📚 Fractions, Decimals and Percentages | 分数、小数和百分数
In Cambridge KS3 Mathematics, fractions, decimals and percentages (FDP) form a core number skill. They are three different ways of describing the same idea: a part of a whole. Mastering conversions between them is essential for algebra, ratio, probability and real-life problem solving.
在剑桥初中数学中，分数、小数和百分数（FDP）构成核心数字技能。它们是描述同一概念——整体的一部分——的三种不同方式。掌握它们之间的转换对代数、比、概率和现实问题解决至关重要。
1. The Big Picture | 概览
A fraction such as 3/4, a decimal such as 0.75 and a percentage such as 75% all describe the same quantity. In KS3 you need to move fluently between these forms because different problems are easier in different forms.
分数如 3/4、小数如 0.75 和百分数如 75% 都描述同一个数量。在 KS3 阶段，你需要在这三种形式之间灵活转换，因为不同的问题在不同形式下更容易处理。
The key conversion rules are: fraction to decimal means dividing the numerator by the denominator; decimal to percentage means multiplying by 100; and percentage to fraction means writing the percentage over 100 and simplifying.
关键转换规则是：分数转小数就是用分子除以分母；小数转百分数就是乘以 100；百分数转分数就是把百分数写在 100 上并化简。
2. Equivalent Fractions | 等值分数
Equivalent fractions represent 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 number.
等值分数虽然看起来不同，但表示相同的值。你可以通过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÷d)/(b÷d)
If you divide both 12 and 16 by 4, you simplify 12/16 to 3/4, which is the simplest form.
字母 a、b、n 和 d 代表整数，且 b、n、d 不能为零。例如，1/2 = 2/4 = 3/6 = 4/8。如果把 12 和 16 都除以 4，就把 12/16 化简为 3/4，也就是最简形式。
Always check whether a fraction can be simplified by finding the highest common factor (HCF) of the numerator and denominator. Simplifying makes later calculations much easier.
始终通过寻找分子和分母的最大公因数（HCF）来检查分数是否可以化简。化简会让后面的计算容易很多。
3. Converting Fractions to Decimals | 分数转小数
y the denominator. The fraction bar is exactly a division sign, so 3/4 means 3 ÷ 4 = 0.75.
要把分数转换为小数，用分子除以分母。分数线就是除号，所以 3/4 表示 3 ÷ 4 = 0.75。
= 0.6. Others give recurring decimals, such as 1/3 = 0.333... and 2/7 = 0.285714285714...
有些分数得到有限小数，如 1/2 = 0.5、1/4 = 0.25 和 3/5 = 0.6。另一些分数得到循环小数，如 1/3 = 0.333... 和 2/7 = 0.285714285714...
In the Cambridge Checkpoint test you may be asked to write a recurring decimal using dot notation, or to round a decimal to a given number of decimal places. Always check the question instruction.
在剑桥 Checkpoint 测试中，你可能会被要求使用点符号表示循环小数，或将小数四舍五入到指定的小数位数。一定要看清题目要求。
4. Converting Decimals to Percentages | 小数转百分数
Percentage means 'out of 100', so converting a decimal to a percentage is simple: multiply the decimal by 100 and add the percent symbol.
百分数表示“每一百份中的多少”，所以把小数转换为百分数很简单：将小数乘以 100 并加上百分号。
0.45 = 0.45 × 100% = 45%
To convert a percentage back to a decimal, divide by 100. For example, 65% = 65 ÷ 100 = 0.65. This is the same as moving the decimal point two places to the left.
要把百分数转回小数，除以 100。例如，65% = 65 ÷ 100 = 0.65。这相当于把小数点向左移动两位。
g with percentages greater than 100, the decimal will be greater than 1. For instance, 125% = 1.25. This is useful for percentage increase problems.
当百分数大于 100 时，小数会大于 1。例如，125% = 1.25。这在百分数增加问题中很有用。
5. Ordering FDP | 分数、小数和百分数排序
To compare or order fractions, decimals and percentages, first change all the values into the same form. Converting to decimals is often the fastest method, especially if a calculator is allowed.
要比较或排序分数、小数和百分数，首先把所有数值转换为同一种形式。转换为小数通常是最快的方法，尤其是在允许使用计算器时。
For example, order 2/5, 0.35 and 40% from smallest to largest. Write them all as decimals: 2/5 = 0.4, 0.35 = 0.35 and 40% = 0.40. The correct order is 0.35, 2/5, 40%.
例如，将 2/5、0.35 和 40% 从小到大排列。把它们都写成小数：2/5 = 0.4、0.35 = 0.35、40% = 0.40。正确顺序是 0.35、2/5、40%。
If the question asks you to use fractions, find a common denominator. For 1/3, 1/4 and 2/5, a common denominator is 60, giving 20/60, 15/60 and 24/60, so the order is 1/4, 1/3, 2/5.
如果题目要求使用分数，就找一个共同分母。对于 1/3、1/4 和 2/5，共同分母是 60，得到 20/60、15/60 和 24/60，所以顺序是 1/4、1/3、2/5。
6. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, the denominators must be the same. If they are not, find a common denominator first. Then add or subtract the numerators and keep the denominator unchanged.
要加减分数，分母必须相同。如果分母不同，先找共同分母。然后加或减分子，分母保持不变。
a/b + c/b = (a + c)/b
ommon multiple of 3 and 4 is 12, so 1/3 = 4/12 and 1/4 = 3/12. Therefore 1/3 + 1/4 = 7/12.
例如，1/3 + 1/4 不能直接相加。3 和 4 的最小公倍数是 12，所以 1/3 = 4/12，1/4 = 3/12。因此 1/3 + 1/4 = 7/12。
roper fractions first. For example, 2 1/3 + 1 1/2 = 7/3 + 3/2 = 14/6 + 9/6 = 23/6 = 3 5/6.
对于带分数，先将其转换为假分数。例如，2 1/3 + 1 1/2 = 7/3 + 3/2 = 14/6 + 9/6 = 23/6 = 3 5/6。
7. Multiplying Fractions | 分数乘法
Multiplying fractions is more direct than adding them: multiply the numerators together and multiply the denominators together. You do not need a common denominator.
分数乘法比分数加法更直接：分子相乘，分母相乘。你不需要共同分母。
a/b × c/d = (a×c)/(b×d)
y a whole number, write the whole number over 1 first: 6 × 3/4 = 6/1 × 3/4 = 18/4 = 4 1/2.
例如，2/3 × 4/5 = 8/15。如果是分数乘以整数，先把整数写成分母为 1 的分数：6 × 3/4 = 6/1 × 3/4 = 18/4 = 4 1/2。
r 3/8 × 4/9, divide 4 by 8 to get 1/2 and divide 3 by 9 to get 1/3, so the product is 1/6.
尽可能在乘法前约分，让数字保持较小。例如 3/8 × 4/9，把 4 和 8 约成 1 和 2，把 3 和 9 约成 1 和 3，所以结果是 1/6。
8. Dividing Fractions | 分数除法
To divide fractions, multiply by the reciprocal of the second fraction. The reciprocal of a/b is b/a. Change the division sign to multiplication and flip the second fraction.
要除以分数，乘以第二个分数的倒数。a/b 的倒数是 b/a。把除号改为乘号并翻转第二个分数。
a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)
hole number by a fraction, write the whole number over 1: 4 ÷ 2/5 = 4/1 × 5/2 = 20/2 = 10.
例如，3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10。整数除以分数时，先把整数写成分母为 1 的分数：4 ÷ 2/5 = 4/1 × 5/2 = 20/2 = 10。
Students often forget to flip the second fraction or accidentally flip the first one. The rule is: keep the first fraction, change the sign, flip the second fraction.
学生经常忘记翻转第二个分数，或不小心翻转第一个分数。规则是：保留第一个分数，改变符号，翻转第二个分数。
9. Percentage of a Quantity | 求一个数的百分数
To find a percentage of a quantity, first convert the percentage to a decimal or a fraction, then multiply by the quantity.
要求一个数的百分之几，先把百分数转换为小数或分数，再乘以这个数。
part = (percentage/100) × total
For example, 15% of 80 = 15/100 × 80 = 0.15 × 80 = 12. Without a calculator, you can find 10% first and then 5%: 10% of 80 is 8, 5% is 4, so 15% is 12.
例如，80 的 15% = 15/100 × 80 = 0.15 × 80 = 12。不用计算器时，可以先求 10% 再求 5%：80 的 10% 是 8，5% 是 4，所以 15% 是 12。
This method is very common in word problems involving discounts, taxes, savings and test scores. Always identify the original total before calculating.
这种方法在涉及折扣、税收、储蓄和考试分数的文字题中非常常见。计算前一定要先确定原始总量。
10. Percentage Increase and Decrease | 百分数增减
For percentage increase, add the percentage to 100%, convert to a decimal and multiply. For percentage decrease, subtract the percentage from 100%, convert to a decimal and multiply.
百分数增加时，把百分数加到 100%，转换为小数再相乘。百分数减少时，从 100% 中减去百分数，转换为小数再相乘。
new amount = original × (1 ± r/100)
Use plus for increase and minus for decrease. For example, increase 80 by 15%: new = 80 × 1.15 = 92. Decrease 80 by 15%: new = 80 × 0.85 = 68.
增加用加号，减少用减号。例如，将 80 增加 15%：新值 = 80 × 1.15 = 92。将 80 减少 15%：新值 = 80 × 0.85 = 68。
to find the percentage change, use the formula: percentage change = change/original × 100% f a price rises from 40 to 50, the change is 10 and the percentage increase is 10/40 × 100% = 25%.
如果需要求百分数变化，使用公式：百分数变化 = 变化量/原始值 × 100%。例如，价格从 40 涨到 50，变化量是 10，百分数增加为 10/40 × 100% = 25%。
11. Common Exam Mistakes and Tips | 常见错误与备考建议
1/3 = 2/5. The correct method is to find a common denominator and add only the numerators.
一个常见错误是在分数相加时把分母也相加，比如写成 1/2 + 1/3 = 2/5。正确方法是找共同分母，只把分子相加。
Another mistake is confusing percentage increase with percentage decrease. Always reread the question to decide whether you need to add or subtract the rate from 100%.
另一个错误是混淆百分数增加和百分数减少。一定要重新读题，判断需要从 100% 中加上还是减去该百分数。
Use these tips in the Checkpoint exam: write all values in the same form before comparing, simplify fractions at the end, and check your answer with a rough estimate. If 15% of 80 should be around 12, an answer of 120 is clearly wrong.
在 Checkpoint 考试中使用这些技巧：比较前把所有值写成同一种形式，最后化简分数，并用粗略估算检查答案。如果 80 的 15% 应该约为 12，那么答案 120 明显是错误的。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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