📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数与百分比
Fractions, decimals and percentages are three different ways of representing parts of a whole. In KS3 mathematics, you must be able to convert between these forms, compare them, and use them in calculations. This topic appears throughout the Cambridge curriculum and is essential for problem solving in real life, from shopping discounts to probability and data handling.
分数、小数和百分比是表示整体中一部分的三种不同方式。在 KS3 数学中，你必须能够在这些形式之间进行转换、比较它们，并在计算中使用它们。这个主题贯穿剑桥课程，对于解决现实生活中的问题至关重要，从购物折扣到概率和数据处理都离不开它。
1. What Are Fractions, Decimals and Percentages? | 什么是分数、小数和百分比？
A fraction shows a part of a whole using a numerator and a denominator. For example, 3/4 means 3 parts out of 4 equal parts.
分数用分子和分母表示整体中的一部分。例如，3/4 表示 4 等份中的 3 份。
A decimal uses a decimal point to show tenths, hundredths, thousandths and so on. The decimal 0.75 means 75 hundredths, which is the same as 75/100.
小数使用小数点来表示十分位、百分位、千分位等。小数 0.75 表示 75 个百分之一，也就是 75/100。
A percentage is a number out of 100. The symbol % means 'per cent' or 'out of 100'. So 75% means 75 out of 100.
百分比是一个以 100 为基准的数。符号 % 表示 '每一百' 或 '百分之'。所以 75% 表示 100 份中的 75 份。
2. Converting Between Fractions and Decimals | 分数与小数的互相转换
n to a decimal, divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.
要将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。
te the decimal over its place value. For example, 0.45 = 45/100, which simplifies to 9/20.
要将有限小数转换为分数，把小数写在其位值之上。例如，0.45 = 45/100，化简后为 9/20。
gebraic methods. For example, if x = 0.333..., then 10x = 3.333..., so 9x = 3 and x = 1/3.
对于循环小数，使用代数方法。例如，设 x = 0.333...，则 10x = 3.333...，因此 9x = 3，所以 x = 1/3。
3. Converting Decimals to Percentages and Back | 小数与百分比的互相转换
To convert a decimal to a percentage, multiply by 100 and add the % sign. For example, 0.62 × 100 = 62%.
要将小数转换为百分比，乘以 100 并加上 % 符号。例如，0.62 × 100 = 62%。
To convert a percentage to a decimal, divide by 100. For example, 8% = 8 ÷ 100 = 0.08.
要将百分比转换为小数，除以 100。例如，8% = 8 ÷ 100 = 0.08。
Remember that percentages greater than 100 are possible, such as 150% = 1.5, and percentages less than 1 are also valid, such as 0.5% = 0.005.
请记住，百分比可以大于 100，例如 150% = 1.5；百分比也可以小于 1，例如 0.5% = 0.005。
4. Converting Fractions to Percentages | 分数转换为百分比
e fraction to a decimal by dividing, then multiply by 100. For example, 5/8 = 0.625 = 62.5%.
要将分数转换为百分比，先将分数除以得到小数，再乘以 100。例如，5/8 = 0.625 = 62.5%。
you can make an equivalent fraction with denominator 100. For example, 7/20 = 35/100 = 35%.
如果分母是 100 的因数，你可以先化为分母为 100 的等值分数。例如，7/20 = 35/100 = 35%。
e number part and the fraction part separately. For example, 2 1/4 = 2 + 0.25 = 2.25 = 225%.
对于带分数，分别转换整数部分和分数部分。例如，2 1/4 = 2 + 0.25 = 2.25 = 225%。
5. Equivalent Fractions and Simplifying | 等值分数与化简
value but have different numerators and denominators. For example, 1/2 = 2/4 = 3/6 = 4/8.
等值分数表示相同的值，但分子和分母不同。例如，1/2 = 2/4 = 3/6 = 4/8。
To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6.
要化简分数，将分子和分母同时除以它们的最大公因数 (HCF)。例如，18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6。
the numerator and denominator by the same non-zero number. For example, 2/5 × 3/3 = 6/15.
要找到等值分数，将分子和分母同时乘以或除以同一个非零数。例如，2/5 × 3/3 = 6/15。
6. Ordering Fractions, Decimals and Percentages | 分数、小数和百分比的比较与排序
To compare fractions, decimals and percentages, convert them all to the same form. Usually, converting to decimals is the easiest method.
要比较分数、小数和百分比，将它们全部转换为同一种形式。通常，转换为小数是最简单的方法。
For example, to order 3/5, 0.58 and 62% from smallest to largest, first convert: 3/5 = 0.6, 0.58 = 0.58 and 62% = 0.62. The order is 0.58, 0.6, 0.62.
例如，要将 3/5、0.58 和 62% 从小到大排序，先转换：3/5 = 0.6，0.58 = 0.58，62% = 0.62。顺序为 0.58、0.6、0.62。
When comparing fractions with the same denominator, the fraction with the larger numerator is larger. For example, 5/9 > 4/9.
当比较分母相同的分数时，分子较大的分数较大。例如，5/9 > 4/9。
7. Adding and Subtracting Fractions | 分数的加减法
r subtract the numerators and keep the denominator the same. For example, 2/7 + 3/7 = 5/7.
要加减分母相同的分数，将分子相加或相减，分母保持不变。例如，2/7 + 3/7 = 5/7。
1/4, the LCD of 3 and 4 is 12, so 1/3 = 4/12 and 1/4 = 3/12. Therefore 4/12 + 3/12 = 7/12.
如果分母不同，先找到最小公分母 (LCD)。例如，计算 1/3 + 1/4，3 和 4 的最小公分母是 12，所以 1/3 = 4/12，1/4 = 3/12。因此 4/12 + 3/12 = 7/12。
For mixed numbers, add or subtract the whole number parts and the fraction parts separately. If the fraction part of the subtraction is not possible, borrow 1 from the whole number.
对于带分数，分别加减整数部分和分数部分。如果减法中分数部分不够减，从整数部分借 1。
8. Multiplying and Dividing Fractions | 分数的乘除法
s together and the denominators together. For example, 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15.
要乘分数，将分子相乘作为新的分子，分母相乘作为新的分母。例如，2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15。
swapping the numerator and denominator. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
要除以一个分数，乘以它的倒数。分数的倒数是将分子和分母互换得到的。例如，3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8。
lt if possible. For example, 6/9 = 2/3 after dividing both numerator and denominator by 3.
如果可能，始终化简结果。例如，6/9 分子分母同时除以 3 后等于 2/3。
9. Percentage of an Amount | 求一个数的百分比
an amount, convert the percentage to a decimal and multiply by the amount. For example, 15% of 80 = 0.15 × 80 = 12.
要求一个数的百分比，将百分比转换为小数，然后乘以这个数。例如，80 的 15% = 0.15 × 80 = 12。
You can also use the fraction method. For example, 25% of 48 = 1/4 × 48 = 12, because 25% is the same as 1/4.
你也可以使用分数方法。例如，48 的 25% = 1/4 × 48 = 12，因为 25% 等于 1/4。
For mental calculation, find 10% first and then scale. For example, to find 30% of 70, find 10% of 70 = 7, then multiply by 3 to get 21.
对于心算，先求 10%，然后按比例放大。例如，要求 70 的 30%，先求 70 的 10% = 7，然后乘以 3 得到 21。
10. Percentage Increase and Decrease | 百分比的增加与减少
To increase an amount by a percentage, add the percentage to 100%, convert to a decimal and multiply. For example, to increase 60 by 20%, use the multiplier 1.20: 60 × 1.20 = 72.
要将一个数增加一个百分比，将百分比加到 100%，转换为小数，然后相乘。例如，将 60 增加 20%，使用乘数 1.20：60 × 1.20 = 72。
To decrease an amount by a percentage, subtract the percentage from 100%, convert to a decimal and multiply. For example, to decrease 80 by 15%, use the multiplier 0.85: 80 × 0.85 = 68.
要将一个数减少一个百分比，从 100% 中减去该百分比，转换为小数，然后相乘。例如，将 80 减少 15%，使用乘数 0.85：80 × 0.85 = 68。
rises from 50 to 65, the change is 15, so the percentage increase is (15 ÷ 50) × 100 = 30%.
百分比变化的公式为：百分比变化 = (变化量 ÷ 原数) × 100。例如，如果价格从 50 上升到 65，变化量为 15，所以百分比增加为 (15 ÷ 50) × 100 = 30%。
11. Reverse Percentages | 逆向百分比
Reverse percentage problems ask you to find the original amount before a percentage increase or decrease. To do this, divide by the multiplier.
逆向百分比问题是要求出在百分比增加或减少之前的原始数量。为此，除以乘数。
For example, if a jacket costs 84 after a 20% tion, the multiplier for the decrease was 0.80. So the original price was 84 ÷ 0.80 = 105.
例如，如果一件夹克降价 20% 后售价为 84，降价乘数为 0.80。所以原价为 84 ÷ 0.80 = 105。
If a salary increases by 5% to 31,500, the multiplier was 1.05. So the original salary was 31,500 ÷ 1.05 = 30,000.
如果工资增加 5% 后为 31,500，乘数为 1.05。所以原始工资为 31,500 ÷ 1.05 = 30,000。
12. Real-Life Applications | 实际应用
tuations such as discounts, interest rates, test scores and probability. For example, a 25% discount on a 200 item saves 200 × 0.25 = 50.
分数、小数和百分比在日常生活中有广泛应用，例如折扣、利率、考试分数和概率。例如，一件 200 元的商品打 75 折（25% 折扣）可节省 200 × 0.25 = 50。
For example, if 18 out of 24 students prefer football, the percentage is 18/24 × 100 = 75%.
在数据处理中，百分比用于描述整体中的比例。例如，如果 24 名学生中有 18 名喜欢足球，百分比为 18/24 × 100 = 75%。
Understanding how to convert between these forms allows you to solve problems efficiently and check your answers using different representations.
理解这些形式之间的转换能让你高效地解决问题，并使用不同的表示方式来检查答案。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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