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

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

In Key Stage 3 Mathematics, one of the most fundamental topics is the relationship between fractions, decimals and percentages. These three forms are simply different ways of representing parts of a whole. Whether you are slicing a cake, calculating a discount or measuring ingredients, you will constantly switch between these representations. Mastering this topic lays the groundwork for ratio, proportion, algebra and probability later in your studies.

在KS3阶段的数学学习中，分数、小数和百分比之间的关系是最基础的主题之一。这三种形式只是表示整体部分的不同方式。无论是切蛋糕、计算折扣还是量取食材，你都会不断在这些表示法之间切换。掌握这一主题将为以后学习比和比例、代数以及概率打下坚实的基础。

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1. Understanding Fractions | 理解分数

A fraction represents a part of a whole. It is written with a numerator (the top number) and a denominator (the bottom number). The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have. For example, ⅗ means you have 3 parts out of 5 equal parts.

分数表示整体的一部分。它由分子（上方的数字）和分母（下方的数字）组成。分母表示整体被分成多少等份，分子表示你拥有其中的几份。例如，⅗ 表示你拥有5等份中的3份。

Proper fractions have a numerator smaller than the denominator, like ⅔. Improper fractions have a numerator larger than or equal to the denominator, such as ⁵/₃. Mixed numbers combine a whole number and a proper fraction, for example 1⅔.

真分数的分子比分母小，如 ⅔。假分数的分子大于或等于分母，如 ⁵/₃。带分数则结合了一个整数和一个真分数，例如 1⅔。

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2. Equivalent Fractions | 等值分数

Equivalent fractions are different fractions that represent the same value. You can find equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. For example, ½ is equivalent to ²/₄, ³/₆ and ⁴/₈ because multiplying top and bottom by 2, 3 or 4 gives the same amount.

等值分数是表示相同数值的不同分数。你可以通过将分子和分母同时乘以或除以相同的非零数字来找到等值分数。例如，½ 等价于 ²/₄、³/₆ 和 ⁴/₈，因为将分子和分母同时乘以2、3或4会得到相同的量。

Understanding equivalent fractions is crucial for simplifying fractions and comparing fractions with different denominators. If the denominators are different, you can use equivalent fractions to rewrite them with a common denominator.

理解等值分数对于化简分数以及比较不同分母的分数至关重要。如果分母不同，你可以利用等值分数将它们改写为具有相同分母的分数。

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3. Simplifying Fractions | 分数的化简

To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). The result is the fraction in its simplest form, where the numerator and denominator have no common factor other than 1. For example, to simplify ⁸/₁₂, the HCF of 8 and 12 is 4, so divide both by 4 to get ⅔.

化简分数时，用分子和分母的最大公因数（HCF）去除它们。结果就是最简分数，此时分子和分母除了1以外没有其他公因数。例如，化简 ⁸/₁₂ 时，8和12的最大公因数是4，因此将两者除以4得到 ⅔。

Simplifying makes fractions easier to understand and compare. Always check if a fraction can be simplified further before using it in calculations or presenting your final answer.

化简可以使分数更易于理解和比较。在进行计算或展示最终答案之前，务必检查分数是否可以进一步化简。

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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. You can do this using short division or long division. For example, ⅗ becomes 0.6 because 3 ÷ 5 = 0.6. Some fractions produce terminating decimals, like ¼ = 0.25, while others produce recurring decimals, such as ⅓ = 0.333…

要将分数转换为小数，用分子除以分母。你可以使用短除法或长除法。例如，⅗ 变成0.6，因为 3 ÷ 5 = 0.6。有些分数产生有限小数，比如 ¼ = 0.25，而另一些则产生循环小数，如 ⅓ = 0.333…

It is useful to memorise the decimal equivalents of common fractions: ½ = 0.5, ⅓ ≈ 0.333, ⅔ ≈ 0.667, ¼ = 0.25, ¾ = 0.75, ⅕ = 0.2, ⅖ = 0.4, ⅗ = 0.6, ⅘ = 0.8.

记住常见分数的小数等价形式很有用：½ = 0.5，⅓ ≈ 0.333，⅔ ≈ 0.667，¼ = 0.25，¾ = 0.75，⅕ = 0.2，⅖ = 0.4，⅗ = 0.6，⅘ = 0.8。

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

To convert a terminating decimal to a fraction, write the decimal as the numerator and place it over a power of 10 corresponding to the number of decimal places. Then simplify if possible. For instance, 0.75 has two decimal places, so write it as ⁷⁵/₁₀₀ and simplify to ¾.

要将有限小数转换为分数，把小数写作分子，分母为与小数位数相对应的10的幂。然后尽可能地化简。例如，0.75有两位小数，因此写成 ⁷⁵/₁₀₀，再化简为 ¾。

For recurring decimals, the process involves a bit of algebra. If x = 0.333…, then 10x = 3.333… Subtract the first equation from the second to get 9x = 3, so x = ⅓. Recurring decimals always correspond to fractions.

对于循环小数，转换过程需要一点代数知识。如果 x = 0.333…，那么 10x = 3.333…。从第二个方程减去第一个方程，得到 9x = 3，因此 x = ⅓。循环小数总是与分数对应。

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6. Understanding Percentages | 理解百分比

A percentage is a way of expressing a number as a fraction of 100. The word ‘percent’ means ‘per hundred’. So 45% means 45 out of 100, or ⁴⁵/₁₀₀. Percentages are extremely useful for comparing proportions, discounts, interest rates and statistics because they always have the same base of 100.

百分比是将一个数表示为100的分数的一种方式。“百分之”的意思就是“每百分之一”。因此45%意味着每100份中的45份，或者说 ⁴⁵/₁₀₀。百分比在比较比例、折扣、利率和统计数据时非常有用，因为它们始终以100为基数。

Since percentages are just fractions with a denominator of 100, many common percentages have simple fraction equivalents: 50% = ½, 25% = ¼, 75% = ¾, 20% = ⅕, 10% = ⅒.

由于百分比只是分母为100的分数，许多常见的百分比都有简单的分数等价形式：50% = ½，25% = ¼，75% = ¾，20% = ⅕，10% = ⅒。

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7. Converting Between Percentages and Fractions | 百分比与分数的互化

To convert a percentage to a fraction, write the percentage as a fraction with denominator 100, then simplify. For example, 60% = ⁶⁰/₁₀₀ = ⅗. To convert a fraction to a percentage, first convert the fraction to a decimal by division, then multiply by 100 and add the % sign. For ⅗, 3 ÷ 5 = 0.6, then 0.6 × 100 = 60%.

要将百分比转换为分数，将百分比写成分母为100的分数，然后化简。例如，60% = ⁶⁰/₁₀₀ = ⅗。要将分数转换为百分比，首先通过除法将分数转换为小数，然后乘以100并加上%符号。对于 ⅗，3 ÷ 5 = 0.6，然后 0.6 × 100 = 60%。

Another method is to find an equivalent fraction with denominator 100. For ⅗, multiply numerator and denominator by 20 to get ⁶⁰/₁₀₀, which directly gives 60%. This method works well when the denominator is a factor of 100.

另一种方法是找到一个分母为100的等值分数。对于 ⅗，将分子和分母同时乘以20得到 ⁶⁰/₁₀₀，直接得出60%。当分母是100的因数时，这种方法很有效。

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8. Converting Between Percentages and Decimals | 百分比与小数的互化

Converting a percentage to a decimal is straightforward: divide the percentage by 100, which means moving the decimal point two places to the left. For example, 85% = 0.85. To convert a decimal to a percentage, multiply by 100, moving the decimal point two places to the right. So 0.23 becomes 23%.

将百分比转换为小数很简单：将百分比除以100，也就是将小数点向左移动两位。例如，85% = 0.85。要将小数转换为百分比，乘以100，即将小数点向右移动两位。因此0.23变成23%。

Always remember to remove or add the % sign appropriately. This conversion is often used when calculating percentage increase or decrease in real-life problems.

务必记得正确地去掉或加上%符号。在实际问题中计算百分比增减时，经常需要用到这种转换。

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9. Ordering and Comparing FDP | 分数、小数和百分比的排序与比较

To order a mix of fractions, decimals and percentages, convert all of them into the same form – usually decimals or percentages. For example, to compare ⅖, 0.45 and 30%, change everything to decimals: ⅖ = 0.4, 30% = 0.3. Then order: 0.3, 0.4, 0.45, which means 30%, ⅖, 0.45.

要对分数、小数和百分比的混合进行排序，需要将它们全部转换成同一种形式——通常是小数或百分比。例如，比较 ⅖、0.45 和 30%，将所有数都转换为小数：⅖ = 0.4，30% = 0.3。然后排序：0.3，0.4，0.45，对应的顺序是 30%，⅖，0.45。

Using a number line can be very helpful for visualising the relative sizes. Mark the converted decimals on a number line from 0 to 1 and compare their positions.

使用数轴对于直观看出相对大小非常有帮助。在0到1的数轴上标出转换后的小数，然后比较它们的位置。

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10. Problem Solving with FDP | 运用分数、小数和百分比解决问题

Many real-world problems involve fractions, decimals and percentages used together. For example, if a shop offers a 20% discount on a £45 item, you can find the sale price by converting 20% to 0.2, multiplying £45 by 0.2 to get £9 discount, and subtracting from the original price to pay £36.

许多实际问题会同时涉及分数、小数和百分比。例如，如果一家商店对一件45英镑的商品提供20%的折扣，你可以将20%转换为0.2，将45英镑乘以0.2得到9英镑的折扣，再从原价中减去，最终支付36英镑。

When baking, a recipe might require ¾ cup of sugar, but you only have a ⅓-cup measure. You can convert both to a common denominator (¹/₁₂) to work out how many scoops you need. ¾ = ⁹/₁₂ and ⅓ = ⁴/₁₂, so you need 2 ¼ scoops of the ⅓-cup measure. These skills are essential for everyday life, science and finance.

在烘焙时，食谱可能需要 ¾ 杯糖，但你只有 ⅓ 杯的量具。你可以将两者转换为同分母（例如12），来计算需要几个量勺。¾ = ⁹/₁₂，⅓ = ⁴/₁₂，所以你需要 2 ¼ 个 ⅓ 杯的量勺。这些技能在日常生活中、科学领域和财务问题上都至关重要。

Mastering the interplay between fractions, decimals and percentages will boost your confidence in mathematics and equip you with practical tools for a wide range of situations.

掌握分数、小数和百分比之间的相互转换将增强你在数学上的自信心，并为你提供应对各种实际情况的实用工具。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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