📚 Fractions, Decimals and Percentages | 分数、小数与百分数
This revision guide covers fractions, decimals and percentages for the Cambridge KS3 Mathematics syllabus. You will learn how to convert between the three forms, carry out calculations with each of them, compare and order values, and apply the ideas to exam-style questions. Work through each section in order, because the skills build on one another: you must be confident with fractions before percentages become easy, and you must be confident with decimals before recurring decimals make sense.
本复习指南涵盖剑桥 KS3 数学课程中的分数、小数与百分数。你将学习这三者之间的互化、进行相关计算、比较与排序数值，并将这些知识应用到考试题型中。请按顺序学习每一节，因为各项技能是层层递进的：只有先熟练掌握分数，百分数才会变得简单；只有先熟练掌握小数，循环小数才能理解透彻。
1. Understanding Fractions | 理解分数
A fraction shows a part of a whole. The numerator is the top number and the denominator is the bottom number. In the fraction 3/4, 3 is the numerator and 4 is the denominator, which tells you that the whole has been split into 4 equal parts and you are looking at 3 of them.
分数表示整体的一部分。分子是上面的数，分母是下面的数。在分数 3/4 中，3 是分子，4 是分母，它表示整体被平均分成 4 份，而你关注的是其中的 3 份。
A proper fraction has a numerator smaller than its denominator, for example 2/5. Its value is always less than 1, because you are taking fewer parts than the whole contains.
真分数的分子小于分母，例如 2/5。它的值总是小于 1，因为你取的份数少于整体所含的份数。
An improper fraction has a numerator equal to or larger than its denominator, for example 7/4 or 5/5. Its value is 1 or more, which is why some exam questions ask you to leave answers as improper fractions rather than mixed numbers.
假分数的分子大于或等于分母，例如 7/4 或 5/5。它的值等于 1 或大于 1，这也是有些考题要求答案写成假分数而非带分数的原因。
A mixed number combines a whole number with a proper fraction, for example 1 3/4. It means 1 whole plus 3 quarters, so its value lies between two consecutive whole numbers.
带分数由一个整数和一个真分数组成，例如 1 3/4。它表示 1 个整体再加 3 个四分之一，所以它的值介于两个相邻整数之间。
Every fraction is also a division. The line in 3/4 means 'divide' and it also explains how to find a fraction of an amount: 3/4 of 20 means 20 ÷ 4 × 3 = 15.
每个分数同时也是一种除法。3/4 中的分数线表示"除以"，所以 3/4 = 3 ÷ 4。这一简单的观念解释了如何把任意分数化成小数，也解释了如何求一个量的几分之几：20 的 3/4 就是 20 ÷ 4 × 3 = 15。
A unit fraction has a numerator of 1, for example 1/2, 1/5 or 1/9. Unit fractions are useful building blocks: 3/5 is just three lots of 1/5, and 7/8 is seven lots of 1/8.
单位分数的分子是 1，例如 1/2、1/5 或 1/9。单位分数是很有用的"积木"：3/5 就是 3 个 1/5，7/8 就是 7 个 1/8。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value even though they look different. You can make an equivalent fraction by multiplying the numerator and the denominator by the same non-zero number, or by dividing both of them by the same non-zero number.
等值分数尽管形式不同，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数，从而得到一个等值分数。
1/2 = 2/4 = 3/6 = 6/12 = 50/100
Multiplying by the same number both above and below the line does not change the value, because you are effectively multiplying the whole fraction by 1 (for example by 2/2 or 5/5).
在分数线上下同乘一个数不会改变数值，因为你实际上是把整个分数乘以 1（例如乘以 2/2 或 5/5）。
To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). The result is the fraction in its simplest form, also called lowest terms.
化简分数时，用分子和分母的最大公因数 (HCF) 同时除它们，所得结果就是最简形式，也叫最简分数。
12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3
You can also simplify in stages. Dividing 12/18 by 2 gives 6/9, and dividing again by 3 gives 2/3. The final answer is the same, but the HCF method is faster and safer in an exam.
你也可以分步化简。把 12/18 先除以 2 得 6/9，再除以 3 得 2/3。最终答案相同，但用最大公因数的方法在考试中更快也更稳妥。
er a common denominator of 40: 3/5 = 24/40 and 5/8 = 25/40, so 5/8 is the larger fraction.
化简不只是为了美观：它能让分数比较变得容易得多。要比较 3/5 和 5/8，把两者都写成公分母 40：3/5 = 24/40，5/8 = 25/40，所以 5/8 更大。
A common mistake is to divide only the numerator or only the denominator. If you divide the top but not the bottom, the value of the fraction changes completely, so always work on both parts at the same time.
一个常见错误是只除分子或只除分母。如果只除上面的数而不除下面的数，分数的值就完全变了，所以一定要同时对分子和分母进行操作。
3. Mixed Numbers and Improper Fractions | 带分数与假分数
To change a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, then place the result over the original denominator.
将带分数化为假分数：整数乘以分母，加上分子，然后将结果写在原分母上。
2 3/5 = (2 × 5 + 3)/5 = 13/5
To change an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.
将假分数化为带分数：用分子除以分母。商为整数部分，余数为新的分子，分母保持不变。
13/5 = 2 3/5 because 13 ÷ 5 = 2 remainder 3
Why bother converting at all? as 1 1/2 × 2 2/3, converting both to improper fractions first is the fastest route to the answer.
为什么要互化呢？带分数更容易阅读，也更容易与整数比较；但假分数在乘法和除法中要方便得多。在像 1 1/2 × 2 2/3 这样的题目中，先把两个数都化成假分数是得到答案最快的途径。
cause 4 × 2 + 1 = 9. In the other direction, 17/6 = 2 5/6, because 17 ÷ 6 = 2 remainder 5.
更多练习示例：4 1/2 = 9/2，因为 4 × 2 + 1 = 9。反方向，17/6 = 2 5/6，因为 17 ÷ 6 = 2 余 5。
r answer by estimating: 3 1/4 should be a little more than 3, and 13/4 = 3.25, which fits.
一个非常常见的错误是把 3 1/4 错写成 3/4，原因是用整数乘了分子而不是分母。用估算来检查答案：3 1/4 应该略大于 3，而 13/4 = 3.25，正好符合。
4. Adding and Subtracting Fractions | 分数的加法与减法
Before adding or subtracting fractions, make sure the denominators are the same. When the denominators match, simply add or subtract the numerators and keep the denominator unchanged.
在加减分数之前，先确认分母相同。当分母相同时，只需把分子相加或相减，分母保持不变。
2/7 + 3/7 = 5/7 5/9 − 2/9 = 3/9 = 1/3
When the denominators are different, find the lowest common multiple (LCM) of the denominators and rewrite each fraction as an equivalent fraction with that common denominator.
当分母不同时，先求出各分母的最小公倍数 (LCM)，再把每个分数改写成分母为该公倍数的等值分数。
1/2 + 1/3 = 3/6 + 2/6 = 5/6
3/4 − 1/6 = 9/12 − 2/12 = 7/12
With mixed numbers, deal with the whole parts and the fraction parts separately, or convert everything to improper fractions first. Both methods give the same answer.
处理带分数时，可以把整数部分和分数部分分别处理，也可以先把所有数化成假分数。两种方法得到的答案相同。
2 1/3 + 1 1/2 = 2 2/6 + 1 3/6 = 3 5/6
. For 3 1/4 − 1 1/2, convert to improper fractions: 13/4 − 3/2 = 13/4 − 6/4 = 7/4 = 1 3/4.
减法有时需要"借位"。对于 3 1/4 − 1 1/2，先化成假分数：13/4 − 3/2 = 13/4 − 6/4 = 7/4 = 1 3/4。
es the answer smaller than one of the fractions being added, which is impossible when both fractions are positive.
千万不要把分母相加。把 1/2 + 1/3 算成 2/5 是错的，因为这样得到的答案会比其中一个加数还小，而当两个分数都是正数时这是不可能的。
5. Multiplying and Dividing Fractions | 分数的乘法与除法
Multiplication is the easiest operation with fractions: multiply the numerators together and multiply the denominators together. Unlike addition, the denominators do not need to be the same.
乘法是分数运算中最简单的一种：分子乘分子，分母乘分母。与加法不同，这里不需要通分。
2/3 × 3/5 = (2 × 3)/(3 × 5) = 6/15 = 2/5
and the 3 and the 9 share a factor of 3, which reduces the calculation to 1/3 × 1/2 = 1/6.
你可以先约分再相乘，让计算更轻松。在 4/9 × 3/8 中，4 与 8 有公因数 4，3 与 9 有公因数 3，计算就简化为 1/3 × 1/2 = 1/6。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is found by swapping the numerator and the denominator.
除以一个分数，等于乘以它的倒数。分数的倒数就是把分子和分母互换位置。
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
The word "of" usually means multiply. So 2/3 of 45 means 2/3 × 45 = 30, which is the same as 45 ÷ 3 × 2.
"of"（的）通常表示乘法。所以 45 的 2/3 就是 2/3 × 45 = 30，也等同于 45 ÷ 3 × 2。
o improper fractions before multiplying or dividing: 1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4.
如果题目中含有带分数，先化成假分数再乘除：1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4。
ember: keep the first fraction as it is, change ÷ to ×, and flip only the second fraction.
最常见的错误是把不该倒的分数倒了，或者把两个分数都倒了。请记住：第一个分数保持不变，把"÷"改成"×"，只把第二个分数倒过来。
6. Understanding Decimals and Place Value | 理解小数与位值
Decimals are another way of writing fractions whose denominators are powers of ten. The first place after the decimal point is tenths, the second is hundredths and the third is thousandths.
小数是分母为 10 的幂的分数的一种写法。小数点后第一位是十分位，第二位是百分位，第三位是千分位。
0.7 = 7/10 0.25 = 25/100 = 1/4 0.125 = 125/1000 = 1/8
A terminating decimal stops after a certain number of digits, for example 0.375. A recurring decimal repeats a digit or a group of digits for ever, for example 0.333... or 0.454545... and these are usually written with a dot or a bar over the repeating part.
有限小数在若干位数字后结束，例如 0.375。循环小数会永远重复某一位数字或某一组数字，例如 0.333... 或 0.454545...，通常会在循环部分上加一个点或一条横线来表示。
When ordering decimals, give every number the same number of decimal places by adding zeros. To compare 0.4, 0.405 and 0.45, write them as 0.400, 0.405 and 0.450, so the order is 0.4, 0.405, 0.45.
给小数排序时，可以用补零的方法让每个数的小数位数相同。比较 0.4、0.405 和 0.45 时，写成 0.400、0.405 和 0.450，所以顺序是 0.4、0.405、0.45。
00 = 360. Dividing by 1000 moves every digit three places to the right: 45 ÷ 1000 = 0.045.
乘以或除以 10 的幂，只是把数字整体移动。乘以 100 就是把每个数字向左移两位：3.6 × 100 = 360。除以 1000 就是把每个数字向右移三位：45 ÷ 1000 = 0.045。
Decimals appear constantly in real contexts such as money and measurement: £3.05, 2.5 kg, 0.75 litres. Estimating first, for example 4
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