Fractions, Decimals and Percentages | 分数、小数与百分数

📚 Fractions, Decimals and Percentages | 分数、小数与百分数

In KS3 Cambridge mathematics, fractions, decimals and percentages are three ways of representing the same value. You need to move confidently between them, calculate with them and apply them to real problems such as discounts, probability and measurement.

在剑桥 KS3 数学中，分数、小数和百分数是表示同一个数值的三种方式。你需要在这三种形式之间熟练转换，并用它们进行计算和解决折扣、概率、测量等实际问题。

1. What Are Fractions? | 什么是分数

A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator tells you how many equal parts the whole is split into, and the numerator tells you how many of those parts are taken.

分数写作 a/b，其中 a 是分子，b 是分母。分母表示整体被平均分成多少份，分子表示取了多少份。

Proper fractions have a numerator that is smaller than the denominator, such as 3/5. Improper fractions have a numerator that is larger than or equal to the denominator, such as 7/4. Mixed numbers combine a whole number with a proper fraction, such as 1 3/4.

真分数的分子小于分母，例如 3/5。假分数的分子大于或等于分母，例如 7/4；带分数由一个整数和一个真分数组成，例如 1 3/4。

Every fraction can also be thought of as a division statement: 3/5 means 3 ÷ 5. This idea is the key to converting any fraction into a decimal.

每个分数也可以看作一个除法算式：3/5 表示 3 ÷ 5。这个思路是把任何分数转换成小数的关键。

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 multiply or divide both the numerator and the denominator by the same non-zero number without changing the value of the fraction.

等值分数虽然形式不同，但数值相同。你可以把分子和分母同时乘或除以同一个非零数，分数的值不变。

To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). For example, 12/18 simplifies to 2/3 because 6 is the HCF of 12 and 18.

化简分数时，用分子和分母的最大公因数（HCF）同时去除。例如，12/18 化简为 2/3，因为 6 是 12 和 18 的最大公因数。

Checking for equivalent fractions is also useful when adding or comparing fractions. Always write the simplest form unless a question asks otherwise.

在分数加减或比较大小时，确认等值分数也很有用。除非题目另有要求，通常都要写出最简形式。

3. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 means 3 ÷ 8, which equals 0.375.

把分数转换成小数，就是用分子除以分母。例如，3/8 表示 3 ÷ 8，等于 0.375。

Some fractions give terminating decimals, such as 1/4 = 0.25. Others give recurring decimals, such as 1/3 = 0.333…, where the digit 3 repeats forever.

有些分数得到有限小数，例如 1/4 = 0.25。有些分数得到循环小数，例如 1/3 = 0.333…，数字 3 无限循环。

The common equivalents below are worth memorising because they appear very often in KS3 questions:

下面这些常见等值关系值得记住，因为它们在 KS3 题目中经常出现：

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%

Using these known facts can make later work on percentages much faster and reduce arithmetic errors.

利用这些已知事实可以更快地完成百分数相关计算，并减少算术错误。

4. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数互化

A percentage means ‘out of 100’. To convert a decimal to a percentage, multiply by 100. For example, 0.63 as a percentage is 0.63 × 100 = 63%.

百分数表示 ‘每 100 份中的多少份’。把小数转换成百分数，要乘以 100。例如，0.63 换算成百分数是 0.63 × 100 = 63%。

To convert a percentage to a decimal, divide by 100. For instance, 45% as a decimal is 45 ÷ 100 = 0.45.

把百分数转换成小数，要除以 100。例如，45% 换算成小数是 45 ÷ 100 = 0.45。

A fraction can be converted to a percentage by making the denominator 100, or by dividing the numerator by the denominator and then multiplying by 100. For example, 3/5 = 0.6, so 3/5 = 60%.

分数换算成百分数，可以把分母变成 100，也可以用分子除以分母后再乘以 100。例如，3/5 = 0.6，所以 3/5 = 60%。

5. Ordering Fractions, Decimals and Percentages | 分数、小数和百分数的排序

To compare fractions, decimals and percentages, first convert all of them into the same form. Decimals are often the easiest form to work with because you can compare place value directly.

要比较分数、小数和百分数，先把它们都转换成同一种形式。使用小数通常最方便，因为可以直接比较数位。

For example, arrange 2/5, 0.45 and 38% in ascending order. Convert them all to decimals: 2/5 = 0.4, 0.45 = 0.45 and 38% = 0.38. The order is 38%, 2/5, 0.45.

例如，将 2/5、0.45 和 38% 按从小到大的顺序排列。把它们都转换成小数：2/5 = 0.4，0.45 = 0.45，38% = 0.38。所以顺序是 38%、2/5、0.45。

When fractions have the same denominator, you can compare the numerators directly. If denominators are different, find a common denominator or use decimals.

当分数分母相同时，可以直接比较分子。如果分母不同，可以找出公分母，或者转换成小数再比较。

6. Adding and Subtracting Fractions | 分数的加法与减法

To add or subtract fractions with the same denominator, keep the denominator and add or subtract the numerators. For example, 2/9 + 4/9 = 6/9 = 2/3.

分母相同的分数相加减时，分母不变，只把分子相加或相减。例如，2/9 + 4/9 = 6/9 = 2/3。

When the denominators are different, first find the lowest common multiple (LCM) of the denominators. Rewrite each fraction with that common denominator, then add or subtract the numerators.

当分母不同时，先找出分母的最小公倍数（LCM）。把每个分数改写为以这个公倍数为分母的等值分数，然后再加减分子。

For mixed numbers, it is often easier to convert them to improper fractions first. After calculating, change the answer back to a mixed number if the question asks for one.

对于带分数，通常先将其转换成假分数再计算。如果题目要求，计算后再把结果改写成带分数。

7. Multiplying and Dividing Fractions | 分数的乘法与除法

To multiply fractions, multiply the numerators together and multiply the denominators together. For example, 2/3 × 4/5 = 8/15.

分数相乘时，分子乘分子，分母乘分母。例如，2/3 × 4/5 = 8/15。

You can simplify before multiplying by cross-cancelling any common factor between a numerator and a denominator. This reduces large numbers and makes the calculation easier.

在相乘之前，可以通过交叉约分来化简分子和分母的公因数。这样可以减小数字，使计算更简单。

To divide by a fraction, multiply by its reciprocal. For example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

除以一个分数，等于乘以它的倒数。例如，2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。

8. Percentages of Amounts | 求一个数的百分之几

To find a percentage of an amount without a calculator, start with easy percentages. For example, 10% of 60 is 6, and 1% of 60 is 0.6. You can combine these to find other percentages.

不用计算器求一个数的百分之几时，可以先从简单的百分数入手。例如，60 的 10% 是 6，60 的 1% 是 0.6。你可以把这些结果组合起来求其他百分数。

Using a decimal multiplier is often faster: multiply the amount by the decimal form of the percentage. For example, 35% of 80 = 0.35 × 80 = 28.

使用小数乘数通常更快：用原数乘以百分数的小数形式。例如，80 的 35% = 0.35 × 80 = 28。

In word problems, identify the original amount and the percentage carefully. Questions may ask for the result, the percentage, or the original amount, so read the wording closely.

在应用题中，要仔细识别原数和百分数。题目可能要求结果、百分数或原数，因此要认真审题。

9. Percentage Increase and Decrease | 百分数增减

To increase an amount by a percentage, multiply by the decimal multiplier 1 + p/100. For example, increasing 50 by 20% gives 50 × 1.2 = 60.

要把一个数增加百分之 p，乘以小数乘数 1 + p/100。例如，将 50 增加 20%，结果是 50 × 1.2 = 60。

To decrease an amount by a percentage, multiply by 1 – p/100. For example, decreasing 80 by 15% gives 80 × 0.85 = 68.

要把一个数减少百分之 p，乘以小数乘数 1 – p/100。例如，将 80 减少 15%，结果是 80 × 0.85 = 68。

This method is especially useful when working with prices, salaries or populations, where you need to apply one percentage change directly.

在处理价格、工资或人口等问题时，这种方法尤其有用，因为可以直接应用一次百分数变化。

10. Real-Life Applications and Common Misconceptions | 实际应用与常见误区

Fractions, decimals and percentages appear in many real-life contexts: discounts in shops, test scores, interest rates, probability and recipe ingredients. Being able to switch between forms helps you choose the best method for each problem.

分数、小数和百分数出现在许多实际情境中：商店折扣、考试分数、利率、概率和食谱配料等。能够在不同形式之间切换，可以帮助你为每个问题选择最佳方法。

One common mistake is to think that 1/3 is the same as 0.3. In fact, 1/3 = 0.333…, so rounding too early can make answers inaccurate. Another misconception is adding denominators when adding fractions, but denominators should stay the same or be changed to a common denominator.

一个常见错误是认为 1/3 等于 0.3。实际上 1/3 = 0.333…，过早四舍五入会导致答案不准确。另一个误区是在分数相加时把分母也相加，但分母应保持不变或化为公分母。

Always check whether your answer is reasonable. For example, finding 90% of a number should give a value smaller than the original number, while increasing by 10% should give a value larger than the original.

做题后要检查答案是否合理。例如，求一个数的 90%，结果应小于原数；而增加 10%，结果应大于原数。

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Fractions, Decimals and Percentages | 分数、小数与百分数