📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分数
Fractions, decimals and percentages are three connected ways of describing parts of a whole. In KS3 Mathematics, you need to move flexibly between these forms, compare quantities and apply them to real-life problems such as discounts, interest and probability.
分数、小数和百分数是描述整体的部分的三种相互关联的方式。在 KS3 数学中，你需要灵活地在这些形式之间转换、比较数量，并将它们应用于折扣、利息和概率等实际问题。
1. Equivalent Fractions and Simplest Form | 等值分数与最简形式
A fraction is made up of a numerator and a denominator. Equivalent fractions have the same value even though they may look different. For example, 1/2, 2/4 and 5/10 all represent the same part of a whole.
分数由分子和分母组成。等值分数虽然看起来不同，但数值相同。例如，1/2、2/4 和 5/10 都表示整体中相同的部分。
To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). This process is sometimes called cancelling down. A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.
化简分数时，将分子和分母同时除以它们的最大公因数 (HCF)。这个过程有时称为约分。当分子和分母除了 1 之外没有其他公因数时，分数就是最简形式。
12/18 = 2/3
Here, both 12 and 18 are divided by 6, so 12/18 simplifies to 2/3. Always check whether a final fraction can be simplified further before writing your answer.
这里，12 和 18 都除以 6，所以 12/18 化简为 2/3。在写出答案前，务必检查最终分数是否可以进一步化简。
2. Converting Fractions to Decimals | 分数转换为小数
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator. Some fractions produce a terminating decimal, while others produce a recurring decimal.
要将分数转换为小数，用分子除以分母。你可以使用短除法或计算器。有些分数得到有限小数，有些则得到循环小数。
3/8 = 0.375
Because 3 divided by 8 is exactly 0.375, this is a terminating decimal. A fraction such as 1/3 gives 0.333… which is a recurring decimal because the 3 repeats forever.
因为 3 除以 8 正好是 0.375，这是一个有限小数。像 1/3 这样的分数得到 0.333…，因为 3 无限循环，所以是循环小数。
Knowing common fraction-decimal equivalences is very helpful. For example, 1/4 = 0.25, 1/2 = 0.5, 3/4 = 0.75 and 1/10 = 0.1. These should be memorised for quick calculations.
熟悉常见的分数-小数等值关系很有帮助。例如，1/4 = 0.25, 1/2 = 0.5, 3/4 = 0.75, 1/10 = 0.1。这些应该记住以便快速计算。
3. Converting Decimals to Fractions | 小数转换为分数
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 the fraction if possible.
要将有限小数转换为分数，根据小数的位数，把它写成分母为 10、100、1000 等的分数。如果可能，再化简分数。
0.25 = 25/100 = 1/4
The decimal 0.25 has two decimal places, so it is written as 25/100. Dividing the numerator and denominator by 25 gives the simplest form 1/4.
小数 0.25 有两位小数，所以写成 25/100。分子和分母同时除以 25，得到最简形式 1/4。
For a mixed decimal such as 1.6, write it as a mixed number first: 1.6 = 1 + 0.6 = 1 + 6/10 = 1 3/5. This is particularly useful when working with measurements.
对于像 1.6 这样的带小数，可以先写成带分数：1.6 = 1 + 0.6 = 1 + 6/10 = 1 3/5。这在处理度量单位时特别有用。
4. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数之间的转换
Percent means ‘out of 100’. To convert a fraction to a percentage, first convert the fraction to a decimal and then multiply by 100. Alternatively, find an equivalent fraction with denominator 100.
百分数表示 ‘每一百份’。将分数转换为百分数，可以先将分数转换为小数，再乘以 100。或者，找到分母为 100 的等值分数。
3/5 = 0.6 = 60%
To convert a percentage to a fraction, write the percentage over 100 and simplify. For example, 45% = 45/100 = 9/20. To convert a decimal to a percentage, multiply by 100.
将百分数转换为分数，把百分数写在 100 上并化简。例如，45% = 45/100 = 9/20。将小数转换为百分数，乘以 100。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333… | 33.3% |
| 3/5 | 0.6 | 60% |
| 7/8 | 0.875 | 87.5% |
These conversions are essential for comparing quantities in different forms, especially in word problems and data handling.
这些转换对于比较不同形式的数量至关重要，尤其是在文字题和数据处理中。
5. Comparing and Ordering Fractions | 比较和排序分数
To compare fractions, you can write them with a common denominator and then compare the numerators. The common denominator is usually the lowest common multiple (LCM) of the original denominators.
比较分数时，可以先通分使分母相同，然后比较分子。公分母通常是原分母的最小公倍数 (LCM)。
For example, to compare 5/8 and 2/3, the LCM of 8 and 3 is 24. So 5/8 = 15/24 and 2/3 = 16/24. Since 16/24 is greater than 15/24, 2/3 is greater than 5/8.
例如，比较 5/8 和 2/3，8 和 3 的最小公倍数是 24。所以 5/8 = 15/24，2/3 = 16/24。因为 16/24 大于 15/24，所以 2/3 大于 5/8。
Another method is to convert each fraction to a decimal or percentage using division. This is often quicker when comparing more than two fractions or when the denominators are large.
另一种方法是利用除法把每个分数转换为小数或百分数。当比较两个以上分数或分母较大时，这种方法通常更快。
6. Adding and Subtracting Fractions | 分数的加法与减法
To add or subtract fractions, they must have the same denominator. Find the lowest common denominator, rewrite each fraction, then add or subtract the numerators while keeping the denominator unchanged.
要加减分数，它们必须有相同的分母。找到最小公分母，重写每个分数，然后将分子相加或相减，分母保持不变。
3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 1 7/12
The lowest common multiple of 4 and 6 is 12, so both fractions are converted into twelfths. The answer 19/12 is an improper fraction, so it is also written as the mixed number 1 7/12.
4 和 6 的最小公倍数是 12，所以两个分数都转换为以 12 为分母。答案 19/12 是假分数，因此也可以写成带分数 1 7/12。
When adding or subtracting mixed numbers, it is often easier to convert them to improper fractions first. For example, 2 1/4 − 3/4 becomes 9/4 − 3/4 = 6/4 = 1 1/2.
加减带分数时，通常先转换为假分数更简单。例如，2 1/4 − 3/4 变成 9/4 − 3/4 = 6/4 = 1 1/2。
7. Multiplying Fractions | 分数的乘法
To multiply fractions, multiply the numerators together and multiply the denominators together. You do not need a common denominator for multiplication.
分数相乘时，分子乘以分子，分母乘以分母。乘法不需要公分母。
2/5 × 3/4 = 6/20 = 3/10
Here, 2 × 3 = 6 and 5 × 4 = 20, giving 6/20. This simplifies to 3/10 by dividing both numerator and denominator by 2. Simplifying before multiplying can make the calculation easier.
这里，2 × 3 = 6，5 × 4 = 20，得到 6/20。分子和分母同时除以 2，化简为 3/10。在相乘之前先约分可以使计算更简单。
When multiplying mixed numbers, convert them to improper fractions first. For example, 2 1/3 × 3/7 = 7/3 × 3/7 = 21/21 = 1.
带分数相乘时，先转换为假分数。例如，2 1/3 × 3/7 = 7/3 × 3/7 = 21/21 = 1。
8. Dividing Fractions | 分数的除法
To divide by a fraction, multiply by its reciprocal. This means you flip the second fraction and change the division sign to a multiplication sign. This rule applies to all fractions, including improper fractions.
除以一个分数等于乘以它的倒数。也就是说，把第二个分数翻转，将除号改为乘号。这个规则适用于所有分数，包括假分数。
5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4
After flipping 2/3 to get 3/2, multiply across: 5 × 3 = 15 and 6 × 2 = 12. The fraction 15/12 simplifies to 5/4, which is written as 1 1/4.
将 2/3 翻转得到 3/2 后，直接相乘：5 × 3 = 15，6 × 2 = 12。分数 15/12 化简为 5/4，写成 1 1/4。
For mixed numbers, convert them to improper fractions before dividing. For example, 1 1/2 ÷ 3 = 3/2 ÷ 3/1 = 3/2 × 1/3 = 3/6 = 1/2.
对于带分数，先转换为假分数再进行除法。例如，1 1/2 ÷ 3 = 3/2 ÷ 3/1 = 3/2 × 1/3 = 3/6 = 1/2。
9. Finding a Percentage of an Amount | 求一个数量的百分数
To find a percentage of an amount without a calculator, first find 10%, 5% and 1% of the amount by dividing by 10, 20 and 100 respectively. Then combine these values to reach the required percentage.
不用计算器求一个数量的百分数，可以先将数量分别除以 10、20 和 100，得到它的 10%、5% 和 1%。然后将这些值组合成所需的百分数。
For example, to find 35% of 240, first find 10% = 24, then 5% = 12. Since 35% = 10% + 10% + 10% + 5%, the result is 24 + 24 + 24 + 12 = 84.
例如，要求 240 的 35%，先求 10% = 24，再求 5% = 12。因为 35% = 10% + 10% + 10% + 5%，所以结果是 24 + 24 + 24 + 12 = 84。
With a calculator, multiply the amount by the percentage as a decimal. For example, 15% of 240 = 0.15 × 240 = 36. This method is faster but you should still understand the non-calculator method.
使用计算器时，将数量乘以百分数对应的小数。例如，240 的 15% = 0.15 × 240 = 36。这种方法更快，但你仍然应该理解非计算器方法。
10. Percentage Increase and Decrease | 百分数的增加与减少
To increase an amount by a percentage, add the percentage to 100% and then multiply the original amount by this new percentage as a decimal. For example, increasing £80 by 25% means finding 125% of £80.
将一个数量增加一定百分数，用 100% 加上该百分数，然后用这个新百分数对应的小数乘以原数量。例如，£80 增加 25% 就是求 £80 的 125%。
125% of £80 = 1.25 × £80 = £100
To decrease an amount by a percentage, subtract the percentage from 100% and multiply. Decreasing £80 by 25% means finding 75% of £80, which is 0.75 × £80 = £60.
将一个数量减少一定百分数，从 100% 中减去该百分数后相乘。£80 减少 25% 就是求 £80 的 75%，即 0.75 × £80 = £60。
The number 1.25 or 0.75 in these examples is called the multiplier. Using multipliers is the most efficient way to solve percentage increase and decrease problems.
这些例子中的 1.25 或 0.75 称为乘数。使用乘数是解决百分数增减问题最有效的方法。
11. Reverse Percentages | 反向百分数
Sometimes you know the final amount after a percentage change and need to find the original amount. To do this, divide the final amount by the multiplier that produced it.
有时你知道百分数变化后的最终金额，需要求原金额。此时，将最终金额除以产生它的乘数。
For example, a jacket costs £64 after a 20% reduction. A 20% reduction means the final price is 80% of the original, so the multiplier is 0.8. Therefore the original price is £64 ÷ 0.8 = £80.
例如，一件夹克降价 20% 后售价为 £64。降价 20% 意味着最终价格是原价的 80%，所以乘数是 0.8。因此原价为 £64 ÷ 0.8 = £80。
Do not simply add 20% to £64, because the 20% reduction was calculated on the original price, not on the reduced price. Reverse percentage problems require dividing by the multiplier, not multiplying.
不要简单地在 £64 上加上 20%，因为 20% 的降价是按原价计算的，而不是按降价后的价格。反向百分数问题需要除以乘数，而不是乘以乘数。
12. Common Misconceptions and Exam Tips | 常见误区与应试技巧
One common mistake is adding the denominators when adding fractions, such as writing 1/2 + 1/3 = 2/5. The correct method is to find a common denominator first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
一个常见错误是在分数相加时把分母也相加，例如写成 1/2 + 1/3 = 2/5。正确的方法是先找到公分母：1/2 + 1/3 = 3/6 + 2/6 = 5/6。
Another common error is confusing 0.5 with 0.05 when converting percentages. Since 50% means 50 out of 100, it is 0.5 as a decimal, not 0.05. Similarly, 5% is 0.05.
另一个常见错误是在转换百分数时
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