📚 Fractions, Decimals and Percentages | 分数、小数和百分数
In KS3 Cambridge Mathematics, the topics of fractions, decimals and percentages form a connected block of number skills. They are used to describe parts of a whole, compare quantities, and solve real-life problems involving money, measurement, and data. Mastering these ideas early builds strong foundations for ratio, proportion, algebra, and probability later in the course.
在 KS3 剑桥数学中，分数、小数和百分数构成一个相互关联的数字技能板块。它们用于描述整体的一部分、比较数量，并解决涉及金钱、测量和数据的实际问题。尽早掌握这些概念能为后续的比、比例、代数和概率学习打下坚实基础。
1. Understanding Fractions | 理解分数
A fraction represents a part of a whole or a part of a set. It is written as a/b, where a is the numerator and b is the denominator. The denominator shows how many equal parts the whole is divided into, while the numerator shows how many of those parts are taken.
分数表示整体或集合的一部分。它写作 a/b，其中 a 是分子，b 是分母。分母表示整体被等分成多少份，分子表示取了多少份。
For example, 3/4 means the whole is cut into 4 equal parts, and 3 of them are shaded or selected. Proper fractions have a numerator smaller than the denominator, improper fractions have a numerator greater than or equal to the denominator, and mixed numbers combine a whole number with a proper fraction.
例如，3/4 表示整体被切成 4 等份，其中 3 份被涂色或选中。真分数的分子小于分母，假分数的分子大于或等于分母，混合数则由一个整数和一个真分数组成。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions look different but have the same value. You can create an equivalent fraction by multiplying or dividing both the numerator and the denominator by the same non-zero whole number. For instance, 1/2 = 2/4 = 3/6 = 50/100.
等值分数看起来不同，但数值相同。将分子和分母同时乘以或除以同一个非零整数，就能得到等值分数。例如，1/2 = 2/4 = 3/6 = 50/100。
Simplifying a fraction means writing it in its lowest terms. Divide the numerator and the denominator by their highest common factor, or HCF. For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6, and 18 ÷ 6 = 3 while 24 ÷ 6 = 4.
化简分数就是把它写成最简形式。用分子和分母的最大公因数（HCF）去除它们。例如，18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6，18 ÷ 6 = 3，24 ÷ 6 = 4。
3. Converting Between Mixed Numbers and Improper Fractions | 混合数与假分数的转换
A mixed number such as 2 3/5 can be converted into an improper fraction. Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. So 2 3/5 = (2 × 5 + 3)/5 = 13/5.
像 2 3/5 这样的混合数可以转换为假分数。将整数乘以分母，加上分子，再把结果放在原分母上方。因此 2 3/5 = (2 × 5 + 3)/5 = 13/5。
To convert an improper fraction back 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. For example, 17/4 = 4 1/4 because 17 ÷ 4 = 4 remainder 1.
将假分数转换回混合数时，用分子除以分母。商是整数部分，余数是新的分子，分母保持不变。例如，17/4 = 4 1/4，因为 17 ÷ 4 = 4 余 1。
4. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, they must have the same denominator. If the denominators are different, find a common denominator first, preferably the lowest common multiple, or LCM. Convert each fraction to an equivalent fraction with that denominator, then add or subtract the numerators.
分数相加或相减时，必须具有相同的分母。如果分母不同，首先要找到一个公分母，最好是分母的最小公倍数（LCM）。把每个分数转换成以该公分母为分母的等值分数，然后将分子相加或相减。
For example, 1/4 + 1/6 has denominators 4 and 6. The LCM of 4 and 6 is 12. So 1/4 = 3/12 and 1/6 = 2/12. Therefore 1/4 + 1/6 = 3/12 + 2/12 = 5/12. Always simplify the final answer if possible.
例如，1/4 + 1/6 的分母是 4 和 6。4 和 6 的最小公倍数是 12。所以 1/4 = 3/12，1/6 = 2/12。因此 1/4 + 1/6 = 3/12 + 2/12 = 5/12。如果可能，最后要化简答案。
5. Multiplying and Dividing Fractions | 分数乘除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. Simplify before or after multiplying if you can. For example, 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15.
分数乘法很简单：分子相乘，分母相乘。可以在乘之前或之后化简。例如，2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15。
Dividing by a fraction means multiplying by its reciprocal. The reciprocal of a/b is b/a, provided a is not zero. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which can be written as 1 7/8. Always flip the second fraction, not the first.
除以一个分数等于乘以它的倒数。a/b 的倒数是 b/a，其中 a 不为零。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8，可以写作 1 7/8。一定要翻转第二个分数，而不是第一个。
6. Decimals and Place Value | 小数与位值
A decimal is another way of writing a fraction whose denominator is a power of ten. Place value extends to the right of the decimal point: tenths, hundredths, thousandths, and so on. For example, 0.37 means 3 tenths and 7 hundredths, or 37/100.
小数是分母为 10 的幂的分数的另一种写法。位值延伸到小数点右边：十分位、百分位、千分位等。例如，0.37 表示 3 个十分之一和 7 个百分之一，即 37/100。
Multiplying a decimal by 10, 100, or 1000 moves the decimal point to the right by 1, 2, or 3 places. Dividing by 10, 100, or 1000 moves the decimal point to the left. For instance, 0.48 × 100 = 48 and 6.2 ÷ 10 = 0.62.
小数乘以 10、100 或 1000 时，小数点分别向右移动 1 位、2 位或 3 位。除以 10、100 或 1000 时，小数点向左移动。例如，0.48 × 100 = 48，6.2 ÷ 10 = 0.62。
7. Converting Fractions to Decimals and Back | 分数与小数的互化
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or long division. For example, 3/8 = 0.375 because 3 ÷ 8 = 0.375. Some fractions convert to repeating decimals, such as 1/3 = 0.333…
将分数转换为小数时，用分子除以分母。可以使用短除法或长除法。例如，3/8 = 0.375，因为 3 ÷ 8 = 0.375。有些分数会转换成循环小数，如 1/3 = 0.333…
To convert a terminating decimal to a fraction, write the decimal as a fraction over a power of ten according to the last digit’s place value, then simplify. For example, 0.65 = 65/100 = 13/20. For repeating decimals, use algebraic methods if needed, but at KS3 you usually work with terminating decimals.
将有限小数转换为分数时，根据最后一位数字的位值把小数写成分母为 10 的幂的分数，然后化简。例如，0.65 = 65/100 = 13/20。循环小数在 KS3 阶段很少涉及，通常只需处理有限小数。
8. Understanding Percentages | 理解百分数
A percentage is a fraction with a denominator of 100. The symbol % means ‘per hundred’ or ‘out of 100’. So 45% means 45 out of 100, or 45/100, or 0.45. Percentages are useful for comparing changes, discounts, interest, and scores.
百分数是一个分母为 100 的分数。符号 % 表示“每一百”或“每 100 份”。所以 45% 表示 100 份中的 45 份，即 45/100 或 0.45。百分数常用于比较变化、折扣、利率和得分。
To work out a percentage, you can think of it as a fraction of a quantity. For example, 20% of 60 means 20/100 × 60 = 12. This connects percentages directly to fraction multiplication.
计算百分数时，可以把它看作数量的几分之几。例如，60 的 20% 就是 20/100 × 60 = 12。这把百分数与分数乘法直接联系起来。
9. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数的转换
There are quick methods for converting among the three forms. To convert a fraction to a percentage, first convert to a decimal, then multiply by 100 and add the % sign. For example, 3/5 = 0.6 = 60%. To convert a percentage to a fraction, write it over 100 and simplify. So 25% = 25/100 = 1/4.
三种形式之间有一些快速转换方法。将分数转换为百分数时，先转换为小数，再乘以 100 并加上 % 符号。例如，3/5 = 0.6 = 60%。将百分数转换为分数时，把它写成分母为 100 的分数并化简。所以 25% = 25/100 = 1/4。
To convert a decimal to a percentage, multiply by 100 and add the % sign. To convert a percentage to a decimal, divide by 100. For example, 0.07 = 7% and 135% = 1.35. You can use a table to remember these relationships:
将小数转换为百分数时，乘以 100 并加上 % 符号。将百分数转换为小数时，除以 100。例如，0.07 = 7%，135% = 1.35。你可以用表格来记住这些关系：
| Fraction | Decimal | Percentage |
| 1/4 | 0.25 | 25% |
| 1/2 | 0.5 | 50% |
| 3/4 | 0.75 | Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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