Grade 6 Topic Series: 900 | 6年级主题系列-900

📚 Grade 6 Topic Series: 900 | 6年级主题系列-900

The number 900 is far more than a simple three-digit value. It appears in a wide range of mathematical contexts, from prime factorisation and perfect squares to geometry, percentages, time, money and algebra. In this article, we will investigate 900 through ten carefully structured sections, each designed to reinforce core IGCSE skills and boost your confidence in solving exam-style problems.

数字 900 远不止是一个简单的三位数。它出现在广泛的数学情境中，从质因数分解、完全平方数，到几何、百分比、时间、金钱和代数。在本文中，我们将通过十个精心设计的章节来研究 900，每一节都旨在强化 IGCSE 的核心技能，并增强你解决考试型问题的信心。

1. Prime Factorisation and the Square Property | 质因数分解与平方性质

To truly understand 900, we begin with its prime factorisation. First, recognise that 900 = 9 × 100. Since 9 = 3² and 100 = 10² = (2 × 5)², we can write 900 as 2² × 3² × 5². Every exponent in this factorisation is even, which immediately tells us that 900 is a perfect square. Indeed, the square root of 900 is 2 × 3 × 5 = 30. This simple fact is tested frequently in non-calculator papers, so always check whether a number with only even prime exponents is a perfect square.

要真正理解 900，我们首先看它的质因数分解。首先，认识到 900 = 9 × 100。因为 9 = 3² 且 100 = 10² = (2 × 5)²，所以我们可以把 900 写成 2² × 3² × 5²。这个分解中每个指数都是偶数，这立刻告诉我们 900 是一个完全平方数。事实上，900 的平方根是 2 × 3 × 5 = 30。这个简单的事实经常在不使用计算器的试卷中考查，所以每当看到质因数指数全为偶数的数时，都要检查它是否为完全平方数。

900 = 2² × 3² × 5² = (2 × 3 × 5)² = 30²

2. Divisors and Divisibility Rules | 因数与整除规则

Because 900 = 2² × 3² × 5², we can find the total number of positive divisors using the formula: for a number pᵃ × qᵇ × rᶜ, the divisor count is (a+1)(b+1)(c+1). Here, the count is (2+1)(2+1)(2+1) = 27. The complete list of divisors includes 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450 and 900. Divisibility rules also apply neatly: 900 is divisible by 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300 and 450, among others.

因为 900 = 2² × 3² × 5²，我们可以用公式计算正因数的总数：对于 pᵃ × qᵇ × rᶜ，因数个数为 (a+1)(b+1)(c+1)。这里，个数为 (2+1)(2+1)(2+1) = 27。完整的因数列表包括 1、2、3、4、5、6、9、10、12、15、18、20、25、30、36、45、50、60、75、90、100、150、180、225、300、450 和 900。整除规则也同样适用：900 能够被 2、3、4、5、6、9、10、12、15、18、20、25、30、36、45、50、60、75、90、100、150、180、225、300 和 450 整除等。

Divisibility by 3: the digit sum 9 + 0 + 0 = 9, so 900 is divisible by 3.
Divisibility by 9: the digit sum 9 is divisible by 9, so 900 is divisible by 9.
Divisibility by 25: the last two digits are 00, so 900 is divisible by 25.

被 3 整除：数字和 9 + 0 + 0 = 9，所以 900 能被 3 整除。
被 9 整除：数字和 9 能被 9 整除，所以 900 能被 9 整除。
被 25 整除：末两位是 00，所以 900 能被 25 整除。

3. Multiples and Number Patterns | 倍数与数字规律

The multiples of 900 are 900, 1800, 2700, 3600, and so on. A useful pattern is that every multiple of 900 is also a multiple of 9 and 100, which means it ends in at least two zeros and has a digit sum divisible by 9. When working with highest common factor (HCF) and lowest common multiple (LCM), 900 often combines with other numbers. For example, for 900 and 360, the HCF is 180 and the LCM is 1800. This is because 900 = 2² × 3² × 5² and 360 = 2³ × 3² × 5, so the HCF takes the lowest exponents and the LCM takes the highest exponents.

900 的倍数是 900、1800、2700、3600 等等。一个有用的规律是：900 的每一个倍数也必然是 9 和 100 的倍数，这意味着它至少以两个零结尾，并且数字和能被 9 整除。在计算最大公因数（HCF）和最小公倍数（LCM）时，900 经常与其他数字组合。例如，对于 900 和 360，HCF 是 180，LCM 是 1800。这是因为 900 = 2² × 3² × 5²，而 360 = 2³ × 3² × 5，所以 HCF 取最低指数，LCM 取最高指数。

4. Percentages and Fractions with 900 | 九百的百分比与分数

Percentages with 900 are extremely common in exam questions. Since 1% of 900 equals 9, we can quickly calculate any percentage. For example, 10% of 900 is 90, 25% of 900 is 225, 50% of 900 is 450, and 75% of 900 is 675. In reverse percentage problems, if 900 represents 72% of a quantity, the original quantity is 900 ÷ 0.72 = 1250. Similarly, expressing fractions as percentages: 900 as a fraction of 1200 is 900/1200 = 3/4 = 75%. These conversion skills are essential for both Paper 1 and Paper 2.

在考试题目中，涉及 900 的百分比非常常见。由于 900 的 1% 等于 9，我们可以快速计算任何百分比。例如，900 的 10% 是 90，25% 是 225，50% 是 450，75% 是 675。在反向百分比问题中，如果 900 代表某个数量的 72%，则原数量为 900 ÷ 0.72 = 1250。类似地，将分数表达为百分比：900 作为 1200 的分数是 900/1200 = 3/4 = 75%。这些转换技能对于 Paper 1 和 Paper 2 都至关重要。

1% of 900 = 9 | 10% of 900 = 90 | 25% of 900 = 225

5. Geometry: Interior Angles of a Heptagon | 几何：七边形内角

One of the most elegant appearances of 900 in geometry is the sum of interior angles of a heptagon. The formula for the sum of interior angles of any n-sided polygon is (n − 2) × 180°. Substituting n = 7 gives (7 − 2) × 180° = 5 × 180° = 900°. Therefore, the interior angles of a heptagon add up to exactly 900°. In a regular heptagon, each interior angle is 900° ÷ 7 = 128.57° (correct to two decimal places). Exam questions often ask you to find one missing angle when the other six are given, so remember this value.

900 在几何中最优雅的出现之一，是七边形内角和。任何 n 边形内角和的公式是 (n − 2) × 180°。代入 n = 7 得到 (7 − 2) × 180° = 5 × 180° = 900°。因此，七边形的内角总和恰好为 900°。在正七边形中，每个内角为 900° ÷ 7 = 128.57°（精确到两位小数）。考试题目经常在已知其余六个角的情况下要求你求一个缺失角，所以请记住这个值。

(7 − 2) × 180° = 5 × 180° = 900°

6. Time and Angle Measurement | 时间与角度测量

900 also connects time and angles. There are 60 minutes in an hour and 60 seconds in a minute, so 900 minutes equals 15 hours, and 900 seconds equals 15 minutes. On a clock, the hour hand rotates 30° per hour because 360° ÷ 12 = 30°. In 900 minutes, which is 15 hours, the hour hand rotates 15 × 30° = 450°. That is equivalent to one full rotation (360°) plus an extra 90°. This type of multi-step conversion appears in IGCSE mathematics papers, especially when dealing with speed, time and circular motion.

900 还连接了时间和角度。1 小时有 60 分钟，1 分钟有 60 秒，所以 900 分钟等于 15 小时，900 秒等于 15 分钟。在钟表上，时针每小时旋转 30°，因为 360° ÷ 12 = 30°。在 900 分钟（即 15 小时）内，时针旋转 15 × 30° = 450°。这相当于一整圈（360°）加上额外的 90°。这种多步骤转换出现在 IGCSE 数学试卷中，尤其是在处理速度、时间和圆周运动时。

7. Money and Currency Conversions | 金钱与货币换算

In financial mathematics, 900 appears frequently. For example, 900 cents equals 9 dollars, and 900 pence equals 9 pounds. If an item costs 900 dollars and you receive a 15% discount, the discount amount is 900 × 0.15 = 135 dollars, leaving a sale price of 900 − 135 = 765 dollars. Currency conversion problems also use numbers like 900. Suppose the exchange rate is 1 US dollar = 0.85 British pounds; then 900 US dollars equals 900 × 0.85 = 765 British pounds. Understanding these conversions helps with proportion and ratio questions.

在金融数学中，900 频繁出现。例如，900 美分等于 9 美元，900 便士等于 9 英镑。如果一件物品售价 900 美元，你获得 15% 的折扣，折扣金额为 900 × 0.15 = 135 美元，售价为 900 − 135 = 765 美元。货币换算问题也常使用像 900 这样的数字。假设汇率是 1 美元 = 0.85 英镑；那么 900 美元等于 900 × 0.85 = 765 英镑。理解这些换算有助于解决比例和比率题目。

8. Algebra: Solving Equations with 900 | 代数：解含九百的方程

Algebraic equations involving 900 test your ability to manipulate squares, roots and quadratics. The simplest equation is x² = 900, which has two solutions: x = 30 or x = −30. This can be written as a factorised quadratic: x² − 900 = 0 becomes (x − 30)(x + 30) = 0. In word problems, a square field of area 900 m² has side length √900 = 30 m. Linear equations may also feature 900, such as 5x + 100 = 900, giving 5x = 800 and x = 160. You should be comfortable solving both linear and quadratic forms.

涉及 900 的代数方程测试你操作平方、根和二次方程的能力。最简单的方程是 x² = 900，它有两个解：x = 30 或 x = −30。这可以写成因式分解的二次式：x² − 900 = 0 变成 (x − 30)(x + 30) = 0。在应用题中，一块面积为 900 平方米的正方形田地的边长为 √900 = 30 米。线性方程也可能出现 900，例如 5x + 100 = 900，得到 5x = 800 和 x = 160。你应该能够自如地解线性和二次两种形式。

x² = 900 → x = ±30

9. Word Problems Involving 900 | 含九百的应用题

Real-world word problems often use 900 as a key number. For instance, a rectangular garden has an area of 900 m². If its length is 45 m, what is its width? The answer is 900 ÷ 45 = 20 m. Another common type: a train travels 900 km in 6 hours. Its average speed is 900 ÷ 6 = 150 km/h. A third example: a sum of money is shared between three people in the ratio 2 : 3 : 4. If the total is 900, the shares are 200, 300 and 400, because 2 + 3 + 4 = 9 parts, and each part is 900 ÷ 9 = 100. These problems require clear reading, a correct formula, and careful arithmetic.

现实世界中的应用题经常以 900 作为关键数字。例如，一块长方形花园的面积为 900 平方米。如果它的长度是 45 米，那么宽度是多少？答案是 900 ÷ 45 = 20 米。另一种常见类型：一列火车在 6 小时内行驶 900 公里。它的平均速度是 900 ÷ 6 = 150 公里/小时。第三个例子：一笔钱按比例 2 : 3 : 4 分给三个人。如果总额是 900，则份额分别是 200、300 和 400，因为 2 + 3 + 4 = 9 份，每份是 900 ÷ 9 = 100。这些问题需要清晰阅读、正确的公式和仔细的算术。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

To avoid losing marks in questions that involve 900, keep these tips in mind. First, always recognise that 900 is a perfect square; missing this can make a question appear harder than it is. Second, when using the interior angle formula, do not confuse the sum of interior angles with each individual angle. Third, when converting 900 minutes to hours, divide by 60, not multiply. Fourth, in percentage problems, check whether you need to write 900 as a percentage of another number or find an unknown total. Finally, always show your method clearly, because credit is awarded for correct reasoning even if the final answer is wrong.

为了避免在涉及 900 的题目中丢分，请记住以下技巧。首先，始终认识到 900 是完全平方数；忽略这一点可能会让题目看起来比实际更难。其次，使用内角和公式时，不要混淆内角和与单个内角。第三，将 900 分钟转换为小时时，要除以 60，而不是乘以 60。第四，在百分比问题中，检查你是需要将 900 写成另一个数的百分比，还是要求未知总数。最后，始终保持步骤清晰，因为即使最终答案错误，正确的推理也能获得分数。

Always factorise before deciding whether a number is a perfect square.
Use the divisor count formula to check how many factors divide 900.
Remember that a heptagon has 7 sides and its interior angle sum is 900°.
Check units carefully when converting between minutes, hours and seconds.

在判断一个数是否是完全平方数之前，先进行质因数分解。
使用因数个数公式来检查能整除 900 的因数有多少个。
记住七边形有 7 条边，其内角和为 900°。
在分钟、小时和秒之间换算时，仔细检查单位。

The number 900 may seem ordinary at first glance, but it is mathematically rich. From its squareness and 27 divisors to its role in heptagon geometry and everyday percentage problems, mastering 900 helps you build a solid foundation for IGCSE mathematics. Review each section, practise similar questions, and you will be ready to handle any exam question involving this powerful number.

数字 900 乍一看可能很普通，但它在数学上内涵丰富。从它的完全平方性质、27 个因数，到它在七边形几何和日常百分比问题中的作用，掌握 900 能帮助你为 IGCSE 数学打下坚实的基础。复习每一节，练习类似题目，你就能轻松处理涉及这个强大数字的任何考试题目。

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Grade 6 Topic Series: 900 | 6年级主题系列-900