Fractions, Decimals and Percentages | 分数、小数与百分比

📚 Fractions, Decimals and Percentages | 分数、小数与百分比

Fractions, decimals and percentages are three different ways of representing parts of a whole. They appear in everyday life, from calculating discounts in shops to measuring ingredients in recipes. In KS3 Mathematics, you will learn how to convert between these forms, compare and order them, and apply them to solve practical problems. Mastering this topic builds a strong foundation for more advanced topics like ratio, proportion, algebra and statistics.

分数、小数和百分比是表示整体之部分的三种不同方式。它们出现在日常生活中，从计算商店折扣到计量食谱配料。在KS3数学中，你将学习如何在这些形式之间转换、比较和排序，并应用它们解决实际问题。掌握这个主题为更高级的课题（如比率、比例、代数和统计）奠定了坚实基础。

1. Understanding Fractions | 理解分数

A fraction represents a part of a whole. It consists of two numbers separated by a horizontal line: the numerator (top) shows how many parts we have, and the denominator (bottom) shows the total number of equal parts the whole is divided into.

分数表示整体的一部分。它由一条横线分隔的两个数字组成：分子（上方）表示我们拥有多少份，分母（下方）表示整体被分成的相等份数的总和。

For example, in the fraction ¾, the numerator 3 tells us we have three parts, and the denominator 4 tells us the whole is divided into four equal parts. Fractions can be proper (numerator < denominator, e.g., ⅖) or improper (numerator ≥ denominator, e.g., ⁷⁄₄).

例如，在分数¾中，分子3告诉我们有三份，分母4告诉我们整体被分成了四等份。分数可以是真分数（分子小于分母，如⅖）或假分数（分子大于或等于分母，如⁷⁄₄）。

Fractions are used to describe quantities that are not whole numbers, such as half a pizza or a quarter of an hour. Understanding the roles of numerator and denominator is the first step to working confidently with fractions.

分数用于描述非整数的量，例如半个披萨或一刻钟。理解分子和分母的作用是自信处理分数的第一步。

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2. Equivalent Fractions | 等价分数

Equivalent fractions are different fractions that represent the same value. For instance, ½, ²⁄₄ and ⁴⁄₈ all name the same amount. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero number.

等价分数是表示相同值但形式不同的分数。例如，½、²⁄₄和⁴⁄₈都表示相同的量。你可以通过将分子和分母同时乘以或除以同一个非零数来创建等价分数。

To form equivalent fractions, multiply top and bottom by the same integer: ½ × ²⁄₂ = ²⁄₄. Conversely, dividing both by a common factor simplifies a fraction without changing its value: ⁴⁄₈ ÷ ⁴⁄₄ = ½. This principle is essential for comparing, adding or subtracting fractions.

要形成等价分数，将分子和分母同时乘以相同的整数：½ × ²⁄₂ = ²⁄₄。反之，将分子和分母同时除以一个公因数可以化简分数而不改变其值：⁴⁄₈ ÷ ⁴⁄₄ = ½。这个原则对于比较、加减分数至关重要。

We often use equivalent fractions when we need a common denominator. For example, to compare ⅔ and ¾, we find equivalent fractions with denominator 12: ⅔ = ⁸⁄₁₂ and ¾ = ⁹⁄₁₂. Then it is clear that ¾ is larger.

当需要公分母时，我们经常使用等价分数。例如，要比较⅔和¾，我们找到分母为12的等价分数：⅔ = ⁸⁄₁₂，¾ = ⁹⁄₁₂。那么显然¾更大。

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3. Simplifying Fractions | 化简分数

Simplifying a fraction means reducing it to its lowest terms. This is done by dividing both the numerator and the denominator by their greatest common factor (GCF). The resulting fraction is equivalent but has smaller numbers, making it easier to read and work with.

化简分数意味着将其化为最简形式。方法是将分子和分母同时除以它们的最大公因数（GCF）。所得的分数等价但数字更小，更易于阅读和运算。

For example, to simplify ¹²⁄₁₆, find the largest number that divides 12 and 16 exactly – that is 4. Then divide: 12 ÷ 4 = 3, 16 ÷ 4 = 4, giving ¾. A fraction is fully simplified when the numerator and denominator have no common factor other than 1.

例如，化简¹²⁄₁₆，找到能整除12和16的最大数——即4。然后相除：12 ÷ 4 = 3，16 ÷ 4 = 4，得到¾。当分子和分母除了1以外没有其他公因数时，分数就完全化简了。

It is good practice to always simplify fractions in final answers. It reduces errors in further calculations and makes comparisons and ordering straightforward. Using divisibility rules can help you find common factors quickly.

在最终答案中始终化简分数是一个好习惯。它减少了进一步计算中的错误，并使比较和排序变得直接。运用整除性规则可以帮助你快速找到公因数。

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4. Mixed Numbers and Improper Fractions | 带分数与假分数

A mixed number combines a whole number with a proper fraction, like 2⅓. An improper fraction has a numerator larger than or equal to the denominator, such as ¹¹⁄₄. Converting between these two forms is a key skill in fraction arithmetic.

带分数将一个整数与一个真分数组合，如2⅓。假分数的分子大于或等于分母，如¹¹⁄₄。在这两种形式之间转换是分数运算中的一项关键技能。

To change an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder goes over the original denominator. For ¹¹⁄₄, 11 ÷ 4 = 2 remainder 3, so 2¾.

要将假分数转换为带分数，用分子除以分母。商成为整数部分，余数写在原分母之上。对于¹¹⁄₄，11 ÷ 4 = 2余3，因此得到2¾。

To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the denominator. For 2⅓: 2 × 3 + 1 = 7, so ⁷⁄₃. This is useful when multiplying or dividing fractions.

要将带分数转换为假分数，将整数乘以分母，加上分子，然后将结果放在分母之上。对于2⅓：2 × 3 + 1 = 7，所以是⁷⁄₃。这在分数的乘除运算中非常有用。

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5. Introduction to Decimals | 小数入门

A decimal is another way of writing a fraction whose denominator is a power of ten. The decimal point separates the whole number part from the fractional part. Each digit after the point represents tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on.

小数是分母为10的幂的分数的另一种写法。小数点将整数部分与小数部分分开。小数点后的每一位分别代表十分位（1/10）、百分位（1/100）、千分位（1/1000）等等。

For instance, 0.7 means 7/10, 0.25 means 25/100 (which simplifies to ¼), and 0.375 means 375/1000 (which simplifies to ⅜). The place value system extends to the right of the decimal point, providing a uniform way to express parts of a whole.

例如，0.7表示7/10，0.25表示25/100（可化简为¼），0.375表示375/1000（可化简为⅜）。位值系统延伸至小数点右侧，提供了一种统一的方式来表示整体的一部分。

Decimals are widely used in measurements, money and data. Understanding decimal place value is crucial for ordering, comparing and converting between decimals and other forms.

小数广泛用于测量、货币和数据中。理解小数位值对于排序、比较以及与其他形式之间的转换至关重要。

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6. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, you simply divide the numerator by the denominator. This can be done using short division or a calculator. The result may be a terminating decimal (e.g., ⅛ = 0.125) or a recurring decimal (e.g., ⅓ = 0.333…, often written as 0.3̅).

要将分数转换为小数，只需用分子除以分母。可以使用短除法或计算器完成。结果可能是有限小数（如⅛ = 0.125）或循环小数（如⅓ = 0.333…，通常写作0.3̅）。

For fractions with denominators that are powers of ten, the conversion is immediate: ⁷⁄₁₀ = 0.7, ²³⁄₁₀₀ = 0.23. For other denominators, memorizing common fraction–decimal equivalents (e.g., ½ = 0.5, ¼ = 0.25, ¾ = 0.75) speeds up problem-solving.

对于分母为10的幂的分数，转换是即时的：⁷⁄₁₀ = 0.7，²³⁄₁₀₀ = 0.23。对于其他分母，记住常见的分数-小数等值（如½ = 0.5，¼ = 0.25，¾ = 0.75）可以加快解题速度。

If the division produces a long string of repeating digits, we use notation like a dot or bar over the repeating block. For example, ²⁄₇ = 0.285714285714… is written as 0.2̅8̅5̅7̅1̅4̅. Understanding terminating and recurring decimals helps when working with exact answers.

如果除法产生一长串重复数字，我们使用点或横线标在循环块上。例如，²⁄₇ = 0.285714285714… 写作0.2̅8̅5̅7̅1̅4̅。理解有限小数和循环小数有助于处理精确答案。

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7. Introduction to Percentages | 百分比入门

A percentage represents a number out of 100. The symbol % means ‘per hundred’. So 45% means 45 out of 100, or 45/100. Percentages are useful for comparing proportions and expressing changes, such as discounts, interest rates and test scores.

百分比表示一个数占100的多少。符号%意味着“每一百”。因此45%表示45 out of 100，或45/100。百分比对于比较比例和表示变化（如折扣、利率和考试成绩）非常有用。

Fractions and decimals can be converted to percentages by making the denominator 100 or by multiplying by 100%. For example, ⅖ = (2 ÷ 5) × 100% = 40%. Similarly, 0.17 = 0.17 × 100% = 17%.

分数和小数可以通过将分母变为100或乘以100%转换为百分比。例如，⅖ = (2 ÷ 5) × 100% = 40%。类似地，0.17 = 0.17 × 100% = 17%。

Percentages greater than 100% indicate a quantity larger than the whole. For instance, 150% means 150/100 = 1.5, which is larger than 1. This concept is used in profit, population growth and magnification.

大于100%的百分比表示比整体更大的量。例如，150%表示150/100 = 1.5，大于1。这一概念用于利润、人口增长和放大倍数中。

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8. Converting Decimals to Percentages | 小数转百分比

Converting a decimal to a percentage is straightforward: multiply the decimal by 100 and add the % sign. This is equivalent to moving the decimal point two places to the right. For example, 0.63 becomes 63%, and 0.08 becomes 8%.

将小数转换为百分比非常简单：将小数乘以100并加上百分号。这相当于将小数点向右移动两位。例如，0.63变为63%，0.08变为8%。

If the decimal is greater than 1, the percentage will exceed 100. For example, 1.25 = 125%. When converting a decimal with many digits, you can round the percentage to a suitable number of decimal places if needed.

如果小数大于1，百分比将超过100。例如，1.25 = 125%。当转换具有许多位的小数时，如有需要，可以将百分比四舍五入到合适的小数位数。

Practice this conversion back and forth until it becomes automatic. Remember: percent means ‘per hundred’, so 0.05 is 5 per hundred, or 5%. Writing a decimal as a percentage is just another way of expressing the same proportion out of 100.

来回练习这种转换直到变得自动。记住：百分比意味着“每百”，因此0.05是每百之五，即5%。将小数写为百分比只是以100为基准表示相同比例的另一方式。

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9. Converting Fractions to Percentages | 分数转百分比

To turn a fraction into a percentage, you can either convert it to a decimal first (by dividing numerator by denominator) and then multiply by 100%, or find an equivalent fraction with denominator 100.

要将分数转换为百分比，你可以先将其转换为小数（用分子除以分母），然后乘以100%，或者找到一个分母为100的等价分数。

For example, ⅜: 3 ÷ 8 = 0.375, then 0.375 × 100% = 37.5%. Alternatively, for fractions like ⅗, we can multiply top and bottom to get a denominator of 100: ⅗ = (3×20)/(5×20) = 60/100 = 60%. The second method works well when the denominator is a factor of 100.

例如，⅜：3 ÷ 8 = 0.375，然后0.375 × 100% = 37.5%。另一种方法，对于像⅗这样的分数，我们可以将分子分母同时乘以一个数使分母变为100：⅗ = (3×20)/(5×20) = 60/100 = 60%。当分母是100的因数时，第二种方法很好用。

Some fraction–percentage equivalents are useful to learn by heart:

• 1/2 = 50%
• 1/4 = 25%
• 3/4 = 75%
• 1/5 = 20%
• 1/10 = 10%
• 1/3 ≈ 33.3%

These common values speed up mental calculations.

一些分数-百分比等值值得熟记：

• 1/2 = 50%
• 1/4 = 25%
• 3/4 = 75%
• 1/5 = 20%
• 1/10 = 10%
• 1/3 ≈ 33.3%

这些常见值能加快心算速度。

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10. Comparing Fractions, Decimals and Percentages | 比较分数、小数与百分比

When you need to compare quantities given in different forms, convert them all to the same representation. Often, converting to decimals is the easiest for comparison because decimal place values allow quick numeric comparison.

当需要比较以不同形式给出的量时，将它们全都转换为相同的表示形式。通常，转换为小数最容易比较，因为小数位值允许快速进行数值比较。

Suppose you want to order 0.3, 35% and ⅖ from least to greatest. Convert: 35% = 0.35, ⅖ = 0.4. Now compare: 0.3 < 0.35 < 0.4. Thus the order is 0.3, 35%, ⅖.

假设你想将0.3、35%和⅖从小到大排序。转换：35% = 0.35，⅖ = 0.4。现在比较：0.3 < 0.35 < 0.4。因此顺序为0.3、35%、⅖。

Alternatively, you can convert everything to percentages or fractions. The key is to use a common form. With practice, you will start to recognise equivalent values and compare them instantly without conversion.

或者，你可以将所有数值转换为百分比或分数。关键是使用统一的形式。通过练习，你会开始识别等值并无需转换即可立即比较它们。

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11. Ordering Fractions, Decimals and Percentages | 排序分数、小数与百分比

Ordering a mixed set of fractions, decimals and percentages requires careful conversion. Write each value in one chosen form, list them, and then order as required – ascending or descending. Always double-check your conversions to avoid mistakes.

对一组混合的分数、小数和百分比进行排序需要仔细转换。将每个值写成选定的一种形式，列出它们，然后按要求排序——升序或降序。始终仔细检查转换以避免错误。

For example, arrange ½, 0.4, 30%, ⅘ from largest to smallest. Convert to decimals: ½ = 0.5, 0.4 remains, 30% = 0.3, ⅘ = 0.8. Order: 0.8, 0.5, 0.4, 0.3 → ⅘, ½, 0.4, 30%.

例如，将½、0.4、30%、⅘从大到小排列。转换为小数：½ = 0.5，0.4不变，30% = 0.3，⅘ = 0.8。排序：0.8, 0.5, 0.4, 0.3 → ⅘, ½, 0.4, 30%。

When ordering fractions with different denominators, finding a common denominator is an alternative. For instance, to order ⅔ and ¾, use denominator 12: ⁸⁄₁₂ and ⁹⁄₁₂, so ⅔ < ¾. This method is especially useful when the fractions are in their original form without decimals.

当对不同分母的分数进行排序时，寻找公分母是另一种方法。例如，要排序⅔和¾，使用分母12：⁸⁄₁₂和⁹⁄₁₂，所以⅔ < ¾。当分数保留原始形式而不使用小数时，此方法尤其有用。

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12. Real-life Applications | 实际应用

Fractions, decimals and percentages are everywhere in daily life. When you shop, a 20% discount means you pay 80% of the original price. If something costs £30, 80% = 0.8 × 30 = £24. Understanding percentages helps you work out the final price quickly.

分数、小数和百分比在日常生活中无处不在。购物时，20%的折扣意味着你支付原价的80%。如果某物售价30英镑，80% = 0.8 × 30 = 24英镑。理解百分比有助于你快速计算出最终价格。

In cooking, recipes often use fractions: ¾ cup of flour, ½ teaspoon of salt. If you want to make half the recipe, you multiply each amount by ½. This involves fraction multiplication, a skill that relies on a solid understanding of the topic.

在烹饪中，食谱常常用到分数：¾杯面粉，½茶匙盐。如果你想制作食谱一半的量，你需要将每种用量乘以½。这涉及分数乘法，这项技能依赖于对该主题的扎实理解。

In statistics, data is frequently presented as percentages or decimals, such as survey results or probabilities. Mastering FDP equivalence allows you to interpret graphs and tables accurately, and to make informed decisions based on numerical information.

在统计学中，数据常常以百分比或小数呈现，例如调查结果或概率。掌握分数、小数和百分比的等值转换，能够让你准确地解读图表和表格，并根据数值信息做出明智的决策。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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