KS3 Year 7 Fractions, Decimals and Percentages: Complete Guide — KS3 Year 7 分数、小数和百分比完全指南

一、理解分数、小数和百分比 | Understanding Fractions, Decimals and Percentages

分数、小数和百分比（英文简称FDP）是KS3阶段数学的核心基础。它们实际上是表示同一个东西的三种不同方式 – 即”整体的一部分”。理解这三种形式以及它们之间的关系，是后续所有数学学习的关键。在Year 7阶段，你需要掌握它们之间的相互转换、大小比较、以及在实际问题中的应用。

Fractions, decimals and percentages (often abbreviated as FDP) are the core foundation of KS3 mathematics. They are, in essence, three different ways of representing the same thing – a part of a whole. Understanding these three forms and the relationships between them is crucial for all subsequent mathematics learning. In Year 7, you need to master converting between them, comparing their sizes, and applying them in real-world problems.

分数由一个分子（numerator）和一个分母（denominator）组成，分母表示整体被分成了几等份，分子表示取了几份。例如，¾表示整体被分成4等份，取了其中的3份。小数则基于十进制位值系统，小数点后的每一位代表十分之一、百分之一、千分之一等。百分比（per cent）字面意思是”每一百”，因此百分数总是以100为基准。

A fraction consists of a numerator and a denominator. 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, ¾ means the whole is divided into 4 equal parts and you have 3 of them. Decimals are based on the base-10 place value system, where each digit after the decimal point represents tenths, hundredths, thousandths, and so on. Percentages literally mean “per hundred”, so percentages are always expressed with 100 as the reference.

二、分数转换为小数 | Converting Fractions to Decimals

将分数转换为小数是KS3 Year 7的重要技能。最简单的方法是将分数理解为除法运算：分子除以分母。例如，¾就是3÷4=0.75。对于分母为10、100、1000的分数，转换非常直接：7/10=0.7，23/100=0.23，119/1000=0.119。但更常见的情况是，你需要进行长除法计算。

Converting fractions to decimals is an important skill in KS3 Year 7. The simplest method is to understand a fraction as a division operation: numerator divided by denominator. For example, ¾ is 3÷4=0.75. For fractions with denominators of 10, 100, or 1000, the conversion is very straightforward: 7/10=0.7, 23/100=0.23, 119/1000=0.119. However, more commonly, you will need to perform long division calculations.

值得注意的是，有些分数转换为小数时会产生有限小数（terminating decimals），如½=0.5、⅕=0.2；而另一些则会产生循环小数（recurring decimals），如⅓=0.333…（通常写作0.3̇）、1/6=0.1666…。判断一个分数是否会产生有限小数的方法是：将分母分解质因数，如果分母的质因数只有2和5，那么这个分数就能化成有限小数。这是因为2和5是10的因数，而十进制体系基于10。

It is worth noting that some fractions produce terminating decimals when converted, such as ½=0.5 and ⅕=0.2, while others produce recurring decimals, such as ⅓=0.333… (usually written as 0.3̇) and 1/6=0.1666…. The method to determine whether a fraction will produce a terminating decimal is to factorise the denominator into prime factors. If the denominator’s prime factors are only 2 and 5, then the fraction can be converted to a terminating decimal. This is because 2 and 5 are factors of 10, and the decimal system is based on 10.

三、小数转换为分数 | Converting Decimals to Fractions

小数转分数需要根据小数的位数来确定分母。一位小数（十分位）的分母为10，两位小数（百分位）的分母为100，三位小数（千分位）的分母为1000，以此类推。转换后务必将分数约简到最简形式。例如，0.25=25/100=¼（约分后）。

Converting decimals to fractions requires determining the denominator based on the number of decimal places. One decimal place (tenths) means denominator 10, two decimal places (hundredths) means denominator 100, three decimal places (thousandths) means denominator 1000, and so on. After conversion, always simplify the fraction to its simplest form. For example, 0.25=25/100=¼ (after simplification).

对于循环小数转换为分数，有一个巧妙的方法。以0.3̇（即0.333…）为例：设x=0.333…，那么10x=3.333…，两式相减得9x=3，所以x=3/9=⅓。对于更复杂的循环小数如0.27̇（即0.272727…），设x=0.272727…，那么100x=27.2727…，相减得99x=27，x=27/99=3/11。这个代数方法在GCSE阶段会深入学习，但Year 7学生也完全可以理解其基本原理。

For recurring decimals, there is a clever method of conversion to fractions. Take 0.3̇ (i.e., 0.333…) as an example: let x=0.333…, then 10x=3.333…, subtract to get 9x=3, so x=3/9=⅓. For more complex recurring decimals like 0.27̇ (i.e., 0.272727…), let x=0.272727…, then 100x=27.2727…, subtract to get 99x=27, x=27/99=3/11. This algebraic method will be studied in depth at GCSE level, but Year 7 students can certainly understand its basic principle.

四、小数与百分比的相互转换 | Converting Between Decimals and Percentages

小数和百分比之间的转换可能是最直观的FDP转换。要将小数转换为百分比，只需将小数点向右移动两位，然后加上百分号。例如：0.45=45%，0.07=7%，1.2=120%。反过来，要将百分比转换为小数，只需去掉百分号后将数字除以100（即将小数点向左移动两位）。例如：67%=0.67，8%=0.08，150%=1.5。

The conversion between decimals and percentages is perhaps the most intuitive of all FDP conversions. To convert a decimal to a percentage, simply move the decimal point two places to the right and add the percent sign. For example: 0.45=45%, 0.07=7%, 1.2=120%. Conversely, to convert a percentage to a decimal, remove the percent sign and divide the number by 100 (i.e., move the decimal point two places to the left). For example: 67%=0.67, 8%=0.08, 150%=1.5.

一个常见的易错点是处理小于1%的百分比。例如，0.5%转换为小数是0.005（不是0.5），½%转换为小数是0.005。同样，当小数小于0.01时，转换后的百分比也会小于1%。例如，0.003=0.3%。Year 7学生需要特别注意小数点位置的准确性，尤其是在处理涉及金钱和测量的问题时。

A common pitfall is handling percentages smaller than 1%. For example, 0.5% converted to a decimal is 0.005 (not 0.5), and ½% as a decimal is 0.005. Similarly, when a decimal is smaller than 0.01, the percentage will also be less than 1%. For example, 0.003=0.3%. Year 7 students need to pay special attention to the accuracy of decimal point placement, especially when dealing with problems involving money and measurement.

五、分数转换为百分比及常见等价值 | Converting Fractions to Percentages and Common Equivalents

将分数转换为百分比有两种常用方法。方法一：先将分数转换为小数（分子÷分母），再将小数转换为百分比。例如，⅜=3÷8=0.375=37.5%。方法二：将分数转化为分母为100的等值分数。例如，7/20=(7×5)/(20×5)=35/100=35%。方法二要求分母必须是100的因数，而方法一适用于所有情况。

There are two common methods for converting fractions to percentages. Method 1: first convert the fraction to a decimal (numerator ÷ denominator), then convert the decimal to a percentage. For example, ⅜=3÷8=0.375=37.5%. Method 2: convert the fraction into an equivalent fraction with a denominator of 100. For example, 7/20=(7×5)/(20×5)=35/100=35%. Method 2 requires the denominator to be a factor of 100, while Method 1 works in all cases.

以下是一些所有Year 7学生都应该记住的常见FDP等价值：½=0.5=50%，¼=0.25=25%，¾=0.75=75%，⅕=0.2=20%，⅖=0.4=40%，⅗=0.6=60%，⅘=0.8=80%，⅛=0.125=12.5%，⅜=0.375=37.5%，⅝=0.625=62.5%，⅞=0.875=87.5%，⅓≈0.333≈33.3%，⅔≈0.667≈66.7%，1/10=0.1=10%，1/20=0.05=5%，1/25=0.04=4%。记住这些等价值可以大大提高解题速度。

Here are the common FDP equivalents that all Year 7 students should memorise: ½=0.5=50%, ¼=0.25=25%, ¾=0.75=75%, ⅕=0.2=20%, ⅖=0.4=40%, ⅗=0.6=60%, ⅘=0.8=80%, ⅛=0.125=12.5%, ⅜=0.375=37.5%, ⅝=0.625=62.5%, ⅞=0.875=87.5%, ⅓≈0.333≈33.3%, ⅔≈0.667≈66.7%, 1/10=0.1=10%, 1/20=0.05=5%, 1/25=0.04=4%. Memorising these equivalents can greatly improve problem-solving speed.

六、比较和排序FDP | Comparing and Ordering FDP

比大小和排序是考试中的常见题型。当分数、小数和百分比混合在一起时，最好的策略是将它们全部转换为同一种形式。通常转换为小数最为方便，因为小数的大小比较非常直观 – 只需从左到右逐位比较即可。例如，要比较⅗、0.58和59%，将它们都转换为小数：⅗=0.6，0.58=0.58，59%=0.59。排序结果为：0.58<0.59<0.6，即0.58<59%<⅗。

Comparing and ordering is a common exam question type. When fractions, decimals and percentages are mixed together, the best strategy is to convert them all into the same form. Converting to decimals is usually the most convenient, as comparing decimal sizes is very intuitive – simply compare digit by digit from left to right. For example, to compare ⅗, 0.58 and 59%, convert them all to decimals: ⅗=0.6, 0.58=0.58, 59%=0.59. The ordering result is: 0.58<0.59<0.6, i.e., 0.58<59%<⅗.

另一种方法是将所有数值转换为百分比，这在处理以百分比为主的问题时特别有效。无论选择哪种方法，关键是保持一致 – 在一次比较中只使用一种形式。Year 7考试中经常出现要求将一组数按升序或降序排列的题目，多加练习可以帮助你在这些题目上做到快速而准确。

Another method is to convert all values to percentages, which is particularly effective when dealing with problems that are primarily percentage-based. Whichever method you choose, the key is to be consistent – use only one form within a single comparison. Year 7 exams frequently feature questions requiring you to arrange a set of numbers in ascending or descending order. Regular practice can help you become both quick and accurate on these questions.

七、求一个数的几分之几 | Finding a Fraction of an Amount

求一个数的几分之几是FDP最实用的应用之一。基本方法是：先用总量除以分母（求出其中的一份是多少），再将结果乘以分子（求出需要的份数）。例如，求60的¾：先算60÷4=15（一份是15），再算15×3=45（三份是45），所以60的¾=45。

Finding a fraction of an amount is one of the most practical applications of FDP. The basic method is: first divide the total by the denominator (to find what one part is worth), then multiply the result by the numerator (to find the required number of parts). For example, to find ¾ of 60: first calculate 60÷4=15 (one part is 15), then calculate 15×3=45 (three parts is 45), so ¾ of 60=45.

对于带分数的情况，先将带分数转换为假分数，再按同样方法计算。例如，求48的2¼（即9/4）：48÷4=12，12×9=108。在应用题中，这种计算经常出现在”打折后价格”、”剩余量”等问题中。例如：”一本书有240页，Jim读了⅝，他还剩多少页没读？”解答：已读=240×⅝=240÷8×5=150页，剩余=240-150=90页。

For mixed numbers, first convert the mixed number to an improper fraction, then calculate using the same method. For example, to find 2¼ (i.e., 9/4) of 48: 48÷4=12, 12×9=108. In word problems, this calculation frequently appears in contexts such as “price after discount” and “remaining amount”. For example: “A book has 240 pages. Jim reads ⅝ of it. How many pages does he have left?” Solution: read=240×⅝=240÷8×5=150 pages, remaining=240-150=90 pages.

八、求一个数的百分之几 | Finding a Percentage of an Amount

求一个数的百分之几同样有标准方法。最常用的方法是”除以100再乘以百分比”：将总量除以100得到1%的值，再乘以所需的百分比。例如，求80的15%：80÷100=0.8（1%是0.8），0.8×15=12，所以80的15%=12。另一种方法是将百分比转换为小数后直接相乘：80×0.15=12。

Finding a percentage of an amount also has a standard method. The most commonly used method is “divide by 100 then multiply by the percentage”: divide the total by 100 to get the value of 1%, then multiply by the required percentage. For example, to find 15% of 80: 80÷100=0.8 (1% is 0.8), 0.8×15=12, so 15% of 80=12. An alternative method is to convert the percentage to a decimal and multiply directly: 80×0.15=12.

使用”10%法”可以使心算更加高效。由于10%是总量的十分之一，你可以很容易地通过10%来推导其他百分比。例如，求350的30%：10%=35，所以30%=35×3=105。同样，5%是10%的一半，1%是10%的十分之一。对于15%，可以计算为10%+5%；对于17.5%，可以计算为10%+5%+2.5%。掌握这些心算技巧可以显著提高解题速度。

Using the “10% method” makes mental calculation much more efficient. Since 10% is one-tenth of the total, you can easily derive other percentages from 10%. For example, to find 30% of 350: 10%=35, so 30%=35×3=105. Similarly, 5% is half of 10%, and 1% is one-tenth of 10%. For 15%, you can calculate it as 10%+5%; for 17.5%, you can calculate it as 10%+5%+2.5%. Mastering these mental arithmetic techniques can significantly improve problem-solving speed.

百分比增减是另一个重要应用。计算增加百分比：先求原数的百分比值，再加到原数上。例如，£200增加15%：15% of £200=£30，新价格=£200+£30=£230。更高效的方法是使用乘数（multiplier）：增加15%等价于乘以1.15，减少15%等价于乘以0.85。£200×1.15=£230。

Percentage increase and decrease is another important application. To calculate a percentage increase: first find the percentage of the original amount, then add it to the original. For example, £200 increased by 15%: 15% of £200=£30, new price=£200+£30=£230. A more efficient method is to use a multiplier: an increase of 15% is equivalent to multiplying by 1.15, and a decrease of 15% is equivalent to multiplying by 0.85. £200×1.15=£230.

九、分数的加减法 | Adding and Subtracting Fractions

同分母分数的加减法很简单：分母保持不变，直接将分子相加或相减。例如，3/8+2/8=5/8，7/10-4/10=3/10。但异分母分数的加减法则需要先找到公分母（common denominator）。通常使用两个分母的最小公倍数（LCM）作为公分母。

Adding and subtracting fractions with the same denominator is straightforward: keep the denominator and simply add or subtract the numerators. For example, 3/8+2/8=5/8, 7/10-4/10=3/10. However, adding and subtracting fractions with different denominators requires first finding a common denominator. Usually, the lowest common multiple (LCM) of the two denominators is used as the common denominator.

找到公分母后，利用等值分数的概念将每个分数转换为以公分母为分母的等值分数，然后再进行加减。例如，计算⅔+¼：2和4的LCM是12（也可直接用8，但12更小）。⅔=8/12，¼=3/12，所以⅔+¼=8/12+3/12=11/12。对于带分数，可以先将其转换为假分数再计算，或者将整数部分和分数部分分开处理。

After finding the common denominator, use the concept of equivalent fractions to convert each fraction to an equivalent fraction with the common denominator, then add or subtract. For example, to calculate ⅔+¼: the LCM of 3 and 4 is 12 (you could also use 8 directly, but 12 is smaller). ⅔=8/12, ¼=3/12, so ⅔+¼=8/12+3/12=11/12. For mixed numbers, you can first convert them to improper fractions, or handle the whole number part and the fractional part separately.

十、FDP在实际生活中的应用 | Real-World Applications of FDP

FDP在日常生活中的应用无处不在。商店打折是百分比最常见的应用场景：原价£45的T恤打八折（20% off），折后价=£45×0.8=£36。如果在此基础上再打15%的学生折扣，最终价格=£36×0.85=£30.60。注意多步折扣不能简单相加（20%+15%≠35%），而需要逐次计算。

FDP applications are everywhere in daily life. Shop discounts are the most common application of percentages: a T-shirt originally priced at £45 with 20% off costs £45×0.8=£36. If there is an additional 15% student discount on top, the final price=£36×0.85=£30.60. Note that multi-step discounts cannot simply be added together (20%+15%≠35%); they must be calculated sequentially.

分数在烹饪和食谱调整中也非常重要。如果一个食谱是为4人设计的，但你需要为6人准备，你需要将所有配料乘以6/4（即1.5倍）。小数则广泛应用于测量和科学计算中：长度、质量、体积的测量通常精确到十分位、百分位或千分位。百分比还广泛应用于金融领域：银行利率、投资回报率、通货膨胀率等都以百分比表示。理解FDP的相互转换关系将使你在各个学科和日常生活中受益。

Fractions are also very important in cooking and recipe adjustment. If a recipe is designed for 4 people but you need to prepare it for 6, you need to multiply all ingredients by 6/4 (i.e., 1.5 times). Decimals are widely used in measurement and scientific calculations: measurements of length, mass, and volume are usually precise to tenths, hundredths, or thousandths. Percentages are also widely applied in finance: bank interest rates, investment returns, inflation rates, and more are all expressed as percentages. Understanding the interconversion relationships of FDP will benefit you across all subjects and in everyday life.

十一、等值分数与分数化简 | Equivalent Fractions and Simplifying Fractions

等值分数（equivalent fractions）是指数值相等但分子分母不同的分数。例如，½=2/4=3/6=4/8=50/100，这些都是等值分数。创建等值分数的方法很简单：将分子和分母同时乘以同一个数（不能为0）。反过来，化简分数（simplifying/cancelling down）就是将分子和分母同时除以它们的最大公因数（HCF），直到分子分母互质（即最大公因数为1），此时分数为最简形式。

Equivalent fractions are fractions that have the same value but different numerators and denominators. For example, ½=2/4=3/6=4/8=50/100 – these are all equivalent fractions. The method for creating equivalent fractions is simple: multiply both the numerator and denominator by the same number (not zero). Conversely, simplifying a fraction (also called cancelling down) involves dividing both the numerator and denominator by their highest common factor (HCF) until the numerator and denominator are coprime (i.e., their HCF is 1), at which point the fraction is in its simplest form.

化简分数是Year 7考试中的必考技能。例如，化简28/42：找28和42的HCF。28的因数有1、2、4、7、14、28；42的因数有1、2、3、6、7、14、21、42。HCF=14，所以28/42=(28÷14)/(42÷14)=2/3。一个快速技巧：如果分子和分母都是偶数，可以先同时除以2。如果都以0或5结尾，可以先除以5。使用质因数分解也可以系统地找到HCF。

Simplifying fractions is an essential skill tested in Year 7 exams. For example, to simplify 28/42: find the HCF of 28 and 42. Factors of 28: 1, 2, 4, 7, 14, 28; factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. HCF=14, so 28/42=(28÷14)/(42÷14)=2/3. A quick tip: if both numerator and denominator are even, divide by 2 first. If both end in 0 or 5, divide by 5 first. Using prime factorisation can also systematically find the HCF.

十二、分数的乘法 | Multiplying Fractions

分数的乘法可能是分数运算中最简单的一种：分子乘分子，分母乘分母。不需要找公分母。例如，⅔×⅗=(2×3)/(3×5)=6/15=⅖（化简后）。计算步骤：先相乘，再化简。如果在相乘之前先进行交叉约分（cross-cancelling），可以避免处理大数字。例如，8/15×5/12：注意到8和12都可以被4整除，5和15都可以被5整除。交叉约分：(8÷4)/(15÷5)×(5÷5)/(12÷4)=2/3×1/3=2/9。

Multiplying fractions is perhaps the simplest of all fraction operations: multiply the numerators together, multiply the denominators together. No common denominator is needed. For example, ⅔×⅗=(2×3)/(3×5)=6/15=⅖ (after simplifying). Steps: first multiply, then simplify. If you use cross-cancelling before multiplying, you can avoid dealing with large numbers. For example, 8/15×5/12: notice that 8 and 12 can both be divided by 4, and 5 and 15 can both be divided by 5. Cross-cancel: (8÷4)/(15÷5)×(5÷5)/(12÷4)=2/3×1/3=2/9.

对于带分数的乘法，先将带分数转换为假分数，再按同样方法相乘。例如，1½×2⅔=3/2×8/3=(3×8)/(2×3)=24/6=4。注意整数也可以看作分母为1的分数（如5=5/1），因此5×⅔=5/1×⅔=10/3=3⅓。Year 7考试中常见的分数乘法应用题包括”求一个分数的几分之几”，这种情况下将两个分数直接相乘即可。

For multiplying mixed numbers, first convert the mixed numbers to improper fractions, then multiply using the same method. For example, 1½×2⅔=3/2×8/3=(3×8)/(2×3)=24/6=4. Note that whole numbers can be viewed as fractions with a denominator of 1 (e.g., 5=5/1), so 5×⅔=5/1×⅔=10/3=3⅓. Common fraction multiplication word problems in Year 7 exams include “finding a fraction of a fraction”, in which case you simply multiply the two fractions directly.

十三、分数的除法 | Dividing Fractions

分数除法的核心技巧是”除一个数等于乘以它的倒数”（Keep-Change-Flip法则）。具体步骤：保持第一个分数不变（Keep），将除号改为乘号（Change），将第二个分数分子分母颠倒得到它的倒数（Flip），然后按分数乘法计算。例如，¾÷⅖=¾×5/2=(3×5)/(4×2)=15/8=1⅞。

The core technique for dividing fractions is “dividing by a number is the same as multiplying by its reciprocal” – the Keep-Change-Flip rule. Steps: Keep the first fraction unchanged, Change the division sign to multiplication, Flip the second fraction (swap its numerator and denominator to get its reciprocal), then multiply as you would for fraction multiplication. For example, ¾÷⅖=¾×5/2=(3×5)/(4×2)=15/8=1⅞.

为什么这个法则成立？从概念上理解：除以½等于乘以2，因为½的倒数是2（一个整体里有2个½）。除以⅓等于乘以3，因为⅓的倒数是3。同样，除以⅖等于乘以5/2。这个逻辑可以扩展到所有分数除法。对于整数与分数的除法，将整数视为分母为1的分数：6÷⅔=6/1×3/2=18/2=9。反过来，分数除以整数：⅗÷4=⅗×¼=3/20。

Why does this rule work? Conceptually: dividing by ½ is the same as multiplying by 2, because the reciprocal of ½ is 2 (there are 2 halves in a whole). Dividing by ⅓ is multiplying by 3, because the reciprocal of ⅓ is 3. Similarly, dividing by ⅖ is multiplying by 5/2. This logic extends to all fraction divisions. For division involving a whole number and a fraction, treat the whole number as a fraction with denominator 1: 6÷⅔=6/1×3/2=18/2=9. Conversely, a fraction divided by a whole number: ⅗÷4=⅗×¼=3/20.

Summary | 总结

分数、小数和百分比（FDP）是KS3 Year 7数学的核心主题。它们是同一概念 – “部分与整体的关系” – 的三种不同表达方式。掌握FDP之间的相互转换是后续所有数学学习的基础，包括比例（ratio）、代数方程、概率和统计。Year 7学生应重点掌握分数与小数互除转换法、小数与百分比小数点移动法、分数与百分比等值分数法，以及理解有限小数与循环小数的区别。通过记忆常见等价值、练习混合排序、掌握”除以分母乘分子”和”10%法”等实用技巧，学生可以在考试和实际生活中灵活运用这些知识。数学学习的关键在于理解概念的本质，而非死记硬背公式。

Fractions, decimals and percentages (FDP) are a core topic of KS3 Year 7 Mathematics. They are three different ways of expressing the same concept – the relationship between a part and a whole. Mastering the interconversion between FDP is the foundation for all subsequent mathematical learning, including ratio, algebraic equations, probability and statistics. Year 7 students should focus on mastering the division method for fraction-decimal conversion, the decimal point movement method for decimal-percentage conversion, the equivalent fraction method for fraction-percentage conversion, as well as understanding the difference between terminating and recurring decimals. By memorising common equivalents, practising mixed ordering, and mastering practical techniques such as “divide by denominator, multiply by numerator” and the “10% method”, students can apply this knowledge flexibly in exams and real-life situations. The key to learning mathematics lies in understanding the essence of concepts, rather than rote memorisation of formulas.

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