KS3 Cambridge Maths: Page 287 Question 1 – Fractions, Decimals and Percentages | KS3 剑桥数学：第287页第1题 – 分数、小数与百分数

📚 KS3 Cambridge Maths: Page 287 Question 1 – Fractions, Decimals and Percentages | KS3 剑桥数学：第287页第1题 – 分数、小数与百分数

Understanding how fractions, decimals and percentages connect is a central skill in the Cambridge Lower Secondary Mathematics course. This article develops the methods needed for a typical page 287 question, including conversions, comparisons, percentage increase and decrease, and word problems.

理解分数、小数和百分数之间的联系是 Cambridge 初中数学课程的核心技能。本文围绕第287页第1题所需的典型方法展开，包括相互转换、比较大小、百分比增减以及应用题。

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1. The Three Representations | 三种表示形式

A fraction, a decimal and a percentage can all describe the same part of a whole. For example, 1/4, 0.25 and 25% are equivalent. Fractions show division, decimals use place value, and percentages are parts out of 100.

分数、小数和百分数都可以表示同一个整体中的部分。例如 1/4、0.25 和 25% 是相等的。分数表示除法，小数使用位值，百分数表示每一百份中的多少。

1/4 = 0.25 = 25%

You can move between these forms by using two key ideas: a fraction bar means division, and a percentage means ‘out of 100’. Keeping these ideas in mind makes conversions much easier.

你可以在这些形式之间切换，关键有两点：分数线表示除法，百分数表示 ‘每一百份’。记住这两点，转换就会容易得多。

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2. Converting Fractions to Percentages | 分数转换为百分数

To convert a fraction to a percentage, divide the numerator by the denominator to get a decimal, then multiply the result by 100. The fraction bar is just a division sign.

要将分数转换为百分数，先用分子除以分母得到小数，然后把结果乘以 100。分数线其实就是除号。

3/8 = 3 ÷ 8 = 0.375 → 0.375 × 100 = 37.5%

Some common conversions are worth memorising because they appear very often. For example, 1/2 is 50%, 1/4 is 25%, 3/4 is 75%, 1/5 is 20% and 1/10 is 10%. These benchmarks help you estimate quickly.

有些常见转换值得记住，因为它们经常出现。例如 1/2 是 50%，1/4 是 25%，3/4 是 75%，1/5 是 20%，1/10 是 10%。这些基准值可以帮助你快速估算。

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3. Converting Decimals to Percentages | 小数转换为百分数

To convert a decimal to a percentage, multiply the decimal by 100. This is the same as moving the decimal point two places to the right, then adding the % symbol.

要将小数转换为百分数，把小数乘以 100。这相当于把小数点向右移动两位，然后加上百分号。

0.85 × 100 = 85% and 1.2 × 100 = 120%

Remember that percentages can be greater than 100. A percentage above 100 simply means the value is larger than the original whole. In growth problems, 120% means the whole plus an extra 20%.

记住，百分数可以大于 100。超过 100 的百分数表示这个值比原来的整体更大。在增长问题中，120% 表示整体再加上额外的 20%。

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4. Converting Percentages to Fractions and Decimals | 百分数转换为分数和小数

To convert a percentage to a fraction, write the percentage over 100 and simplify if possible. To convert a percentage to a decimal, divide the percentage by 100.

要将百分数转换为分数，把百分数写成分母为 100 的分数并尽可能化简；要转换为小数，则把百分数除以 100。

65% = 65/100 = 13/20 and 65% = 0.65

When simplifying, look for the highest common factor of the numerator and the denominator. For 65/100, the highest common factor is 5, so dividing both parts by 5 gives 13/20.

化简时，要寻找分子和分母的最大公因数。对于 65/100，最大公因数是 5，所以分子分母同时除以 5 得到 13/20。

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5. Comparing Fractions, Decimals and Percentages | 比较分数、小数和百分数

To compare values given in different forms, first convert them all into the same form. Decimals are usually the easiest to compare because you can look at place value digit by digit.

要比较不同形式的数，首先把它们都转换成同一种形式。小数通常最容易比较，因为可以逐位查看位值。

Order 2/5, 0.39 and 41%: 2/5 = 0.40, 41% = 0.41, so 0.39 < 0.40 < 0.41

Writing all three values as decimals shows the correct order clearly. This method also helps when answering questions about which value is the largest or smallest.

把三个数都写成小数可以清楚地看出正确顺序。这种方法也有助于回答哪个数最大或最小的问题。

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6. Percentage Increase and Decrease | 百分数增减

To increase an amount by p%, multiply the amount by (1 + p/100). To decrease an amount by p%, multiply the amount by (1 – p/100). The number you multiply by is called the multiplier.

要将一个数增加 p%，用这个数乘以 (1 + p/100)；要减少 p%，则乘以 (1 – p/100)。乘上的这个数叫做乘数或倍数。

£120 increased by 15% = 120 × 1.15 = £138
£80 decreased by 12% = 80 × 0.88 = £70.40

Using a multiplier is much faster than finding the percentage of the amount and then adding or subtracting. For a decrease, the multiplier is always less than 1; for an increase, it is always greater than 1.

使用乘数比先求百分数再加或减要快得多。对于减少，乘数总是小于 1；对于增加，乘数总是大于 1。

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7. Reverse Percentages | 逆向百分数

If you know the value after a percentage change, you can find the original value by dividing the known amount by the multiplier. This is called a reverse percentage calculation.

如果已知百分数变化后的值，可以用已知量除以乘数来求原值。这叫做逆向百分数计算。

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