📚 Fractions, Decimals and Percentages for KS3 Cambridge Maths | KS3剑桥数学：分数、小数与百分比
In the Cambridge KS3 Mathematics curriculum, fractions, decimals and percentages are treated as three connected ways of describing the same idea: parts of a whole. This topic builds fluency in converting between forms, comparing values, performing calculations and applying these skills to real-world problems. A strong grasp of these number forms is essential for later work in ratio, proportion, algebra and probability.
在剑桥 KS3 数学课程中，分数、小数和百分比被视为同一概念的三种相互关联的表达方式：整体的部分。本专题帮助学生熟练地在三种形式之间转换、比较数值、进行计算，并将这些技能应用于实际问题。扎实掌握这些数的形式对于后续学习比、比例、代数和概率至关重要。
1. Understanding Fractions | 理解分数
A fraction is written as a numerator over a denominator. The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts are being considered. For example, in the fraction 3/4, the whole is divided into 4 equal parts and 3 parts are selected.
分数写作分子在上、分母在下。分母表示整体被分成多少等份，分子表示取其中的几份。例如，在分数 3/4 中，整体被分成 4 等份，取其中的 3 份。
Fractions can represent parts of a shape, parts of a set, or a point on a number line. A proper fraction has a numerator smaller than the denominator, such as 5/8. An improper fraction has a numerator greater than or equal to the denominator, such as 9/4. A mixed number combines a whole number with a proper fraction, such as 2 1/3.
分数可以表示图形的一部分、集合的一部分或数轴上的一个点。真分数的分子小于分母，例如 5/8。假分数的分子大于或等于分母，例如 9/4。带分数由一个整数和一个真分数组成，例如 2 1/3。
Proper: 3/5 Improper: 7/4 Mixed: 1 3/4
2. Equivalent Fractions and Simplifying | 等值分数与化简
tor and denominator by the same non-zero whole number. For example, 1/2 = 2/4 = 3/6 = 4/8.
等值分数看似不同但数值相同。你可以将分子和分母同时乘以或除以同一个非零整数来得到等值分数。例如，1/2 = 2/4 = 3/6 = 4/8。
Simplifying a fraction means dividing the numerator and denominator by their highest common factor until no common factor remains except 1. The fraction 12/18 simplifies to 2/3 because the highest common factor of 12 and 18 is 6.
化简分数是指用分子和分母的最大公因数同时除以它们，直到除 1 以外没有其他公因数为止。分数 12/18 化简为 2/3，因为 12 和 18 的最大公因数是 6。
12/18 = (12 ÷ 6)/(18 ÷ 6) = 2/3
3. Decimals and Place Value | 小数与位值
A decimal is another way to write a fraction whose denominator is a power of 10. The digits after the decimal point represent tenths, hundredths, thousandths and so on. For example, 0.35 means 3 tenths plus 5 hundredths, which is equal to 35/100.
小数是分母为 10 的幂的分数的另一种写法。小数点后的数字分别表示十分位、百分位、千分位等。例如，0.35 表示 3 个十分之一加 5 个百分之一，等于 35/100。
Place value is important when comparing and rounding decimals. The decimal 0.7 is greater than 0.699 because 7 tenths is larger than 6 tenths plus 9 hundredths plus 9 thousandths. Rounding 3.276 to two decimal places gives 3.28 because the digit in the thousandths place is 6, so the hundredths digit is rounded up.
位值在比较和四舍五入小数时非常重要。小数 0.7 大于 0.699，因为 7 个十分之一大于 6 个十分之一加 9 个百分之一加 9 个千分之一。将 3.276 四舍五入到两位小数得到 3.28，因为千分位上的数字是 6，所以百分位数字向上进位。
| Tenths: 0.1 = 1/10 | Hundredths: 0.01 = 1/100 | Thousandths: 0.001 = 1/1000 |
4. Converting Between Fractions and Decimals | 分数与小数互化
decimals such as 1/3 = 0.333... which can be written with a dot above the repeating digit.
要将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。一些分数产生有限小数，而另一些产生循环小数，例如 1/3 = 0.333...，可以在重复数字上方加点表示。
pending on the number of decimal places, then simplify. For example, 0.45 = 45/100 = 9/20.
要将有限小数转换为分数，根据小数位数将小数写成分母为 10、100 或 1000 的分数，然后化简。例如，0.45 = 45/100 = 9/20。
3/8 = 0.375 and 0.45 = 9/20
5. Percentages and Their Meaning | 百分比及其意义
A percentage is a fraction with denominator 100. The symbol % means 'out of 100'. For example, 47% means 47 out of 100, which is equal to 47/100 or 0.47. Percentages are useful for comparing proportions because they give a common scale.
百分比是分母为 100 的分数。符号 % 表示“每 100 份中的份数”。例如，47% 表示 100 份中的 47 份，等于 47/100 或 0.47。百分比对于比较比例非常有用，因为它们提供了统一的尺度。
Percentages can be greater than 100 when a quantity increases beyond its original value. For example, 150% means 150 out of 100, or 1.5 times the original amount. Percentages less than 1 are also possible, such as 0.5% which equals 0.005.
当某个量超过原值时，百分比可以大于 100。例如，150% 表示 100 份中的 150 份，即原值的 1.5 倍。小于 1 的百分比也存在，例如 0.5% 等于 0.005。
6. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比的互化
rms are interchangeable. To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, 0.85 = 85%. To convert a percentage to a decimal, divide by 100. For example, 32% = 0.32.
这三种形式可以相互转换。要将小数转换为百分比，乘以 100 并加上 % 符号。例如，0.85 = 85%。要将百分比转换为小数，除以 100。例如，32% = 0.32。
by the denominator to obtain a decimal, then multiply by 100. For example, 3/5 = 0.6 = 60% a percentage to a fraction, write it as a fraction over 100 and simplify. For example, 65% = 65/100 = 13/20.
要将分数转换为百分比，先用分子除以分母得到小数，再乘以 100。例如，3/5 = 0.6 = 60%。要将百分比转换为分数，将其写成分母为 100 的分数并化简。例如，65% = 65/100 = 13/20。
| Fraction: 1/4 | Decimal: 0.25 | Percentage: 25% |
| Fraction: 3/10 | Decimal: 0.3 | Percentage: 30% |
| Fraction: 7/20 | Decimal: 0.35 | Percentage: 35% |
7. Comparing and Ordering | 比较和排序
To compare fractions, decimals and percentages, convert all values to the same form. Often decimals are the easiest to compare because you can compare place values directly. For example, to order 2/5, 0.44 and 39%, convert them all to decimals: 0.4, 0.44 and 0.39. In ascending order they are 0.39, 0.4, 0.44, which is 39%, 2/5, 0.44.
要比较分数、小数和百分比，将所有数值转换为同一种形式。通常小数最容易比较，因为可以直接比较位值。例如，要对 2/5、0.44 和 39% 排序，将它们都转换为小数：0.4、0.44 和 0.39。按升序排列为 0.39、0.4、0.44，即 39%、2/5、0.44。
use 24 as the common denominator: 2/3 = 16/24 and 5/8 = 15/24, so 2/3 is greater than 5/8.
比较不同分母的分数时，先找到公分母，转换每个分数，然后比较分子。例如，比较 2/3 和 5/8，用 24 作为公分母：2/3 = 16/24，5/8 = 15/24，所以 2/3 大于 5/8。
8. Adding and Subtracting Fractions | 分数的加减法
nd keep the denominator. For example, 2/9 + 4/9 = 6/9 = 2/3, and 7/10 - 3/10 = 4/10 = 2/5.
要将同分母分数相加或相减，只需将分子相加或相减，分母保持不变。例如，2/9 + 4/9 = 6/9 = 2/3，7/10 - 3/10 = 4/10 = 2/5。
umbers, add or subtract the whole number parts and fraction parts separately, borrowing if necessary.
如果分母不同，先找出最小公分母，将每个分数改写为等值分数，然后再进行分子加减。例如，1/4 + 2/5 = 5/20 + 8/20 = 13/20。对于带分数，分别加减整数部分和分数部分，必要时进行借位。
1/4 + 2/5 = 5/20 + 8/20 = 13/20
9. Multiplying and Dividing Fractions | 分数的乘除法
= 3/10. Mixed numbers should first be converted to improper fractions before multiplying.
要将分数相乘，将分子相乘、分母相乘，然后尽可能化简。例如，3/4 × 2/5 = 6/20 = 3/10。带分数在相乘前应先转换为假分数。
g its numerator and denominator. For example, 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4.
要除以一个分数，乘以它的倒数。分数的倒数通过交换其分子和分母得到。例如，5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4。
5/6 ÷ 2/3 = 5/6 × 3/2 = 5/4 = 1 1/4
10. Percentage of Amounts and Problems | 百分数应用与问题解决
t, convert the percentage to a decimal and multiply by the amount. For example, to find 15% of 240, calculate 0.15 × 240 = 36. Alternatively, find 10% first and then adjust: 10% of 240 is 24, 5% is 12, so 15% is 24 + 12 = 36.
要求一个数的百分之几，将百分比转换为小数再乘以这个数。例如，求 240 的 15%，计算 0.15 × 240 = 36。也可以先求 10% 再调整：240 的 10% 是 24，5% 是 12，所以 15% 是 24 + 12 = 36。
Percentage increase and decrease problems are common. To increase 80 by 25%, find 25% the original amount: 80 + 20 = 100. Alternatively, multiply by 1.25. To decrease 60 by 30%, find 30% of 60 which is 18, then subtract: 60 - 18 = 42, or multiply by 0.70.
百分比增减问题很常见。将 80 增加 25%，先求 80 的 25% 是 20，然后加到原数上：80 + 20 = 100。也可以乘以 1.25。将 60 减少 30%，先求 60 的 30% 是 18，然后相减：60 - 18 = 42，或者乘以 0.70。
- Find 20% of 150: 0.20 × 150 = 30
- Increase 40 by 15%: 40 × 1.15 = 46
- Decrease 72 by 25%: 72 × 0.75 = 54
- 求 150 的 20%：0.20 × 150 = 30
- 将 40 增加 15%：40 × 1.15 = 46
- 将 72 减少 25%：72 × 0.75 = 54
In word problems, identify the original amount, the percentage change and whether the change is an increase or decrease. Then choose the correct multiplication or division operation. Drawing a bar model can help visualise the relationship between the part and the whole.
在应用题中，要确定原量、百分比变化以及变化是增加还是减少。然后选择正确的乘法或除法运算。画条形模型有助于直观表示部分与整体的关系。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图