Fractions, Decimals and Percentages | 分数、小数和百分数

📚 Fractions, Decimals and Percentages | 分数、小数和百分数

Fractions, decimals and percentages are three ways of representing parts of a whole. In KS3 Cambridge Mathematics, you are expected to move fluently between these forms and use them to solve real-world problems, from discounts in shops to probability and data handling. This revision guide explains the key methods step by step and gives you the tools to avoid common mistakes.

分数、小数和百分数是表示整体部分的三种方式。在剑桥 KS3 数学中，你需要在这三种形式之间熟练转换，并用它们解决现实问题，例如商店折扣、概率和数据处理。本复习指南逐步解释关键方法，并帮助你避免常见错误。

1. Understanding the Basics | 理解基本概念

A fraction has a numerator (top number) and a denominator (bottom number). A decimal uses a decimal point to show tenths, hundredths and so on. A percentage is a number out of 100, written with the % symbol.

分数有分子（上面的数）和分母（下面的数）。小数用小数点表示十分位、百分位等。百分数是表示每 100 份中的数量，用 % 符号书写。

For example, 1/2 means 1 part out of 2 equal parts, 0.5 means 5 tenths, and 50% means 50 out of 100. These three all represent the same amount.

例如，1/2 表示 2 等份中的 1 份，0.5 表示 5 个十分之一，50% 表示每 100 份中的 50 份。这三者表示相同的量。

You should remember key equivalents by heart: 1/4 = 0.25 = 25%, 1/2 = 0.5 = 50%, 3/4 = 0.75 = 75%, 1/10 = 0.1 = 10%.

你应该熟记关键等价关系：1/4 = 0.25 = 25%，1/2 = 0.5 = 50%，3/4 = 0.75 = 75%，1/10 = 0.1 = 10%。

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2. Converting Fractions to Decimals | 分数转小数

To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator if allowed.

要把分数转换为小数，用分子除以分母。可以使用短除法，如果允许也可以用计算器。

For example, 3/8 means 3 ÷ 8. Since 8 goes into 3 zero times, add a decimal point and zeros: 8 goes into 30 three times (24), remainder 6; bring down 0 to make 60; 8 goes into 60 seven times (56), remainder 4; bring down 0 to make 40; 8 goes into 40 five times exactly. So 3/8 = 0.375.

例如，3/8 表示 3 ÷ 8。8 进入 3 是 0 次，加上小数点并补零：8 进入 30 三次（24），余 6；落下 0 得 60；8 进入 60 七次（56），余 4；落下 0 得 40；8 进入 40 五次正好除尽。因此 3/8 = 0.375。

Some fractions produce recurring decimals, such as 1/3 = 0.333… or 2/11 = 0.1818… You can write a dot over the repeating digit or digits.

有些分数会得到循环小数，例如 1/3 = 0.333… 或 2/11 = 0.1818… 可以在重复的数字上加点表示循环。

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3. Converting Decimals to Percentages | 小数转百分数

To convert a decimal to a percentage, multiply by 100 and add the % sign. This is the same as moving the decimal point two places to the right.

要把小数转换为百分数，乘以 100 并加上 % 符号。这相当于把小数点向右移动两位。

For example, 0.45 × 100 = 45, so 0.45 = 45%. For 0.075, moving the decimal point two places gives 7.5, so 0.075 = 7.5%.

例如，0.45 × 100 = 45，所以 0.45 = 45%。对于 0.075，小数点向右移动两位得到 7.5，所以 0.075 = 7.5%。

If the decimal is greater than 1, the percentage is greater than 100%. For instance, 1.2 = 120%. This is useful for increases or comparing quantities.

如果小数大于 1，百分数就大于 100%。例如，1.2 = 120%。这在表示增加或比较数量时很有用。

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4. Converting Percentages to Fractions | 百分数转分数

To convert a percentage to a fraction, write the percentage as a fraction over 100, then simplify if possible.

要把百分数转换为分数，将百分数写成分母为 100 的分数，然后尽可能化简。

For example, 35% = 35/100. Divide numerator and denominator by 5 to get 7/20. So 35% = 7/20.

例如，35% = 35/100。分子和分母同时除以 5 得到 7/20。所以 35% = 7/20。

If the percentage includes a decimal or fraction, first write an equivalent fraction over 100. For instance, 12.5% = 12.5/100 = 25/200 = 1/8.

如果百分数包含小数或分数，先写成等价的以 100 为分母的分数。例如，12.5% = 12.5/100 = 25/200 = 1/8。

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5. Ordering and Comparing | 排序与比较

To compare fractions, decimals and percentages, convert them all to the same form. Percentages or decimals are often easiest for ordering.

要比较分数、小数和百分数，先把它们都转换成同一种形式。通常用百分数或小数排序最方便。

Example: Put 3/5, 0.55, 58% and 0.6 in ascending order. Convert: 3/5 = 0.60 = 60%, 0.55 = 55%, 58% = 58%, 0.6 = 60%. Ascending order is 0.55 (55%), 58%, 3/5 (60%), 0.6 (60%).

示例：将 3/5、0.55、58% 和 0.6 从小到大排列。转换后：3/5 = 0.60 = 60%，0.55 = 55%，58% = 58%，0.6 = 60%。从小到大为 0.55（55%）、58%、3/5（60%）、0.6（60%）。

A common mistake is to look only at the digits after the decimal point and ignore place value, for example thinking 0.7 is smaller than 0.25. Always compare tenths first, then hundredths.

一个常见错误是只看小数点后的数字而忽略位值，例如认为 0.7 比 0.25 小。一定要先比较十分位，再比较百分位。

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6. Finding a Percentage of an Amount | 求一个数的百分之几

To find a percentage of an amount without a calculator, find 10% first, then scale. For 10%, divide by 10. For 5%, divide by 20 or halve 10%. For 1%, divide by 100.

不用计算器求一个数的百分之几时，先求 10%，再按比例计算。10% 是除以 10。5% 是除以 20 或取 10% 的一半。1% 是除以 100。

Example: Find 35% of 240. 10% of 240 = 24, so 30% = 3 × 24 = 72. 5% = half of 10% = 12. Then 35% = 30% + 5% = 72 + 12 = 84.

示例：求 240 的 35%。240 的 10% = 24，所以 30% = 3 × 24 = 72。5% = 10% 的一半 = 12。因此 35% = 30% + 5% = 72 + 12 = 84。

Using a calculator, convert the percentage to a decimal and multiply. For example, 35% of 240 = 0.35 × 240 = 84.

使用计算器时，先把百分数转换为小数再相乘。例如，240 的 35% = 0.35 × 240 = 84。

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7. Percentage Increase and Decrease | 百分数增减

A percentage increase adds a percentage of the original amount. A percentage decrease subtracts it. Always identify the original amount clearly.

百分数增加是在原数的基础上加上原数的百分之几。百分数减少是从原数中减去原数的百分之几。一定要明确哪个是原数。

Example: Increase 120 by 15%. Find 15% of 120: 10% = 12, 5% = 6, so 15% = 18. Add to original: 120 + 18 = 138.

示例：将 120 增加 15%。求 120 的 15%：10% = 12，5% = 6，所以 15% = 18。加到原数上：120 + 18 = 138。

For a decrease, subtract the increase amount. Example: Decrease 80 by

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