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

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

Fractions, decimals and percentages are three interconnected ways of representing parts of a whole. At Key Stage 3, you need to move between these forms confidently, understanding their equivalence and applying them to real problems. This article will guide you through each representation, show you clear conversion methods, and help you compare values with ease.

分数、小数和百分比是表示整体的一部分的三种相互关联的方式。在关键阶段3，你需要自信地在这些形式之间进行转换，理解它们的等价性，并将其应用于实际问题。本文将引导你了解每种表示方法，向你展示清晰的转换方法，并帮助你轻松比较数值。

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1. Understanding Fractions | 理解分数

A fraction represents a part of a whole or a number of equal parts. It consists of a numerator (top number) and a denominator (bottom number). For example, 3/4 means 3 parts out of 4 equal parts. Fractions can be proper (numerator < denominator), improper (numerator >= denominator), or mixed numbers.

分数代表整体的一部分或若干等份。它由分子（顶部的数）和分母（底部的数）组成。例如，3/4 表示 4 等份中的 3 份。分数可以是真分数（分子 < 分母）、假分数（分子 ≥ 分母）或带分数。

Key terms: equivalent fractions (e.g., 1/2 = 2/4 = 4/8), simplified form (dividing numerator and denominator by the same number), and unit fractions (numerator = 1).

关键术语：等值分数（如 1/2 = 2/4 = 4/8）、最简形式（分子和分母除以同一个数）以及单位分数（分子为1）。

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2. Understanding Decimals | 理解小数

Decimals are another way to write fractions, especially those with denominators 10, 100, 1000, etc. The decimal point separates the whole number from the fractional part. For instance, 0.75 means 75 hundredths, which is the same as 75/100. Each place after the point represents tenths, hundredths, thousandths, and so on.

小数是另一种书写分数的方式，特别是分母为10、100、1000等的分数。小数点将整数部分与小数部分分开。例如，0.75 表示百分之七十五，与 75/100 相同。小数点后的每一位分别代表十分位、百分位、千分位等。

Place value chart:

位值表：

Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
100	10	1		1/10	1/100	1/1000

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3. Understanding Percentages | 理解百分比

A percentage is a fraction with a denominator of 100. The word ‘percent’ means ‘out of 100’. 50% means 50 out of 100, or 50/100, which simplifies to 1/2. Percentages are useful for comparing proportions and are commonly used in finance, statistics, and daily life.

百分比是分母为100的分数。“百分比”这个词的意思是“每100中的”。50% 表示每100中有50，即 50/100，可简化为 1/2。百分比在比较比例时很有用，常用于金融、统计和日常生活。

Symbol: %. For example, 45% = 45/100 = 0.45 = 9/20.

符号：%。例如，45% = 45/100 = 0.45 = 9/20。

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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. For example, 3/4 = 3 ÷ 4 = 0.75. If the denominator is a power of 10 (10, 100, 1000), simply write the numerator with the correct number of decimal places. For example, 7/10 = 0.7, 23/100 = 0.23.

要将分数转换为小数，用分子除以分母。例如，3/4 = 3 ÷ 4 = 0.75。如果分母是10的幂（10、100、1000），只需写出分子并正确放置小数点。例如，7/10 = 0.7，23/100 = 0.23。

Some fractions produce recurring decimals, such as 1/3 = 0.333… which can be written as 0.3 with a dot above the 3, or 1/6 = 0.1666… = 0.16 with a dot above the 6. These are important at KS3 level.

有些分数会产生循环小数，例如 1/3 = 0.333… 可写作 0.3̇，或者 1/6 = 0.1666… = 0.16̇。这些在 KS3 阶段也很重要。

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5. Converting Decimals to Fractions | 小数转换为分数

To convert a terminating decimal to a fraction, use place value. Write the digits after the decimal point as the numerator, and the denominator as 1 followed by as many zeros as there are decimal places. Then simplify if possible. Example: 0.65 = 65/100 = 13/20 (divide by 5).

要将有限小数转换为分数，利用位值。将小数点后的数字作为分子，分母为1后面跟上与小数位数相同的零的个数。然后尽可能化简。例子：0.65 = 65/100 = 13/20（除以5）。

For recurring decimals, a different method is needed, but at KS3, you mainly handle simple recurring decimals like 0.1̇ = 1/9, 0.3̇ = 1/3, etc. Always check for mixed numbers: 1.2 = 1 1/5 = 6/5.

对于循环小数，需要另一种方法，但在 KS3 阶段，你主要处理简单的循环小数，如 0.1̇ = 1/9，0.3̇ = 1/3 等。记住可以转为带分数：例如 1.2 = 1 1/5 = 6/5。

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6. Converting Fractions to Percentages | 分数转换为百分比

Method 1: Convert the fraction to an equivalent fraction with denominator 100. Then the numerator is the percentage. For example, 3/5 = ?/100 → 3/5 = 60/100, so 60%. This works well when the denominator is a factor of 100 (e.g., 2, 4, 5, 10, 20, 25, 50).

方法一：将分数转换为分母为100的等值分数。那么分子就是百分比。例如，3/5 = ?/100 → 3/5 = 60/100，所以是 60%。当分母是100的因数（如2、4、5、10、20、25、50）时，这种方法效果很好。

Method 2: If Method 1 is not easy, divide the numerator by the denominator and multiply by 100%. Example: 5/8. 5 ÷ 8 = 0.625, then 0.625 × 100% = 62.5%. Remember to add the percentage sign.

方法二：如果方法一不方便，将分子除以分母再乘以100%。例如：5/8。5 ÷ 8 = 0.625，然后 0.625 × 100% = 62.5%。记得加上百分号。

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7. Converting Decimals to Percentages and Vice Versa | 小数与百分比的相互转换

Decimal to percentage: multiply by 100%. For example, 0.7 × 100% = 70%. A shortcut is to move the decimal point two places to the right. 0.03 → 3%. 1.25 → 125%.

小数转百分比：乘以100%。例如，0.7 × 100% = 70%。一个快捷方法是将小数点向右移动两位。0.03 → 3%。1.25 → 125%。

Percentage to decimal: divide by 100. So move the decimal point two places to the left. 85% = 85 ÷ 100 = 0.85. 6% = 0.06. 120% = 1.2.

百分比转小数：除以100。也就是将小数点向左移动两位。85% = 85 ÷ 100 = 0.85。6% = 0.06。120% = 1.2。

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8. Ordering and Comparing | 比较与排序

When given a mix of fractions, decimals and percentages, the easiest way to compare them is to convert all to one form—usually decimals or percentages. Then order them on a number line or list them from least to greatest.

当给出分数、小数和百分比的混合时，最简单的比较方法是将它们全部转换为同一种形式——通常是小数或百分比。然后在数轴上排序，或从小到大列出。

Example: Which is largest: 40%, 0.35, 2/5? Convert all to decimals: 40% = 0.4, 0.35 stays, 2/5 = 0.4. So 0.4 and 0.4 are equal, and larger than 0.35. Thus 40% = 2/5 > 0.35.

例子：哪个最大：40%，0.35，2/5？全部转为小数：40% = 0.4，0.35 不变，2/5 = 0.4。所以 0.4 和 0.4 相等，且大于 0.35。因此 40% = 2/5 > 0.35。

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9. Common Equivalent Values to Memorise | 需要记忆的常见等值

Some equivalents appear so often that memorising them saves time. Here is a table of key conversions:

一些等值出现得非常频繁，记住它们可以节省时间。以下是关键转换表：

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
3/5	0.6	60%
4/5	0.8	80%
1/8	0.125	12.5%
3/8	0.375	37.5%
5/8	0.625	62.5%
7/8	0.875	87.5%
1/10	0.1	10%
1/3	0.333… (0.3̇)	33 1/3%
2/3	0.666… (0.6̇)	66 2/3%

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10. Real-world Applications | 实际应用

These conversions appear everywhere: calculating discounts (e.g., 20% off a price), working out test scores (e.g., 18/20 = 90%), adjusting recipes (e.g., 3/4 cup of flour), and reading statistics (e.g., 0.65 probability = 65% chance). Being fluent in conversions helps you make quick decisions and check calculations.

这些转换无处不在：计算折扣（例如价格打8折）、计算考试得分（例如18/20 = 90%）、调整食谱（例如3/4杯面粉）以及阅读统计数据（例如0.65的概率 = 65%的几率）。熟练转换有助于你快速做出决定和检查计算。

For example, a shop offers a 25% discount on a £40 item. You can find 25% = 0.25 × 40 = £10 off, so you pay £30. Alternatively, knowing 25% = 1/4, you can divide £40 by 4 = £10 discount.

例如，一家商店对一件40英镑的商品提供25%的折扣。你可以算出 25% = 0.25 × 40 = 10英镑的折扣，所以支付30英镑。或者，知道 25% = 1/4，你可以将40英镑除以4得到10英镑的折扣。

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11. Practice and Tips | 练习与技巧

Here are some practice problems to test your understanding:

以下是一些测试你理解程度的练习题：

• Convert 0.45 to a fraction in simplest form. (Answer: 9/20)
• Convert 7/25 to a percentage. (Answer: 28%)
• Which is larger: 3/8 or 35%? (Answer: 3/8 = 37.5% > 35%)
• Express 0.05 as a percentage and as a fraction. (Answer: 5% and 1/20)
• Order from smallest to largest: 0.3, 28%, 1/5. (Answer: 1/5 (0.2), 28% (0.28), 0.3)

Tip: always simplify fractions to their lowest terms unless asked otherwise. Use a number line if visualising comparisons.

提示：除非另有要求，始终将分数化简到最简形式。如果需要进行比较的视觉化，可使用数线。

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12. Summary | 总结

Mastering fractions, decimals and percentages means being able to switch between forms and recognise their equivalence. The key skills are dividing for fraction-to-decimal, multiplying by 100 for decimal-to-percentage, and using equivalent fractions or division for fraction-to-percentage. With practice, these conversions become automatic, allowing you to tackle more complex problems with confidence.

掌握分数、小数和百分比意味着能够在不同形式之间切换并认识它们的等价性。关键技能包括：分数转小数用除法，小数转百分比乘以100，分数转百分比使用等值分数或除法。通过练习，这些转换会变得自动化，使你能够自信地应对更复杂的问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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