Fractions, Decimals and Percentages — 分数、小数和百分比完全指南

分数、小数和百分比是英国 KS3（Year 7）数学课程中最重要、也最实用的基础模块之一。它们本质上是”同一个数量”的三种不同写法，学会在它们之间自由转换，是后续学习比例、代数、概率和统计的敲门砖。本文从分数的基本结构讲起，逐步覆盖等价分数、化简、假分数与带分数、分数四则运算、小数的位值与运算、三者互化、求百分比的方法，以及生活中的实际应用，并配以大量可直接上手的例题。

Fractions, decimals and percentages form one of the most important and practical foundation modules in the UK KS3 (Year 7) mathematics curriculum. They are, in essence, three different ways of writing the same quantity, and learning to convert freely between them is the gateway to later work on ratio, algebra, probability and statistics. This article starts from the basic structure of a fraction and works through equivalent fractions, simplifying, improper fractions and mixed numbers, the four operations on fractions, decimal place value and arithmetic, converting between the three forms, methods for finding percentages, and real-life applications, with plenty of worked examples you can try straight away.

一、分数的三个部分：分子、分母与分数线 | The Three Parts of a Fraction: Numerator, Denominator and Fraction Bar

分数由三个部分组成：分子（numerator）、分母（denominator）和分数线（fraction bar）。分数线把一个整体”分割”成若干等份；分母写在分数线下方，表示整体被平均分成了几份；分子写在分数线上面，表示我们取走了几份。例如在 3/4 中，分母 4 表示把整体分成 4 等份，分子 3 表示取走其中 3 份。理解这个基本结构，是学习所有分数运算的第一步。

A fraction has three parts: the numerator, the denominator and the fraction bar. The fraction bar splits a whole into equal parts. The denominator, written below the bar, tells us how many equal parts the whole has been divided into; the numerator, written above the bar, tells us how many of those parts we are taking. 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 them. Understanding this basic structure is the first step in every fraction calculation.

在 Year 7 阶段，你还会遇到”单位分数”（unit fraction），即分子为 1 的分数，如 1/2、1/3、1/10。单位分数是分数的”积木”，任何分数都可以看作若干个单位分数相加。例如 3/4 = 1/4 + 1/4 + 1/4。养成用单位分数思考的习惯，能让后面的加减法变得简单许多。

In Year 7 you will also meet “unit fractions”, which are fractions with a numerator of 1, such as 1/2, 1/3 and 1/10. Unit fractions are the building blocks of fractions; any fraction can be seen as several unit fractions added together. For example, 3/4 = 1/4 + 1/4 + 1/4. Getting into the habit of thinking with unit fractions makes later addition and subtraction much easier.

二、等价分数与化简：用最大公因数约分 | Equivalent Fractions and Simplifying: Using the Highest Common Factor

等价分数（equivalent fractions）是数值相同但写法不同的分数。例如 1/2、2/4、4/8 都表示同样的数量。产生等价分数的规则很简单：把分子和分母同时乘以或除以同一个非零数，分数的大小不变，这就是分数的”基本性质”。它也解释了为什么 1/2 和 2/4 可以用等号连接。

Equivalent fractions are fractions that have the same value but are written differently. For example, 1/2, 2/4 and 4/8 all represent the same quantity. The rule for producing equivalent fractions is simple: multiply or divide both the numerator and the denominator by the same non-zero number, and the value of the fraction stays the same. This is the fundamental property of fractions, and it explains why 1/2 and 2/4 can be joined by an equals sign.

化简分数（simplify）就是把分数写成最简单的形式，即分子分母互质（没有 1 以外的公因数）。方法是用分子和分母的最大公因数（HCF，highest common factor）同时除以两者。例如化简 12/18：12 和 18 的 HCF 是 6，所以 12 ÷ 6 = 2，18 ÷ 6 = 3，得到 2/3。化简后的分数叫”最简分数”（fraction in lowest terms）。

Simplifying a fraction means writing it in its simplest form, where the numerator and denominator have no common factor other than 1 (they are coprime). The method is to divide both by their highest common factor (HCF). For example, to simplify 12/18: the HCF of 12 and 18 is 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving 2/3. A simplified fraction is said to be “in lowest terms”.

三、假分数与带分数：两种写法的互相转换 | Improper Fractions and Mixed Numbers: Converting Between the Two Forms

分数分两类：真分数（proper fraction，分子小于分母，如 3/4）和假分数（improper fraction，分子大于或等于分母，如 7/4）。假分数的值大于或等于 1。带分数（mixed number）则用一个整数加一个真分数来表示，如 1 3/4 表示”1 个整体再加 3/4″。在计算乘除法时，假分数通常比带分数更顺手。

Fractions fall into two types: proper fractions (where the numerator is smaller than the denominator, such as 3/4) and improper fractions (where the numerator is larger than or equal to the denominator, such as 7/4). An improper fraction has a value of 1 or more. A mixed number uses a whole number plus a proper fraction, such as 1 3/4, which means “one whole plus three quarters”. For multiplication and division, improper fractions are usually easier to work with than mixed numbers.

两者互化的方法是考试常考题型。假分数转带分数：用分子除以分母，商是整数部分，余数是新分子，分母不变。例如 7/4：7 ÷ 4 = 1 余 3，所以 7/4 = 1 3/4。带分数转假分数：整数部分乘分母再加分子，作为新分子，分母不变。例如 2 3/5：(2 × 5) + 3 = 13，所以 2 3/5 = 13/5。

Converting between the two forms is a common exam question. Improper to mixed: divide the numerator by the denominator; the quotient is the whole number, the remainder is the new numerator, and the denominator stays the same. For example, 7/4: 7 ÷ 4 = 1 remainder 3, so 7/4 = 1 3/4. Mixed to improper: multiply the whole number by the denominator, add the numerator, and put the result over the original denominator. For example, 2 3/5: (2 × 5) + 3 = 13, so 2 3/5 = 13/5.

四、分数加减法：先通分再运算 | Adding and Subtracting Fractions: Find a Common Denominator First

分数加减法的核心原则：分母相同才能直接加减。同分母分数相加减，只需把分子相加减，分母保持不变。例如 3/8 + 2/8 = 5/8。这是所有分数加减运算的基础，务必先掌握。

The core rule of adding and subtracting fractions is that you can only add or subtract directly when the denominators are the same. With a common denominator, simply add or subtract the numerators and keep the denominator unchanged. For example, 3/8 + 2/8 = 5/8. This is the foundation of all fraction addition and subtraction, so master it first.

分母不同时，需要先”通分”（find a common denominator），即找到两个分母的公倍数（通常用最小公倍数 LCM，lowest common multiple），把两个分数改写成等价分数，再做加减。例如 1/3 + 1/4：3 和 4 的 LCM 是 12，所以 1/3 = 4/12，1/4 = 3/12，相加得 7/12。

When the denominators differ, you must first find a common denominator, that is, a common multiple of the two denominators (usually the lowest common multiple, LCM). Rewrite both fractions as equivalent fractions, then add or subtract. For example, 1/3 + 1/4: the LCM of 3 and 4 is 12, so 1/3 = 4/12 and 1/4 = 3/12, and the sum is 7/12.

带分数的加减有两种做法：一是把带分数转成假分数再运算；二是整数部分和分数部分分别加减，最后把结果合并。做完之后别忘了检查结果是否能化简。例如 1 1/2 + 2 1/3，先转成假分数 3/2 + 7/3，通分后 9/6 + 14/6 = 23/6 = 3 5/6。

There are two ways to add or subtract mixed numbers: convert them to improper fractions first, or add or subtract the whole parts and the fraction parts separately and then combine the results. When you finish, remember to check whether the answer can be simplified. For example, 1 1/2 + 2 1/3 becomes the improper fractions 3/2 + 7/3; with a common denominator this is 9/6 + 14/6 = 23/6 = 3 5/6.

五、分数乘除法：交叉约分与倒数规则 | Multiplying and Dividing Fractions: Diagonal Cancelling and the Reciprocal Rule

分数乘法比加减法更简单：分子乘分子，分母乘分母，最后化简。例如 2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2。运算前可以”交叉约分”（cancel diagonally）来简化，比如 2 和 4 可以约掉公因数 2，让数字变小后再相乘。

Multiplying fractions is simpler than adding: multiply the numerators together and the denominators together, then simplify. For example, 2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2. You can “cancel diagonally” before multiplying to make the numbers smaller, for example cancelling a common factor of 2 between 2 and 4.

分数除法只有一条规则：除以一个分数等于乘以它的倒数（reciprocal）。把除号变成乘号，同时把除数”上下颠倒”。例如 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。记住口诀”除变乘，除数颠倒”。

Division of fractions has one single rule: dividing by a fraction is the same as multiplying by its reciprocal. Change the division sign to a multiplication sign and flip the divisor upside down. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. Remember the saying: “change the division to multiplication, and flip the second fraction”.

若除数或乘数中含有带分数或整数，先把它们转成假分数，再套用上面的规则。例如 2 ÷ 3/4 = 2/1 × 4/3 = 8/3 = 2 2/3。

If a mixed number or whole number appears in a division or multiplication, convert it to an improper fraction first, then apply the rules above. For example, 2 ÷ 3/4 = 2/1 × 4/3 = 8/3 = 2 2/3.

六、求一个数的几分之几：分数应用题的核心模型 | Finding a Fraction of an Amount: The Core Model of Fraction Word Problems

“求一个数的几分之几”是分数应用题的万能模型。方法只有一步：先除以分母求出 1 份（unit），再乘以分子求出所需份数。例如求 24 的 3/4：24 ÷ 4 = 6（1 份是 6），6 × 3 = 18（3 份是 18），所以 24 的 3/4 是 18。

“Finding a fraction of an amount” is the universal model behind fraction word problems. The method is a single step: first divide by the denominator to find one part (the unit), then multiply by the numerator to find the number of parts you need. For example, to find 3/4 of 24: 24 ÷ 4 = 6 (one part is 6), and 6 × 3 = 18 (three parts is 18), so 3/4 of 24 is 18.

这个方法也反过来用：如果已知某数的 2/5 是 12，求这个数，就先除以分子再乘分母。12 ÷ 2 = 6，6 × 5 = 30，所以这个数是 30。掌握”先除后乘”和”先乘后除”两个方向，就能应对绝大多数分数文字题。

The method also works in reverse: if 2/5 of a number is 12 and you need to find the whole number, divide by the numerator then multiply by the denominator. 12 ÷ 2 = 6, then 6 × 5 = 30, so the number is 30. Once you master both directions (divide then multiply, and its reverse), you can handle the vast majority of fraction word problems.

七、小数的位值与比较：从十分位到千分位 | Decimal Place Value and Comparison: From Tenths to Thousandths

小数（decimal）用小数点（decimal point）把整数部分和小数部分分开。小数点右边每一位都有固定含义：第一位是十分位（tenths），第二位是百分位（hundredths），第三位是千分位（thousandths）。例如 0.47 表示 4 个十分之一加 7 个百分之一。理解位值（place value）是掌握小数运算的关键。

A decimal uses a decimal point to separate the whole part from the fractional part. Each position to the right of the point has a fixed meaning: the first is tenths, the second is hundredths and the third is thousandths. For example, 0.47 means 4 tenths plus 7 hundredths. Understanding place value is the key to mastering decimals.

比较小数大小时，不要看数字的长度，而要看最高位的大小：先比整数部分，再从左到右逐位比较小数部分。例如 0.7 大于 0.68，因为十分位上 7 大于 6。必要时可以在末尾补 0 使位数对齐，例如把 0.7 看成 0.70，比较就更直观。

When comparing decimals, do not look at the length of the number; look at the value of the highest place. Compare the whole parts first, then compare the decimal places from left to right. For example, 0.7 is larger than 0.68 because 7 tenths is more than 6 tenths. If it helps, add trailing zeros to line up the places, for example treating 0.7 as 0.70 makes the comparison easier to see.

八、小数的加减与乘除：对齐小数点与数位数 | Adding, Subtracting, Multiplying and Dividing Decimals: Align the Point, Count the Digits

小数加减法的关键是对齐小数点，再按整数相加减，最后把小数点垂直落下。例如计算 3.45 + 2.7：把 2.7 写成 2.70 对齐后相加，得 6.15。小数点对齐就等于相同数位对齐。

The key to adding and subtracting decimals is to align the decimal points, then add or subtract as you would with whole numbers, and finally bring the point straight down. For example, to work out 3.45 + 2.7: write 2.7 as 2.70, line up the points and add to get 6.15. Aligning the points is the same as aligning the place values.

小数乘法先忽略小数点，按整数相乘，最后数出两个因数中小数位数的总和，从积的右边数起点上小数点。例如 0.4 × 0.6：4 × 6 = 24，两个因数共 2 位小数，所以答案是 0.24。小数除法先把除数变成整数（同时把被除数的小数点也移动相同的位数），再按整数除法计算。

To multiply decimals, ignore the points and multiply as whole numbers first, then count the total number of decimal places in the two factors and place the point in the product, counting from the right. For example, 0.4 × 0.6: 4 × 6 = 24, and the two factors have 2 decimal places in total, so the answer is 0.24. To divide decimals, first turn the divisor into a whole number (moving the dividend’s point the same number of places), then divide as with whole numbers.

九、分数、小数、百分比三者互化 | Converting Between Fractions, Decimals and Percentages

分数、小数、百分比（percentage）是同一数量的三种写法，学会互化是 KS3 的核心技能。分数转小数：用分子除以分母，如 3/4 = 0.75。小数转分数：把小数写成”十分之几、百分之几”再化简，如 0.75 = 75/100 = 3/4。

Fractions, decimals and percentages are three ways of writing the same quantity, and converting between them is a core KS3 skill. Fraction to decimal: divide the numerator by the denominator, for example 3/4 = 0.75. Decimal to fraction: write the decimal as tenths, hundredths and so on, then simplify, for example 0.75 = 75/100 = 3/4.

百分比（百分数）就是”分母为 100 的分数”，百分号 % 表示”每一百份”。小数转百分比：乘以 100 再加 % 号，如 0.45 = 45%。百分比转小数：除以 100（去掉 % 号，小数点左移两位），如 45% = 0.45。分数转百分比：先转小数，再乘 100。

A percentage is simply a fraction with denominator 100; the % sign means “out of one hundred”. Decimal to percentage: multiply by 100 and add the % sign, for example 0.45 = 45%. Percentage to decimal: divide by 100 (remove the % sign and move the point two places left), for example 45% = 0.45. Fraction to percentage: convert to a decimal first, then multiply by 100.

下面这张表列出了常见分数、小数、百分比的对应关系，建议背熟，考试能省很多时间。

The table below lists the conversions between the most common fractions, decimals and percentages. Learn them by heart; they save a lot of time in exams.

分数 Fraction	小数 Decimal	百分比 Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%
1/3	0.333…	33.3%
2/3	0.666…	66.7%

十、求一个数的百分比：三条通用方法 | Finding a Percentage of an Amount: Three Reliable Methods

求一个数的百分之几是 Year 7 的高频题型，有三条通用方法。方法一：先求 1%，再乘所需份数。例如求 200 的 15%：200 的 1% 是 2，15% 就是 2 × 15 = 30。

Finding a percentage of an amount is a very common Year 7 question type, and there are three reliable methods. Method one: find 1% first, then multiply by the number of parts you need. For example, to find 15% of 200: 1% of 200 is 2, so 15% is 2 × 15 = 30.

方法二：把百分比直接变成小数，再相乘。例如 15% of 200 = 0.15 × 200 = 30。方法三：把百分比写成分数再计算，尤其适合 50%、25%、10% 这类”友好百分比”。例如 25% of 80 = 1/4 × 80 = 20。三条方法结果相同，选你最有把握的一条即可。

Method two: turn the percentage straight into a decimal and multiply. For example, 15% of 200 = 0.15 × 200 = 30. Method three: write the percentage as a fraction and calculate, which works especially well for “friendly” percentages like 50%, 25% and 10%. For example, 25% of 80 = 1/4 × 80 = 20. All three methods give the same answer; pick the one you are most confident with.

十一、比较与排序：把一切变成同一种形式 | Comparing and Ordering: Turning Everything into One Form

比较不同形式的数（分数、小数、百分比混在一起）时，最有效的策略是”统一形式”：把它们全部化成小数（或全化成分数、全化成百分比），再从小到大排序。因为小数有直观的位值，通常把一切都转成小数最方便。

When comparing numbers in different forms (fractions, decimals and percentages mixed together), the most effective strategy is to use one form: convert them all to decimals (or all to fractions, or all to percentages), then order them from smallest to largest. Because decimals have intuitive place value, converting everything to decimals is usually the most convenient.

例如把 3/5、0.55、58% 排序：3/5 = 0.6，58% = 0.58，所以从小到大是 0.55 < 0.58 < 0.6，即 0.55 < 58% < 3/5。排序完成后，答案要写回原来的形式，不要只写换算后的小数。

For example, to order 3/5, 0.55 and 58%: 3/5 = 0.6 and 58% = 0.58, so from smallest to largest we have 0.55 < 0.58 < 0.6, that is 0.55 < 58% < 3/5. When you finish ordering, write the answer back in the original forms, not just the converted decimals.

十二、生活中的分数小数百分比与常见误区 | Fractions, Decimals and Percentages in Real Life, and Common Mistakes

分数、小数、百分比在真实生活中无处不在：商店折扣（discount）、银行利息（interest）、食谱配料比例、考试成绩百分比、地图比例尺等。掌握它们，你才能真正”用数学解决实际问题”。

Fractions, decimals and percentages appear everywhere in real life: shop discounts, bank interest, recipe proportions, exam score percentages and map scales. Mastering them lets you genuinely “use maths to solve real problems”.

典型应用题：一件衣服原价 80 元，按原价的 75% 出售，现价是多少？用方法二：0.75 × 80 = 60 元。再如：一次考试 50 题，做对 42 题，正确率是多少？42/50 = 84/100 = 84%。这类”文字题”的关键是把题目翻译成数学算式。

A typical word problem: a shirt originally costs 80 yuan and is sold at 75% of the original price. What is the sale price? Using method two: 0.75 × 80 = 60 yuan. Another example: a test has 50 questions and you answer 42 correctly; what is the percentage score? 42/50 = 84/100 = 84%. The key to these word problems is translating the wording into a mathematical expression.

最后提醒几个常见误区，考试时务必避开：一是分数加减时不先通分，直接把分子分母分别相加；二是化简时只约掉分子或分母一方；三是比较小数时误以为”位数越多越大”；四是把 0.5 和 50% 当成两个不同的数。牢记”先通分、整体约、对齐位、统一形式”，这些坑都能躲开。

Finally, watch out for these common mistakes in exams: adding fractions without finding a common denominator first; simplifying by cancelling only the numerator or only the denominator; thinking a longer decimal is always larger; and treating 0.5 and 50% as two different numbers. Remember “common denominator first, cancel the whole fraction, align the places, and convert to one form”, and you will avoid all of these traps.

十三、小数的四舍五入：保留到指定数位 | Rounding Decimals: Keeping a Given Number of Decimal Places

四舍五入（rounding）是处理小数时的常用技能，目的是把冗长的小数简化到指定精度。规则：看要保留的最后一位的右边一位数字，若它大于等于 5 就”进一”，否则直接舍去。例如把 3.746 保留两位小数：看第三位小数 6，6 ≥ 5，所以 3.746 ≈ 3.75。

Rounding is a common skill when working with decimals; its purpose is to shorten a long decimal to a specified precision. The rule: look at the digit immediately to the right of the last place you want to keep; if it is 5 or more, round up, otherwise leave it. For example, to round 3.746 to two decimal places: look at the third decimal digit 6, and since 6 ≥ 5, we have 3.746 ≈ 3.75.

常见的保留方式有三种：保留整数（到个位）、保留一位小数（到十分位）、保留两位小数（到百分位）。例如 7.82 保留整数：看十分位 8，8 ≥ 5，所以 7.82 ≈ 8。注意四舍五入后要写”约等于”符号 ≈，而不是等号。

There are three common levels of rounding: to the nearest whole number (units), to one decimal place (tenths), and to two decimal places (hundredths). For example, to round 7.82 to the nearest whole number: look at the tenths digit 8, and since 8 ≥ 5, we have 7.82 ≈ 8. Note that after rounding you should write the “approximately equal to” sign ≈, not an equals sign.

四舍五入在钱和测量中特别常用，因为金额通常只保留到”分”（两位小数），长度、重量等测量结果也要按精度取整。一个实用的综合题：把 2/3 化成小数，再保留两位小数。2/3 = 0.666…，看第三位小数 6，6 ≥ 5，所以 2/3 ≈ 0.67。这正好把前面的分数化小数与四舍五入串了起来。

Rounding is especially common with money and measurement, since amounts are usually kept to two decimal places (pence) and measurements are rounded to a given precision. A useful combined exercise: convert 2/3 to a decimal, then round it to two decimal places. 2/3 = 0.666…, and the third decimal digit is 6, which is 5 or more, so 2/3 ≈ 0.67. This neatly links the earlier fraction-to-decimal conversion with rounding.

Summary | 总结

本文系统梳理了 KS3（Year 7）阶段分数、小数、百分比的核心知识：分数的结构与单位分数、等价分数与化简、假分数与带分数的互化、分数加减乘除的规则、求一个数的几分之几、小数的位值与四则运算、三者互化、求百分比的三条方法、比较排序的统一策略，以及生活中的应用与常见误区。掌握这些，你就掌握了分数世界的”通用语言”。

This article has systematically covered the core KS3 (Year 7) knowledge of fractions, decimals and percentages: the structure of fractions and unit fractions, equivalent fractions and simplifying, converting between improper fractions and mixed numbers, the rules of fraction arithmetic, finding a fraction of an amount, decimal place value and arithmetic, converting between the three forms, three methods for finding percentages, the unified strategy for comparing and ordering, and real-life applications with common mistakes. Master these, and you have mastered the common language of the fraction world.

学习建议：先背熟常见分数与小数百分比的对应表，再反复练习通分和互化，最后用文字应用题检验理解。每个知识点都配一个小例子亲手算一遍，比单纯阅读更有效。

Study tips: first memorise the table of common conversions, then practise finding common denominators and converting forms repeatedly, and finally test your understanding with word problems. Work through one small example by hand for every topic; it is far more effective than reading alone.

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