📚 IGCSE Number Skills: Fractions, Decimals and Percentages | IGCSE 数字技能：分数、小数与百分数
Welcome to this IGCSE Mathematics revision guide for topic T-1-1061. Fractions, decimals and percentages are three connected ways of representing parts of a whole. Being able to move confidently between these forms is essential for many IGCSE questions, from basic calculations to real-life financial problems.
欢迎学习针对 T-1-1061 的 IGCSE 数学复习指南。分数、小数和百分数是表示整体中部分数量的三种相互关联的方式。从基础计算到现实生活中的金融问题，能否在这些形式之间熟练转换对于许多 IGCSE 题目都至关重要。
This guide covers the definitions you must know, the conversions that appear in almost every paper, the four operations with fractions and decimals, and the percentage applications that carry the most marks. Every section ends with the kind of wording an examiner is likely to use, so you can practise recognising what the question is really asking.
本指南涵盖你必须掌握的定义、几乎每份试卷都会出现的互化、分数与小数的四则运算，以及占分最多的百分数应用。每一节都会点出考官可能使用的表述方式，帮助你练习识别题目真正在问什么。
1. Fractions: The Basic Idea | 分数：基本概念
A fraction is written as one number over another, for example 3/4. The denominator, the bottom number, tells you how many equal parts the whole has been split into, and the numerator, the top number, tells you how many of those parts you are looking at.
分数写作一个数置于另一个数之上的形式，例如 3/4。分母（下面的数）表示整体被平均分成了多少份，分子（上面的数）表示你关注的是其中多少份。
, so 3/4 means 3 ÷ 4 = 0.75. This single idea links fractions to decimals and explains why you can always convert a fraction into a decimal by division.
在 IGCSE 题目中，你应当认识到分数也表示除法。分数线意味着分子除以分母，所以 3/4 就是 3 ÷ 4 = 0.75。这一观念把分数与小数联系起来，也解释了为什么你总能通过除法把分数化成小数。
and then multiply by the numerator. For example, 3/4 of 240 = 240 ÷ 4 × 3 = 60 × 3 = 180.
求某个量的几分之几时，先除以分母，再乘以分子。例如 3/4 of 240 = 240 ÷ 4 × 3 = 60 × 3 = 180。
a/b of N = N ÷ b × a
2. Types of Fractions: Proper, Improper and Mixed | 分数的类型：真分数、假分数与带分数
Proper fractions have a numerator smaller than the denominator, such as 2/5, so their value is less than 1. Improper fractions have a numerator greater than or equal to the denominator, such as 9/4 or 7/7, so their value is 1 or more. A mixed number combines a whole number with a proper fraction, such as 2 1/3.
真分数的分子小于分母，例如 2/5，其值小于 1。假分数的分子大于或等于分母，例如 9/4 或 7/7，其值为 1 或更大。带分数由一个整数和一个真分数组成，例如 2 1/3。
write the remainder over the denominator. For 9/4: 9 ÷ 4 = 2 remainder 1, so 9/4 = 2 1/4.
把假分数化成带分数时，用分子除以分母，再把余数写成分母之上的分数。对于 9/4：9 ÷ 4 = 2 余 1，所以 9/4 = 2 1/4。
minator, add the numerator, and keep the same denominator. For 2 1/3: (2 × 3 + 1)/3 = 7/3.
把带分数化成假分数时，用整数乘以分母，加上分子，分母保持不变。对于 2 1/3：(2 × 3 + 1)/3 = 7/3。
a b/c = (a × c + b) / c
In calculations involving multiplication or division, always convert mixed numbers into improper fractions first. Adding and subtracting can sometimes be done with the whole parts and fraction parts separately, but the improper-fraction method is safer and less likely to lose marks.
在涉及乘法或除法的计算中，务必先把带分数化为假分数。加减法有时可以把整数部分和分数部分分开处理，但化为假分数的方法更稳妥，也更不容易丢分。
3. Equivalent Fractions and Simplifying | 等值分数与化简
nominator by the same non-zero number, or by dividing both by the same non-zero number. For example, 1/2 = 2/4 = 5/10 = 50/100.
等值分数表示整体中相同的比例。你可以把分子和分母同时乘以同一个非零数，或同时除以同一个非零数，得到等值分数。例如 1/2 = 2/4 = 5/10 = 50/100。
by their highest common factor. For 24/36, the highest common factor is 12, so 24/36 = 2/3.
化简就是把分数写成最简形式，即分子与分母除 1 以外没有公因数。把分子和分母同时除以它们的最大公因数。对于 24/36，最大公因数是 12，所以 24/36 = 2/3。
you need to check whether a statement is true, or to find a missing numerator or denominator.
若交叉相乘的积相等，两个分数就相等：当且仅当 a × d = b × c 时，a/b = c/d。当你需要判断某个等式是否成立，或求缺失的分子、分母时，这个检验非常有用。
An IGCSE answer that is numerically correct but not in lowest terms may lose a mark in a "give your answer in its simplest form" question. Get into the habit of checking for common factors at the end of every fraction calculation.
在 IGCSE 中，如果答案数值正确但不是最简形式，遇到"以最简形式给出答案"的题目可能会被扣分。养成在每次分数计算结束后检查公因数的习惯。
4. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
Percent means "per hundred", so a percentage is simply a fraction with a denominator of 100. 37% on gives you both conversions: divide by 100 to go from a percentage to a decimal, and multiply by 100 to go from a decimal to a percentage.
百分数意为"每一百"，所以百分数其实就是分母为 100 的分数。37% = 37/100 = 0.37。这一个定义就给出了两个方向的互化：由百分数化小数要除以 100，由小数化百分数要乘以 100。
0.36 = 36/100 = 9/25, and 0.125 = 125/1000 = 1/8. To convert a fraction to a percentage, convert to a decimal first and then multiply by 100.
分数化小数，用分子除以分母。小数化分数，看最后一位数字的数位：0.36 = 36/100 = 9/25，0.125 = 125/1000 = 1/8。分数化百分数，先化成小数，再乘以 100。
Some conversions are worth memorising because they appear repeatedly in non-calculator papers.
有些互化值得背下来，因为它们在不允许使用计算器的试卷中反复出现。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | 33.3…% |
| 2/3 | 0.666… | 66.6…% |
| 1/10 | 0.1 | 10% |
ecimals are recurring, which means a digit or block of digits repeats forever. 1/3 = 0.333… and 5/11 = 0.454545… Examiners usually accept a rounded answer such as 0.33 or 0.455 unless the question says "exact value".
有些小数是循环小数，即某一位或某一组数字永远重复。1/3 = 0.333…，5/11 = 0.454545… 除非题目要求"精确值"，考官通常接受 0.33 或 0.455 这样的近似答案。
5. Ordering and Comparing Values | 大小比较与排序
To compare fractions, decimals and percentages, convert everything into the same form. Decimals are usually the easiest, because you can then compare place value directly. Rewrite 0.62, 3/5, 65% and 0.605 as decimals: 0.62, 0.6, 0.65 and 0.605.
要比较分数、小数和百分数，先把它们全部化成同一种形式。通常小数最方便，因为可以直接比较数位。把 0.62、3/5、65% 和 0.605 都化成小数：0.62、0.6、0.65 和 0.605。
Among these, 0.6 is the smallest, followed by 0.605, then 0.62, and 0.65 is the largest. So the order is 3/5, 0.605, 0.62, 65%.
其中 0.6 最小，接着是 0.605，然后是 0.62，最大的是 0.65。所以顺序是 3/5、0.605、0.62、65%。
Since 60% ommon trap is a percentage that looks large because of the digits but is actually small: 9% is much less than 0.9, because 9% = 0.09.
因为 60% = 0.6，所以比较百分数与比较它们除以 100 后得到的小数是同一回事。常见的陷阱是某个百分数看起来数字很大但实际很小：9% 比 0.9 小得多，因为 9% = 0.09。
6. Adding and Subtracting Fractions | 分数的加法和减法
n the denominators are different, first rewrite both fractions with a common denominator, ideally the lowest common multiple of the denominators.
分母相同时，分子相加或相减，分母保持不变：3/7 + 2/7 = 5/7。分母不同时，先把两个分数改写为同分母分数，最好使用分母的最小公倍数。
ommon multiple of 3 and 4 is 12, so 2/3 = 8/12 and 1/4 = 3/12, giving 8/12 + 3/12 = 11/12.
对于 2/3 + 1/4，3 和 4 的最小公倍数是 12，所以 2/3 = 8/12，1/4 = 3/12，得到 8/12 + 3/12 = 11/12。
2 1/2 + 1 1/3, convert both to improper fractions: 5/2 + 4/3 = 15/6 + 8/6 = 23/6 = 3 5/6.
减法用同样的方法：3/4 − 2/5 = 15/20 − 8/20 = 7/20。在 2 1/2 + 1 1/3 这样的带分数题目中，把两者都化为假分数：5/2 + 4/3 = 15/6 + 8/6 = 23/6 = 3 5/6。
Always simplify the final answer, and write mixed numbers with a clear gap between the whole number and the fraction so a marker cannot misread 3 5/6 as 35/6.
最后一定要化简答案，并在带分数的整数与分数之间留出清晰的间隔，以免阅卷老师把 3 5/6 误读为 35/6。
7. Multiplying and Dividing Fractions | 分数的乘法和除法
ors before multiplying, which keeps the numbers small: 4/9 × 3/8 = (4 × 3)/(9 × 8) = 12/72 = 1/6, or by cancelling, 4/9 × 3/8 = 1/6 directly.
分数相乘，分子乘分子、分母乘分母：2/3 × 4/5 = 8/15。更高效的做法是相乘前先约去公因数，这样数字更小：4/9 × 3/8 = (4 × 3)/(9 × 8) = 12/72 = 1/6，或者直接约分得到 4/9 × 3/8 = 1/6。
y its reciprocal, that is, turn the second fraction upside down. So a/b ÷ c/d = a/b × d/c.
除以一个分数，就是乘以它的倒数，也就是把第二个分数上下颠倒。因此 a/b ÷ c/d = a/b × d/c。
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
t dividing by a fraction smaller than 1 always makes the answer larger, which is a useful check on your working.
整数可以写成以 1 为分母的分数，所以 6 ÷ 3/4 = 6/1 × 4/3 = 24/3 = 8。注意，除以一个小于 1 的分数总会让结果变大，这是检验计算是否正确的好方法。
8. Working with Decimals | 小数的运算
oth numbers have the same number of decimal places: 4.7 + 0.38 becomes 4.70 + 0.38 = 5.08.
小数的加减法要对齐小数点，保持各数位对齐。用 0 补足缺少的数位，使两个小数位数相同：4.7 + 0.38 写成 4.70 + 0.38 = 5.08。
of decimal places in the question. For 0.4 × 0.03, multiply 4 × 3 = 12, and since there are one plus two equals three decimal places in total, the answer is 0.012.
小数相乘时，先忽略小数点，按整数相乘，再补回与题目中小数位数总和相同的小数位。对于 0.4 × 0.03，先算 4 × 3 = 12，因为一共有 1 + 2 = 3 位小数，所以答案是 0.012。
oth numbers by the same power of 10. For 4.8 ÷ 0.6, multiply both by 10 to get 48 ÷ 6 = 8.
小数除法中，把两个数同乘相同的 10 的幂，使除数变成整数。对于 4.8 ÷ 0.6，两者同乘 10 得 48 ÷ 6 = 8。
Rounding is also tested. To round to two decimal
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