📚 Fractions, Decimals, Percentages and Ratio | 分数、小数、百分数与比
Understanding fractions, decimals, percentages and ratio is essential for success in IGCSE Mathematics. These topics appear throughout the Number strand and are used in algebra, geometry, statistics and real-life financial questions. This revision article covers the key methods, links between the forms and common exam techniques.
掌握分数、小数、百分数和比是 IGCSE 数学取得好成绩的基础。这些内容贯穿数字模块，并广泛用于代数、几何、统计和实际财务问题。本文总结关键方法、不同形式之间的联系以及常见考试技巧。
1. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
iply by 100. To convert a percentage to a fraction, write it over 100 and simplify, such as 65% = 65/100 = 13/20.
分数化小数，用分子除以分母。例如 3/8 = 3 ÷ 8 = 0.375。小数化百分数，将小数乘以 100。百分数化分数，先写成 100 分之几再化简，如 65% = 65/100 = 13/20。
3... can be written using dot notation. Recognising equivalent forms helps when ordering and comparing values in IGCSE exam questions.
循环小数如 1/3 = 0.333... 可用点标记表示。识别等价形式有助于在 IGCSE 考试题目中进行排序和比较数值。
2. Adding, Subtracting, Multiplying and Dividing Fractions | 分数的四则运算
For example, 1/4 + 1/6 = 3/12 + 2/12 = 5/12. Always simplify the final answer if possible.
分数加减时，先找到公分母。例如 1/4 + 1/6 = 3/12 + 2/12 = 5/12。如果可能，最后一定要化简答案。
35. To divide fractions, multiply by the reciprocal: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
分数相乘时，分子乘分子、分母乘分母：2/5 × 3/7 = 6/35。分数相除时，乘以倒数：3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
For mixed numbers, convert them into improper fractions before multiplying or dividing. This reduces sign and whole-number errors.
处理带分数时，在进行乘除前先将其化为假分数。这样可以减少符号和整数部分的错误。
3. Decimal Calculations and Rounding | 小数运算与舍入
When adding or subtracting decimals, align the decimal points in vertical columns. For multiplication, ignore the decimal points initially, multiply the integers, then insert the decimal point so the result has the total number of decimal places.
小数加减时，在竖式中将小数点对齐。小数乘法可以先忽略小数点按整数相乘，再根据原来的小数位数总和点上小数点。
10 to make the divisor a whole number. For example, 6.4 ÷ 0.8 is equivalent to 64 ÷ 8 = 8.
小数除法可将被除数和除数同时乘以 10 的幂，使除数成为整数。例如 6.4 ÷ 0.8 等价于 64 ÷ 8 = 8。
Rounding to a number of decimal places counts digits after the decimal point. Rounding to significant figures counts from the first non-zero digit. Always consider the digit immediately to the right when deciding whether to round up or down.
按小数位舍入是从小数点后开始数位。按有效数字舍入是从第一个非零数字开始数位。判断是否进位时，始终观察右邻的一位数字。
4. Percentage Increase and Decrease | 百分数的增加与减少
For a percentage increase, use a multiplier greater than 1. For a percentage decrease, use a multiplier between 0 and 1. This method is safer than calculating the increase or decrease separately.
百分数增加使用大于 1 的乘数，百分数减少使用 0 到 1 之间的乘数。使用乘数法比先算出增减量再相加或相减更安全。
New value = original value × (1 + r/100)
New value = original value × (1 - r/100)
For example, increase 80 by 15%: 80 × 1.15 = 92. Decrease 80 by 15%: 80 × 0.85 = 68. The same method works for VAT, discounts and population change.
例如将 80 增加 15%：80 × 1.15 = 92。将 80 减少 15%：80 × 0.85 = 68。同样方法适用于增值税、折扣和人口变化。
5. Reverse Percentages | 逆向百分数
Reverse percentage questions give a value after a percentage change and ask for the original value. Identify the correct multiplier for the final allowed percentage, then divide by that multiplier.
逆向百分数题目给出百分数变化后的值，要求原值。先确定最终对应百分数的乘数，再用该乘数去除给出的值。
For example, a jacket is reduced by 20% in a sale and now costs $64. Since $64 represents 80% of the original price, the original price is 64 ÷ 0.80 = $80.
例如一件夹克降价 20% 后售价 64 美元。由于 64 美元代表原价的 80%，原价为 64 ÷ 0.80 = 80 美元。
Do not add 20% nown original amount, not the final amount. A common incorrect answer would be 64 × 1.20 = $76.80.
不要将售价直接增加 20%，因为百分数是以未知原价为基础，而不是以最终售价为基础。常见错误答案是 64 × 1.20 = 76.80 美元。
6. Ratio and Proportional Division | 比与按比例分配
A ratio compares two or more quantities. To simplify a ratio, divide all parts by their highest common factor. For example, 24 : 36 simplifies to 2 : 3 by dividing both sides by 12.
比用于比较两个或多个量。化简比时，将所有部分除以它们的最大公因数。例如 24 : 36 两边同时除以 12，化简为 2 : 3。
e part is 60 ÷ 5 = 12 dollars. The shares are 3 × 12 = 36 dollars and 2 × 12 = 24 dollars.
按给定比例分配金额时，先求总份数。例如将 60 美元按 3 : 2 分配。总份数为 3 + 2 = 5，因此一份为 60 ÷ 5 = 12 美元。各部分分别为 3 × 12 = 36 美元和 2 × 12 = 24 美元。
A ratio can also be written as fractions of the total. In the above example, the first share is 3/5 of 60 and the second share is 2/5 of 60.
比也可以写为占总量的分数。在上例中，第一部分占 60 的 3/5，第二部分占 60 的 2/5。
7. Scales, Maps and Units | 比例尺、地图与单位
A map scale is a ratio between map distance and actual distance. A scale of 1 : 25000 means that 1 cm on the map represents 25000 cm in real life, which is 250 m.
地图比例尺表示图上距离与实际距离的比。比例尺 1 : 25000 表示图上 1 cm 代表实际 25000 cm，即 250 m。
inutes. For speed, density and pressure problems, always use consistent units before substituting into a formula.
单位换算时要记住 1 km = 1000 m，1 m = 100 cm，1 小时 = 60 分钟。在速度、密度和压强问题中，代入公式前必须使用一致的单位。
× 25000 cm = 100000 cm = 1 km. This type of calculation is often set in IGCSE exam papers.
如果一段路在 1 : 25000 地图上的长度为 4 cm，实际长度为 4 × 25000 cm = 100000 cm = 1 km。此类计算在 IGCSE 试卷中经常出现。
8. Practical Applications: Discounts, Tax and Interest | 实际应用：折扣、税收与利息
When a percentage change is followed by another percentage change, apply the multipliers in order. A 15% discount followed by 10% tax is not the same as a 5% net reduction.
当百分数变化接连发生时，应依次使用乘数。先打 15% 的折扣再加 10% 的税，并不等于净减 5%。
For example, an item priced at $200 has a 15% discount and then 10% sales tax. The final price is 200 × 0.85 = 170, then 170 × 1.10 = $187.
例如一件商品原价 200 美元，先打 15% 折扣，再收 10% 销售税。最终价格为 200 × 0.85 = 170，再 170 × 1.10 = 187 美元。
al, R is the annual interest rate as a percentage, and T is the time in years. Compound interest repeats the multiplier method each year.
单利可用 I = PRT/100 计算，其中 P 为本金，R 为年利率（百分数），T 为时间（年）。复利则每年重复使用乘数法。
9. Estimation and Accuracy | 估算与精度
Estimation is a powerful checking tool. Round each number in a calculation to one significant figure, then perform the simplified calculation mentally.
估算是很有用的检查工具。将算式中的每个数舍入到一位有效数字，再进行简化的心算。
0. If your calculator answer is very different from 200, check the decimal point or input.
例如估算 49.6 × 2.14 ÷ 0.52。舍入为 50 × 2 ÷ 0.5 = 200。如果计算器答案与 200 相差很大，应检查小数点或输入。
Upper and lower bounds arise when a number is given correct to a certain degree of accuracy. For example, a length of 6 cm correct to the nearest cm has lower bound 5.5 cm and upper bound 6.5 cm.
当一个数按一定精确度给出时，会产生上界和下界。例如长度 6 cm 精确到最接近的 cm，其下界为 5.5 cm，上界为 6.5 cm。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
A very common mistake is adding denominators when adding fractions. Remember that only the numerators are added after finding a common denominator.
一个非常常见的错误是分数相加时把分母也相加。记住在找到公分母后，只把分子相加。
Another common error is confusing 20% increase with 20% of the final value in reverse percentage questions. Always identify the multiplier that corresponds to the final value.
另一个常见错误是在逆向百分数问题中混淆 20% 增加与最终值的 20%。务必先确定与最终值对应的乘数。
In ratio problems, do not forget to find the total number of parts before dividing. Always show clear working and check that your answer is reasonable in the context of the question.
在比例问题中，分配前不要忘记先求总份数。始终展示清晰步骤，并结合题意判断答案是否合理。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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