📚 Fractions, Decimals and Percentages | 分数、小数和百分数
Fractions, decimals and percentages are three different ways of representing parts of a whole. In KS3 Cambridge Mathematics, you must be able to convert between these forms, compare them, and use them confidently in calculations. This article explains the key concepts, common methods and typical mistakes to avoid.
分数、小数和百分数是表示“整体的一部分”的三种不同方式。在剑桥初中数学课程中，你需要能够在这些形式之间进行转换、比较大小，并熟练地运用它们进行计算。本文将讲解关键概念、常用方法以及需要避免的常见错误。
1. What Are Fractions? | 什么是分数？
A fraction is written as a numerator over a denominator, for example 3/4. The denominator shows how many equal parts the whole is divided into, and the numerator shows how many of those parts are taken.
分数由分子和分母组成，例如 3/4。分母表示整体被平均分成多少份，分子表示取了多少份。
A proper fraction has a numerator smaller than its denominator, such as 2/5. An improper fraction has a numerator greater than or equal to its denominator, such as 7/4. A mixed number combines a whole number with a proper fraction, such as 1 3/4.
真分数的分子小于分母，例如 2/5。假分数的分子大于或等于分母，例如 7/4。带分数由一个整数和一个真分数组成，例如 1 3/4。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions show the same value but use different numerators and denominators. You can create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number.
等值分数表示相同的数值，但使用不同的分子和分母。你可以将分子和分母同时乘以或除以同一个非零数来得到等值分数。
a/b = (a × n)/(b × n) = (a ÷ n)/(b ÷ n)
To simplify a fraction, divide both parts by their highest common factor, or HCF. For example, 12/18 becomes 2/3 because both 12 and 18 can be divided by 6.
化简分数时，将分子和分母同时除以它们的最大公因数，即 HCF。例如，12/18 化简为 2/3，因为 12 和 18 都可以除以 6。
3. Mixed Numbers and Improper Fractions | 带分数与假分数
A mixed number can be converted to an improper fraction by multiplying the whole number by the denominator, adding the numerator, and placing the result over the original denominator.
带分数可以转换为假分数：用整数乘以分母，再加上分子，把结果放在原分母上。
1 3/4 = (1 × 4 + 3)/4 = 7/4
rator over the same denominator. For example, 11/3 = 3 2/3 because 11 ÷ 3 = 3 remainder 2.
将假分数转换为带分数时，用分子除以分母。商为整数部分，余数作为新分子，分母不变。例如，11/3 = 3 2/3，因为 11 ÷ 3 = 3 余 2。
4. Converting Fractions to Decimals | 分数化小数
inator. You can use short division or long division. For example, 3/8 means 3 ÷ 8 = 0.375.
将分数转换为小数时，用分子除以分母。你可以使用短除法或长除法。例如，3/8 就是 3 ÷ 8 = 0.375。
as 1/3 = 0.333... which can be written as 0.3 with a dot over the 3, or as 0.333 repeated.
有些分数能化为有限小数，例如 1/4 = 0.25。另一些分数化为循环小数，例如 1/3 = 0.333...，可以写成数字 3 上方加点的形式。
If the denominator is 10, 100 or 1000, the conversion is easier because the fraction is already expressed in tenths, hundredths or thousandths.
如果分母是 10、100 或 1000，转换会更容易，因为分数已经以十分位、百分位或千分位表示。
5. Converting Decimals to Fractions | 小数化分数
To convert a terminating decimal to a fraction, write the decimal as a fraction with denominator 10, 100, 1000 and so on, depending on the number of decimal places. Then simplify if possible.
将有限小数转换为分数时，根据小数位数把小数写成分母为 10、100、1000 等的分数，然后化简。
ecimal places, so 0.65 = 65/100. Dividing both numerator and denominator by 5 gives 13/20.
例如，0.65 有两位小数，所以 0.65 = 65/100。分子分母同时除以 5，得到 13/20。
you should recognise common recurring decimals such as 0.333... = 1/3 and 0.666... = 2/3.
对于循环小数，转换过程更复杂。但是在初中阶段，你应该掌握常见的循环小数，例如 0.333... = 1/3，0.666... = 2/3。
6. Percentages and Their Fraction/Decimal Forms | 百分数及其分数、小数形式
A percentage is a number out of 100. The symbol % means "per hundred". For example, 35% means 35 out of 100, or 35/100.
百分数表示一个数占 100 的多少。符号 % 的意思是“每一百”。例如，35% 表示 100 份中的 35 份，即 35/100。
To convert a percentage to a fraction, write it over 100 and simplify. To convert a percentage to a decimal, divide by 100 by moving the decimal point two places to the left.
将百分数转换为分数时，先写成分母为 100 的分数，再化简。将百分数转换为小数时，除以 100，即把小数点向左移动两位。
| Percentage | Fraction | Decimal |
|---|---|---|
| 50% | 1/2 | 0.5 |
| 25% | 1/4 | 0.25 |
| 10% | 1/10 | 0.1 |
| 75% | 3/4 | 0.75 |
7. Ordering Fractions, Decimals and Percentages | 分数、小数和百分数的大小比较
To compare fractions, decimals and percentages, convert them all to the same form. Decimals are usually the easiest for ordering, so change any fractions or percentages to decimals first.
要比较分数、小数和百分数，先把它们转换成同一种形式。小数通常最容易比较大小，因此可以先将分数或百分数化为小数。
For example, to order 2/5, 0.45 and 38%, convert: 2/5 = 0.4, 0.45 = 0.45, and 38% = 0.38. In ascending order: 0.38, 0.4, 0.45, so 38%, 2/5, 0.45.
例如，要比较 2/5、0.45 和 38%，先转换：2/5 = 0.4，0.45 = 0.45，38% = 0.38。从小到大排列：0.38，0.4，0.45，即 38%，2/5，0.45。
You can also compare fractions by finding a common denominator. For example, compare 2/3 and 3/4 by writing them as 8/12 and 9/12, so 3/4 is larger.
你也可以通过找公分母来比较分数。例如，比较 2/3 和 3/4，可以写成 8/12 和 9/12，因此 3/4 更大。
8. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, they must have the same denominator. If they do not, find the lowest common denominator first.
分数的加减法必须先化为同分母。如果分母不同，首先要找到最小的公分母。
1/4 + 1/6: the lowest common multiple of 4 and 6 is 12, so rewrite as 3/12 + 2/12 = 5/12.
例如，1/4 + 1/6：4 和 6 的最小公倍数是 12，因此改写为 3/12 + 2/12 = 5/12。
For mixed numbers, add or subtract the whole number parts separately, then deal with the fraction parts. You may need to borrow from the whole number when subtracting.
对于带分数，可以先分别计算整数部分，再计算分数部分。在减法中，有时需要从整数部分借位。
1/2 + 1/3 = 3/6 + 2/6 = 5/6
9. Multiplying and Dividing Fractions | 分数的乘法与除法
To multiply fractions, multiply the numerators together and multiply the denominators together. Then simplify if possible.
分数相乘时，分子与分子相乘，分母与分母相乘，然后化简。
2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is found by swapping the numerator and denominator.
除以一个分数，等于乘以这个分数的倒数。分数的倒数就是把分子和分母交换位置。
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Always simplify your final answer, and leave it as an improper fraction or mixed number according to the question's instructions.
最后一定要化简答案，并根据题目要求保留假分数或带分数的形式。
10. Finding Percentages of Amounts | 求一个数的百分之几
f an amount, write the percentage as a fraction or decimal, then multiply. For example, 15% of 80 means 0.15 × 80 = 12.
求一个数的百分之几时，先把百分数写成分数或小数，再相乘。例如，80 的 15% 就是 0.15 × 80 = 12。
You can also use benchmark percentages to make mental calculations easier. For example, 10% is found by dividing by 10; 5% is half of 10%; 1% is dividing by 100.
你可以使用基准百分数进行心算。例如，10% 就是除以 10；5% 是 10% 的一半；1% 是除以 100。
For harder percentages, break them into parts. To find 17% of 60, think of 10% + 5% + 2%: 6 + 3 + 1.2 = 10.2.
对于较难的百分数，可以拆分成多个部分。求 60 的 17%，可以看成 10% + 5% + 2%：6 + 3 + 1.2 = 10.2。
11. Real-Life Applications and Common Mistakes | 实际应用与常见错误
Fractions, decimals and percentages appear in shopping discounts, test scores, recipes and data analysis. For example, a 20% discount on a £45 item saves £9, so the new price is £36.
分数、小数和百分数出现在购物折扣、考试成绩、食谱和数据分析中。例如，一件 45 英镑的商品打八折，节省 9 英镑，新价格为 36 英镑。
A common mistake is adding denominators when adding fractions. Remember that 1/4 + 1/4 is 2/4, not 2/8. The denominator stays the same when fractions already have a common denominator.
一个常见错误是在分数相加时把分母也相加。请记住，1/4 + 1/4 = 2/4，而不是 2/8。当分数已经有相同分母时，分母保持不变。
Another mistake is confusing 0.5, 5% and 1/2. They are not all equal: 0.5 = 50%, but 5% = 0.05 and 1/2 = 50%. Always check the form before comparing.
另一个常见错误是混淆 0.5、5% 和 1/2。它们并不相等：0.5 = 50%，而 5% = 0.05，1/2 = 50%。比较之前一定要确认形式。
When converting a decimal to a percentage, multiply by 100 rather than divide. For example, 0.6 becomes 60%, not 0.006%.
将小数转换为百分数时，应乘以 100，而不是除以。例如，0.6 应转换为 60%，而不是 0.006%。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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