📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比
Fractions, decimals and percentages are three different ways of representing parts of a whole. In the Cambridge KS3 mathematics curriculum, you are expected to move confidently between these three forms and use them to solve problems in real-life contexts such as discounts, interest and measurement.
分数、小数和百分比是表示整体部分的三种不同方式。在剑桥 KS3 数学课程中，你需要能够在三种形式之间灵活转换，并运用它们解决折扣、利息和测量等实际情境中的问题。
1. Understanding Equivalent Fractions | 理解等值分数
An equivalent fraction is a fraction written with different numbers but representing the same value. For example, 1/2, 2/4 and 5/10 all show the same amount. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.
等值分数是指用不同数字书写但表示相同数值的分数。例如 1/2、2/4 和 5/10 都表示相同的数量。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。
1/2 = (1 × 2)/(2 × 2) = 2/4 = (2 ÷ 2)/(4 ÷ 2) = 1/2
A fraction is in its simplest form when the numerator and denominator have no common factor other than 1. For example, 8/12 simplifies to 2/3 because both 8 and 12 can be divided by 4.
当分子和分母除 1 外没有其他公因数时，分数就是最简形式。例如 8/12 化简为 2/3，因为 8 和 12 都能被 4 整除。
2. Converting Fractions to Decimals | 分数与小数的转换
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 means 3 ÷ 4, which gives 0.75. This method works for both proper and improper fractions.
要将分数转换为小数，用分子除以分母。例如 3/4 就是 3 ÷ 4，得到 0.75。这个方法对真分数和假分数都适用。
3/4 = 3 ÷ 4 = 0.75
To convert a terminating decimal to a fraction, count the decimal places and write the decimal as a fraction over 10, 100 or 1000. Then simplify the fraction if possible.
要将有限小数转换为分数，先数小数点后的位数，把小数写成分母为 10、100 或 1000 的分数，然后再化简。
0.6 = 6/10 = 3/5
Repeating decimals such as 0.333… need a different method, but at KS3 level you will mostly meet terminating decimals.
循环小数如 0.333… 需要另用方法，但在 KS3 阶段你大多会遇到有限小数。
3. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比之间的转换
Percent means ‘out of 100’. To convert a fraction to a percentage, first convert it to a decimal and then multiply by 100. For example, 2/5 = 0.4 = 40%. To convert a percentage to a fraction, write it over 100 and simplify.
百分比表示“以 100 为基数”。要将分数转换为百分比，先转换为小数，再乘以 100。例如 2/5 = 0.4 = 40%。要将百分比转换为分数，先写成分母为 100 的分数，再化简。
2/5 = 0.4 = 40%
25% = 25/100 = 1/4
The table below lists some common conversions that are useful to memorise for quick mental work.
下表列出了一些常见转换，熟记后有助于快速心算。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
Knowing these common conversions by heart speeds up your work in KS3 tests and in later topics such as probability and ratio.
熟记这些常见转换可以加快你在 KS3 考试以及后续概率和比例等主题中的解题速度。
4. Ordering and Comparing Fractions | 分数的大小比较与排序
To compare fractions, rewrite them with a common denominator or convert them to decimals. The common denominator is usually the lowest common multiple of the denominators. For example, compare 2/3 and 3/5: the LCM of 3 and 5 is 15, so 2/3 = 10/15 and 3/5 = 9/15, therefore 2/3 > 3/5.
比较分数时，可以先把分数化为同分母，或转换为小数。公分母通常是各分母的最小公倍数。例如比较 2/3 和 3/5：3 和 5 的最小公倍数是 15，所以 2/3 = 10/15，3/5 = 9/15，因此 2/3 > 3/5。
2/3 = 10/15 and 3/5 = 9/15, so 2/3 > 3/5
Another method is to use equivalent decimals: 2/3 ≈ 0.667 and 3/5 = 0.6, so 2/3 is larger. This method is especially useful when you are allowed a calculator.
另一种方法是使用等值小数：2/3 ≈ 0.667，3/5 = 0.6，所以 2/3 较大。这种方法在允许使用计算器时特别有用。
When ordering more than two fractions, convert them all to decimals or give them all the same denominator, then arrange them from smallest to largest or largest to smallest as required.
当需要排序两个以上的分数时，可以将它们全部转换为小数，或全部化为同分母，然后按照要求从小到大或从大到小排列。
5. Adding and Subtracting Fractions | 分数的加减法
Adding and subtracting fractions require a common denominator. If the denominators are different, find the lowest common denominator and rewrite each fraction before adding or subtracting the numerators. Never add or subtract the denominators.
分数的加减法要求分母相同。如果分母不同，先求最小公分母，把每个分数化成同分母，再对分子进行加减。千万不要对分母进行加减。
2/3 + 1/4 = 8
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