3/4. So 25% of 120 = 1/4 × 120 = 30.

你也可以使用分数：25% 是 1/4，50% 是 1/2，75% 是 3/4。因此 120 的 25% = 1/4 × 120 = 30。

10% of a number = number ÷ 10

7. Percentage Increase and Decrease | 百分数增减

A percentage increase means adding a percentage of the original value. For example, increasing 80 by 15% gives 80 + 0.15 × 80 = 92.

百分数增加表示在原值上增加一个百分比。例如，80 增加 15% 得到 80 + 0.15 × 80 = 92。

A percentage decrease means subtracting. For example, decreasing 80 by 15% gives 80 – 0.15 × 80 = 68.

百分数减少表示在原值上减去一个百分比。例如，80 减少 15% 得到 80 – 0.15 × 80 = 68。

For a multiplier method, increase by 15% uses 1.15, while decrease by 15% uses 0.85.

使用乘数法时，增加 15% 用 1.15，减少 15% 用 0.85。

New value = original value × multiplier

Multipliers can be found by converting the percentage to a decimal and adding or subtracting from 1. For example, a 30% increase has multiplier 1.30, and a 30% decrease has multiplier 0.70.

乘数可以将百分数化为小数后与 1 相加或相减得到。例如，增加 30% 的乘数为 1.30，减少 30% 的乘数为 0.70。

8. Reverse Percentages | 逆向百分数

In a reverse percentage question, you are given the value after a percentage change and asked to find the original amount. Divide by the multiplier instead of multiplying.

在逆向百分数问题中，已知百分数变化后的值，要求原来的量。此时要用除以乘数，而不是乘以乘数。

Example: If a price increases by 15% to 92, the original price is 92 ÷ 1.15 = 80.

示例：如果价格增加 15% 后为 92，则原价为 92 ÷ 1.15 = 80。

If a price decreases by 20% to 48, the original price is 48 ÷ 0.8 = 60. This is a common KS3 exam skill.

如果价格减少 20% 后为 48，则原价为 48 ÷ 0.8 = 60。这是 KS3 考试中常见的技能。

Original value = final value ÷ multiplier

9. Word Problems and Applications | 应用题与实际应用

Word problems often combine fractions, decimals and percentages. Read carefully to identify the original amount, the operation and the required form of the answer.

应用题通常综合分数、小数和百分数。仔细阅读题目，确定原数量、运算以及答案所需的形式。

Example: A class has 30 students. 40% are boys. How many boys are there?

示例：一个班级有 30 名学生，40% 是男生。有多少名男生？
Answer: 40% of 30 = 0.4 × 30 = 12 boys.

答案：30 的 40% = 0.4 × 30 = 12 名男生。

Another example: A shop reduces a £45 item by 20%. What is the sale price?

另一个示例：商店将一件 45 英镑的商品降价 20%。折扣价是多少？

Sale price = 45 × 0.8 = £36.

折扣价 = 45 × 0.8 = 36 英镑。

When a question asks for the result as a fraction, decimal or percentage, make sure your final answer is in that exact form. For example, ‘give your answer as a fraction in its simplest form’ means you must simplify fully.

当题目要求以分数、小数或百分数作答时，确保最终答案为该形式。例如，“答案用最简分数表示”意味着必须完全约分。

10. Common Misconceptions | 常见误区

Do not confuse 0.5 with 5%. 0.5 is 50%, while 0.05 is 5%. The number of decimal places matters.

不要混淆 0.5 和 5%。0.5 是 50%，而 0.05 才是 5%。小数位数很重要。

When increasing and decreasing by the same percentage, you do not return to the original value. For example, increasing 100 by 10% gives 110, but decreasing 110 by 10% gives 99.

按相同百分数先增加再减少，不会回到原值。例如，100 增加 10% 得 110，但 110 减少 10% 得 99。

Always simplify fractions in final answers unless the question says otherwise. For example, 4/8 should be written as 1/2.

除非题目另有要求，最终答案中的分数一定要化为最简形式。例如，4/8 应写成 1/2。

Finally, avoid using percentages greater than 100% to describe a part of a whole. A part of a whole cannot exceed the whole unless comparing two different quantities.

最后，描述整体的一部分时不要使用超过 100% 的百分数。整体的一部分不能超过整体，除非是在比较两个不同的量。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数