📚 Mastering Fractions, Decimals and Percentages for Cambridge KS3 Maths | 掌握剑桥KS3数学中的分数、小数与百分数
In the Cambridge KS3 Mathematics curriculum, fractions, decimals and percentages are not isolated topics. They form a connected number system that appears in almost every problem-solving context, from sharing quantities to calculating discounts. A single value can be written in three different ways, and the skill examiners reward most is the ability to move fluently between those three representations and then choose the most efficient one for the job. This article brings together the key skills you need: equivalent fractions, conversions in both directions, ordering and comparing, the four operations, percentage of amounts, percentage change, and the common exam-style questions that test all of it.
在剑桥KS3数学课程中，分数、小数和百分数并不是孤立的主题。它们构成了一个相互关联的数字体系，几乎出现在每一个问题解决情境中——从分配数量到计算折扣。同一个数值可以用三种不同的方式书写，而考官最看重的技能，正是你能否在这三种表示形式之间自如转换，并挑选出最高效的那一种来解决问题。本文将汇总你需要掌握的关键技能：等值分数、双向转换、大小比较与排序、四则运算、百分数的计算、百分数增减，以及考查以上全部内容的常见考试题型。
1. One Value, Three Forms | 一个数值，三种形式
The foundation of this whole topic is the idea that a fraction, a decimal and a percentage are simply three different labels for the same point on the number line. The value 1/2, the value 0.5 and the value 50% are not three numbers - they are one number written three ways. Once you accept this, every conversion question becomes a question about changing the label, not changing the value.
整个主题的基础在于这样一个观念：分数、小数和百分数，只是数轴上同一个点的三种不同标签。1/2、0.5 和 50% 并不是三个数，而是同一个数的三种写法。一旦你接受了这一点，所有转换题都会变成"换标签"而不是"换数值"的问题。
Cambridge KS3 places this topic in the Number strand, and it is deliberately revisited every year because it feeds into so many other areas: ratio and proportion, probability, statistics, algebra with algebraic fractions, and later IGCSE work on standard form and compound interest. A weak grasp here creates problems in topics that look completely unrelated.
剑桥KS3把这一主题放在"数"的板块中，并且刻意每年复现，因为它支撑着太多其他领域：比与比例、概率、统计、含代数分式的代数，以及之后IGCSE中的标准形式和复利。这里基础不牢，就会在看似毫无关联的主题上出问题。
- Fractions tell you how many equal parts of a whole you have. | 分数告诉你，一个整体中你拥有多少等份。
- Decimals use place value in base 10, which makes them ideal for written calculation and for calculators. | 小数使用十进制位值，因此非常适合笔算和计算器运算。
- Percentages are fractions with a denominator of 100, which makes them ideal for comparison and for money problems. | 百分数就是分母为100的分数，因此非常适合比较和金钱类问题。
2. Understanding Equivalent Fractions | 理解等值分数
Equivalent fractions have the same value even though they look different. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero whole number. For example, 1/2, 2/4, 3/6 and 12/24 are all equal because each one represents the same shaded part of a whole.
等值分数虽然看起来不同，但数值相同。你可以将分子和分母同时乘以或除以同一个非零整数，从而得到一个等值分数。例如 1/2、2/4、3/6 和 12/24 都相等，因为它们表示同一个整体中被涂色的相同部分。
This works because multiplying the top and the bottom by the same number is really multiplying the whole fraction by a special form of 1, such as 2/2 or 5/5. Multiplying by 1 never changes a value, so the fraction keeps its size while its appearance changes.
这样做之所以成立，是因为分子分母同乘一个数，本质上是把整个分数乘上了一个特殊形式的 1，比如 2/2 或 5/5。乘以 1 不改变数值，所以分数的大小不变，只是外观变了。
| Fraction | × or ÷ | Equivalent |
| 1/2 | × 6 / × 6 | 6/12 |
| 3/4 | × 5 / × 5 | 15/20 |
| 18/24 | ÷ 6 / ÷ 6 | 3/4 |
nt only when a × d = b × c. For 2/4 and 12/24, 2 × 24 = 48 and 4 × 12 = 48, so they match.
一个很实用的检查习惯是交叉相乘：两个分数 a/b 与 c/d 等值，当且仅当 a × d = b × c。对 2/4 和 12/24 来说，2 × 24 = 48，4 × 12 = 48，两者相等。
3. Simplifying to Lowest Terms | 约分至最简形式
e numerator and the denominator by their highest common factor (HCF). For 18/24 the HCF is 6, so 18/24 = 3/4. For 45/60 the HCF is 15, so 45/60 = 3/4 as well.
约分就是用尽可能小的分子和分母来书写一个分数，同时保持数值不变。约分时，把分子和分母同时除以它们的最大公因数（HCF）。18/24 的最大公因数是 6，所以 18/24 = 3/4。45/60 的最大公因数是 15，所以 45/60 也等于 3/4。
A fraction is in lowest terms when the only common factor of the numerator and denominator is 1. You can also simplify in stages - divide by 2, then by 3, then by 5 - and you will still arrive at the same answer, though it takes longer and gives more chances to make an arithmetic slip.
当分子和分母的公因数只有 1 时，这个分数就处于最简形式。你也可以分步约分——先除以 2，再除以 3，再除以 5——最终结果同样正确，只是更耗时，也更容易出现计算失误。
e left unsimplified, for example writing 6/8 instead of 3/4. The same rule applies to mixed numbers: 4/3 should be written as 1 1/3, and 7/4 + 1/4 = 8/4 should be written as 2.
除非题目另有说明，最终答案一律要写成最简形式。剑桥的评分标准常常会对"数值正确但未约分"的答案扣分，例如把 3/4 写成 6/8。带分数也同理：4/3 应写成 1 1/3，而 7/4 + 1/4 = 8/4 应写成 2。
4. Converting Fractions to Decimals | 将分数转换为小数
e denominator, adding decimal places and zeros as needed. 3/8 = 0.375, 7/20 = 0.35, and 5/16 = 0.3125.
每个分数都是一次除法。分数 3/8 表示 3 ÷ 8。所以把分数化成小数，其实就是做一次短除法：用分子除以分母，必要时补上小数位和 0。3/8 = 0.375，7/20 = 0.35，5/16 = 0.3125。
it or block of digits repeats forever. The most familiar example is 1/3 = 0.333... , written with a dot or bar over the repeating digit in exam notation. 2/3 = 0.666... , 1/6 = 0.1666... and 1/7 = 0.142857142857... are all recurring.
有些分数得到有限小数，即在小数点后某一位停止；另一些则得到循环小数，某个数字或数字块会无限重复。最常见的例子是 1/3 = 0.333…，在考试书写中用点或横线标出循环的数字。2/3 = 0.666…，1/6 = 0.1666…，1/7 = 0.142857142857… 都是循环小数。
A fast test: a fraction in lowest terms gives a terminating decimal only when its denominator has no prime factors other than 2 and 5. That is why eighths, sixteenths, twentieths and twenty-fifths terminate, while thirds, sevenths and ninths recur.
一个快速的判断方法：处于最简形式的分数，只有当分母除了 2 和 5 以外没有其他质因数时，才会得到有限小数。这就是为什么分母为 8、16、20、25 的分数是有限小数，而分母为 3、7、9 的是循环小数。
You should memorise these common conversions, because they save time in every paper:
你应该记住下面这些常见转换，它们能在每张试卷上为你省下时间：
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 1/3 | 0.333... | 33.3%... |
5. Converting Decimals to Fractions | 将小数转换为分数
into a fraction, read the decimal using place value and then simplify. The decimal 0.75 is "seventy-five hundredths" ich simplifies to 3/4. The decimal 0.6 is 6/10 = 3/5. The decimal 0.125 is 125/1000 = 1/8.
要把有限小数化为分数，先按位值把这个小数读出来，然后再约分。小数 0.75 读作"百分之七十五"，所以是 75/100，约分后为 3/4。小数 0.6 是 6/10 = 3/5。小数 0.125 是 125/1000 = 1/8。
- 1 decimal place → denominator 10 | 1 位小数 → 分母为 10
- 2 decimal places → denominator 100 | 2 位小数 → 分母为 100
- 3 decimal places → denominator 1000 | 3 位小数 → 分母为 1000
the first equation from the second gives 9x = 4, so x = 4/9. The same method gives 0.777... = 7/9 and 0.363636... = 36/99 = 4/11.
对于循环小数，有一个让KS3学生很感兴趣的巧妙代数技巧。要把 0.444… 转换成分数，设它为 x。那么 10x = 4.444…，用第二个式子减去第一个式子得 9x = 4，所以 x = 4/9。同样的方法可得 0.777… = 7/9，0.363636… = 36/99 = 4/11。
+ 35/100 = 2 7/20. Keep the whole-number part separate and convert only the decimal part.
带整数部分的小数同理：2.35 = 2 + 35/100 = 2 7/20。把整数部分单独留下，只转换小数部分即可。
6. Percentages as Fractions and Decimals | 百分数与分数、小数
The word "per cent" means "out of one hundred", so 35% eans 35 out of 100. That gives two instant conversions: divide by 100 to get a decimal (35% = 0.35), and write over 100 to get a fraction (35% = 35/100 = 7/20).
"percent" 这个词的意思是"每一百"，所以 35% 字面意思就是 100 份中的 35 份。由此可得到两个即时转换：除以 100 得到小数（35% = 0.35），写成以 100 为分母的分数得到分数形式（35% = 35/100 = 7/20）。
ther way is just as easy. To change a decimal into a percentage, multiply by 100 (0.08 = 8% 00, or scale the fraction so the denominator becomes 100. For instance, 7/20 = 35/100 = 35%.
反方向同样简单。把小数化为百分数，乘以 100（0.08 = 8%）。把分数化为百分数，可以先化成小数再乘以 100，也可以把分数扩大到分母为 100。例如 7/20 = 35/100 = 35%。
fraction → ÷ numerator by denominator → decimal → × 100 → percentage
分数 → 分子除以分母 → 小数 → × 100 → 百分数
Percentages greater than 100 and percentages less than 1 both appear in KS3 questions. 150% = 1.5 = 3/2, and 0.5% = 0.005 = 1/200. Do not assume a percentage must be a whole number between 0 and 100.
大于 100% 和小于 1% 的百分数都会在KS3题目中出现。150% = 1.5 = 3/2，0.5% = 0.005 = 1/200。不要以为百分数一定是 0 到 100 之间的整数。
7. Ordering and Comparing Across the Three Forms | 跨三种形式的大小比较与排序
When a question asks you to arrange values such as 0.6, 3/5, 62% lue column by column is quick and reliable. In this example the values become 0.6, 0.6, 0.62 and 0.58, giving the order 0.58, 0.6 = 3/5, 62%.
当题目要求你把 0.6、3/5、62% 和 0.58 这样的数值排序时，最稳妥的方法是把它们全部转换成同一种形式。小数通常是最快的选择，因为逐位比较小数位既迅速又可靠。在这个例子里，各值变为 0.6、0.6、0.62 和 0.58，排序结果为 0.58、0.6 = 3/5、62%。
Alternatively, you can convert everything to fractions with a common denominator. For 2/3 and 3/5, the lowest common denominator is 15, giving 10/15 and 9/15, so 2/3 is larger. This method is often better when the values are awkward decimals such as 0.666... , because it keeps the answer exact.
另一种方法，是把所有数化为同分母的分数。比较 2/3 和 3/5 时，最小公分母是 15，得到 10/15 和 9/15，所以 2/3 更大。当数值是 0.666… 这类难以书写的小数时，这个方法更好，因为它保持答案精确。
- Convert to decimals when the numbers are simple or a calculator is allowed. | 数值简单或允许使用计算器时，转换为小数。
- Convert to fractions with a common denominator when you need an exact order or an exact answer. | 需要精确排序或精确答案时，转换为同分母分数。
- Sketching the values on a number line is a quick sanity check when two values look very close. | 当两个数值看起来非常接近时，把它们标在数轴上是快速的检验方法。
8. Adding and Subtracting Fractions | 分数的加减运算
minators - 3/8 + 2/8 is not 5/16. This is the single most common error in the whole topic.
分母相同时，只需把分子相加或相减，分母保持不变：3/8 + 2/8 = 5/8。千万不要把分母相加——3/8 + 2/8 不等于 5/16。这是整个主题中最常见的错误。
- then add or subtract. For 2/3 + 1/4, the LCM of 3 and 4 is 12, so 2/3 = 8/12 and 1/4 = 3/12, giving 8/12 + 3/12 = 11/12. For 5/6 - 1/4, the LCM is 12, giving 10/12 - 3/12 = 7/12.
分母不同时，先把分数改写为同分母（最好是两分母的最小公倍数），再相加或相减。计算 2/3 + 1/4 时，3 和 4 的最小公倍数是 12，所以 2/3 = 8/12，1/4 = 3/12，得 8/12 + 3/12 = 11/12。计算 5/6 - 1/4 时，最小公倍数是 12，得 10/12 - 3/12 = 7/12。
numbers and fractions separately, borrowing from the whole number when the fraction part is too small to subtract.
带分数需要多一步。可以先把它们化为假分数——1 1/2 + 2 3/4 变成 3/2 + 11/4 = 6/4 + 11/4 = 17/4 = 4 1/4；也可以把整数部分和分数部分分开计算，当分数部分不够减时，从整数部分借 1。
a/b ± c/d = (a × d ± c × b) ÷ (b × d)
9. Multiplying and Dividing Fractions | 分数的乘除运算
5 = 8/15. There is no need for a common denominator. You can make the work lighter by cancelling common factors before multiplying (this is sometimes called cross-cancelling).
分数乘法是四种运算中最简单的：分子乘分子，分母乘分母。所以 2/3 × 4/5 = 8/15，完全不需要通分。如果在相乘之前先约掉公因数（有时称为交叉约分），计算会更轻松。
/2 = 15/8 = 1 7/8. A helpful way to remember why this works is to ask how many 2/5 fit into 3/4, or to think of "dividing by a half is the same as multiplying by two".
除法要使用倒数。除以一个分数，等于乘以它的倒数——也就是把第二个分数上下颠倒。所以 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。理解其原理的一个好办法是问：3/4 里面有几个 2/5？或者想"除以 1/2 等于乘以 2"。
4 = 12/12 = 1. A common trap is cancelling the 3 and the 4 across the multiplication sign in a way that ignores which number is the numerator.
带分数在乘除之前必须先化为假分数。计算 1 1/3 × 3/4 时，先把 1 1/3 写成 4/3，再算 4/3 × 3/4 = 12/12 = 1。一个常见的陷阱是：约分时弄错了哪个数是分子、哪个数是分母。
Watch out for "of". In a question such as "find 3/4 of 60", the word "of" sing this link connects fraction arithmetic directly to percentages of amounts in Section 11.
要留意 "of" 这个词。在"求 60 的 3/4"这类题目里，"of" 意味着相乘：3/4 × 60 = 45。认识到这一联系，就把分数运算与第 11 节的"求一个量的百分之几"直接贯通起来。
10. Decimal Arithmetic and Place Value | 小数运算与位值
akes the alignment obvious: 3.60 + 1.45 = 5.05. A missing zero is the most common cause of a wrong answer.
小数加减法的关键是把小数点对齐，让相同数位处于同一列。把 3.6 写成 3.60 就能看清对齐方式：3.60 + 1.45 = 5.05。少写一个 0 往往是答案出错的根本原因。
ginal numbers and put that many decimal places back into the answer. For 0.4 × 0.3, the digits give 12 and there are two decimal places in total, so the answer is 0.12 - not 1.2 and certainly not 12.
小数乘法，先忽略小数点，按整数相乘，然后数出两个原数小数位数的总和，在答案中补回同样的小数位数。计算 0.4 × 0.3 时，数字部分得 12，小数位共两位，所以答案是 0.12，而不是 1.2，更不是 12。
30. This uses the equivalence idea from Section 2 - multiplying both numbers by the same amount preserves the ratio.
小数除法，把被除数和除数同时扩大相同的 10 的幂，直到除数变成整数。计算 4.5 ÷ 0.15 时，两者同乘 100，得到 450 ÷ 15 = 30。这里用到的是第 2 节的等值思想——两数同乘一个数，比值不变。
Always estimate first. For 0.4 × 0.3, thinking "0.4 is nearly a half and a half of 0.3 is 0.15" at 12 is wildly wrong. Estimation costs three seconds and catches most place-value errors.
一定要先估算。计算 0.4 ×
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