📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数
In KS3 Cambridge Mathematics, you must be able to move confidently between fractions, decimals and percentages. These three forms describe the same idea: a part of a whole. This revision guide covers the key methods, conversions and problem-solving skills you need.
在 KS3 剑桥数学中，你必须能够自信地在分数、小数和百分数之间进行转换。这三种形式描述的是同一个概念：整体的一部分。本复习指南涵盖你所需的关键方法、转换技巧和解题能力。
1. Fractions: Parts of a Whole | 分数：整体的部分
A fraction has a numerator on top and a denominator on the bottom. The denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many parts you have.
分数的上方是分子，下方是分母。分母表示整体被平均分成的份数，分子表示你取了多少份。
For example, 3/4 means the whole is split into 4 equal parts, and you take 3 of them. If a cake is cut into 4 slices and you eat 3, you have eaten 3/4 of the cake.
例如，3/4 表示整体被分成 4 等份，你取了其中的 3 份。如果一个蛋糕被切成 4 块，你吃了 3 块，那么你就吃了这个蛋糕的 3/4。
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 or dividing both the numerator and denominator by the same non-zero number.
等值分数虽然看起来不同，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，得到等值分数。
simplify a fraction, divide the numerator and denominator by their highest common factor, or HCF.
例如，1/2 = 2/4 = 3/6 = 12/24。化简分数时，用分子和分母的最大公因数（HCF）同时除以它们。
12/24 simplifies to 1/2 because the HCF of 12 and 24 is 12, and 12 ÷ 12 = 1, 24 ÷ 12 = 2.
因此 12/24 化简为 1/2，因为 12 和 24 的最大公因数是 12，12 ÷ 12 = 1，24 ÷ 12 = 2。
3. Mixed Numbers and Improper Fractions | 带分数与假分数
A mixed number contains a whole number and a proper fraction, such as 2 1/3. An improper fraction has a numerator larger than its denominator, such as 7/3.
带分数包含一个整数和一个真分数，例如 2 1/3。假分数的分子大于分母，例如 7/3。
over the original denominator. For 2 1/3: 2 × 3 + 1 = 7, so the improper fraction is 7/3.
将带分数化为假分数时，用整数乘以分母，再加上分子，把结果放在原分母上方。对于 2 1/3：2 × 3 + 1 = 7，因此假分数是 7/3。
o a mixed number, divide 7 by 3: the quotient is 2 and the remainder is 1, so 7/3 = 2 1/3.
将 7/3 化回带分数时，用 7 除以 3：商是 2，余数是 1，因此 7/3 = 2 1/3。
4. Converting Fractions to Decimals | 分数转小数
the numerator by the denominator. For a simple fraction like 3/4, calculate 3 ÷ 4 = 0.75.
要将分数转换为小数，用分子除以分母。对于简单的分数如 3/4，计算 3 ÷ 4 = 0.75。
you should recognise common decimal equivalents like 1/2 = 0.5, 1/4 = 0.25 and 1/5 = 0.2.
有些分数会产生循环小数，例如 1/3 = 0.333...，可以在重复的数字上方加点表示。在 KS3 中，你应当熟记常见的等值小数，如 1/2 = 0.5、1/4 = 0.25 和 1/5 = 0.2。
5. Converting Decimals to Percentages | 小数转百分数
Percent means 'out of 100'. To change a decimal to a percentage, multiply the decimal by 100 and add the % sign.
百分数表示“每一百份中的多少”。要将小数化为百分数，把小数乘以 100，再加上百分号。
For example, 0.45 × 100 = 45%, and 0.06 × 100 = 6%. To change a percentage back to a decimal, divide by 100: 72% = 0.72.
例如，0.45 × 100 = 45%，0.06 × 100 = 6%。要将百分数化回小数，除以 100：72% = 0.72。
| Decimal | 小数 | Percentage | 百分数 |
|---|---|
| 0.1 | 10% |
| 0.25 | 25% |
| 0.5 | 50% |
| 0.75 | 75% |
| 1.0 | 100% |
6. Converting Percentages to Fractions | 百分数转分数
percentage as a fraction with denominator 100, then simplify if possible. For example, 45% = 45/100 = 9/20 after dividing both terms by 5.
把百分数写成分母为 100 的分数，然后在可能的情况下化简。例如，45% = 45/100，将分子分母同时除以 5 后得到 9/20。
For a percentage like 16 2/3% proper fraction: 16 2/3 = 50/3. Then divide by 100, which gives 50/3 ÷ 100 = 50/300 = 1/6.
对于像 16 2/3% 这样的百分数，先把带分数写成假分数：16 2/3 = 50/3。再除以 100，得到 50/3 ÷ 100 = 50/300 = 1/6。
7. Ordering and Comparing | 排序与比较
When comparing fractions, decimals and percentages, change every value into the same form. It is usually easiest to convert all values to decimals or to percentages.
在比较分数、小数和百分数时，要把每个值都化为相同的形式。通常最容易的做法是把所有值都转换为小数或百分数。
For example, to order 0.4, 3/5 and 35% from smallest to largest, convert: 0.4 = 40%, 3/5 = 0.6 = 60%, and 35%. The correct order is 35%, 0.4, 3/5.
例如，要将 0.4、3/5 和 35% 从小到大排序，先转换：0.4 = 40%，3/5 = 0.6 = 60%，35% 保持不变。正确顺序是 35%、0.4、3/5。
8. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, they must have the same denominator. Find the lowest common multiple of the denominators, rewrite each fraction, then add or subtract the numerators.
分数相加或相减时，它们必须有相同的分母。找出分母的最小公倍数，重写每个分数，然后对分子进行加减。
4 and 6 is 12. Rewrite: 1/4 = 3/12 and 1/6 = 2/12. Add the numerators: 3/12 + 2/12 = 5/12.
对于 1/4 + 1/6，4 和 6 的最小公倍数是 12。重写为：1/4 = 3/12，1/6 = 2/12。分子相加：3/12 + 2/12 = 5/12。
For mixed numbers, add or subtract the whole numbers first, then deal with the fraction parts. You may need to borrow from the whole number when subtracting.
对于带分数，先加减整数部分，再处理分数部分。做减法时，有时需要从整数部分借位。
9. Multiplying and Dividing Fractions | 分数乘除法
ultiply the denominators together. For example, 2/3 × 3/5 = 6/15, which simplifies to 2/5.
分数相乘时，分子乘分子，分母乘分母。例如，2/3 × 3/5 = 6/15，化简后为 2/5。
To divide fractions, keep the first fraction, change the division sign to multiplication, and flip the second fraction. This is often called 'keep-change-flip'.
分数相除时，保留第一个分数，把除号变为乘号，再将第二个分数取倒数。这通常称为“保留-变号-取倒数”。
For 3/4 ÷ 2/5, write 3/4 × 5/2 = 15/8, which equals 1 7/8 as a mixed number.
对于 3/4 ÷ 2/5，写作 3/4 × 5/2 = 15/8，化为带分数等于 1 7/8。
10. Finding Percentages of Amounts | 求一个数的百分比
convert the percentage to a decimal or a fraction, then multiply. For example, to find 15% of 60, calculate 0.15 × 60 = 9.
求一个数的百分之几时，先把百分数化成小数或分数，再相乘。例如，求 60 的 15%，计算 0.15 × 60 = 9。
For non-calculator work, use a useful method: 10% = divide by 10, 5% = half of 10%, and 1% = divide by 100. Combine these to build other percentages.
在非计算器题目中，可以使用简单方法：10% 等于除以 10，5% 等于 10% 的一半，1% 等于除以 100。把这些组合起来就能算出其他百分比。
To find 17% of 200 without a calculator: 10% = 20, 5% = 10, 1% = 2, so 17% = 20 + 10 + 2 + 2 = 34.
不借助计算器求 200 的 17%：10% = 20，5% = 10，1% = 2，因此 17% = 20 + 10 + 2 + 2 = 34。
11. Ratio and Proportion Links | 比与比例的联系
ls, the fraction of boys is 12/20 = 3/5, and the ratio of boys to girls is 12 : 8 = 3 : 2.
分数、百分数和比都是比较数量的方法。例如，一个班级有 12 名男生和 8 名女生，男生的占比是 12/20 = 3/5，男生与女生的比是 12 : 8 = 3 : 2。
To share an amount in a given ratio, find the total number of parts, divide the amount by that total, then multiply by each part of the ratio.
按给定比例分配一个数量时，先求出总份数，用总量除以总份数，再分别乘以比例中的每一份。
12. Common Errors and Exam Tips | 常见错误与考试技巧
A common mistake is adding the denominators when adding fractions. Remember: only add the numerators after the denominators are the same. Never add denominators.
一个常见错误是在分数相加时把分母也相加。请记住：只有在分母相同之后才能把分子相加，绝不能把分母相加。
Another common error is writing 0.5% as 0.5 when converting. Half of one percent is 0.005 as a decimal, not 0.5. Always divide the percentage by 100.
另一个常见错误是在转换时把 0.5% 写成 0.5。百分之零点五写成小数是 0.005，而不是 0.5。转换时始终要把百分数除以 100。
In exam questions, underline whether the answer should be a fraction, decimal or percentage. Show all working clearly, especially when simplifying fractions or converting mixed numbers.
在考试题中，要划出答案应以分数、小数还是百分数表示。清晰写出所有解题步骤，尤其是在化简分数或转换带分数时。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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