📚 Direct Proportion and the Unitary Method | 正比例与单位法
Direct proportion is one of the most practical and frequently tested topics in the Cambridge KS3 mathematics syllabus. It describes a relationship where two quantities increase or decrease at the same rate, so that their ratio stays constant. Mastering the unitary method for solving direct proportion problems gives you a reliable toolkit for everything from scaling recipes to converting currencies.
正比例是剑桥 KS3 数学大纲中最实用、最常考的主题之一。它描述了两个量以相同速率增加或减少的关系，使得它们的比值保持不变。掌握用来解决正比例问题的单位法，能为你提供从调整食谱配方到兑换货币等各种场景的可靠工具。
1. What is Direct Proportion? | 什么是正比例？
Two quantities are in direct proportion if, when one is multiplied by a factor, the other is multiplied by exactly the same factor. This means their ratio remains constant. If you double the number of items you buy, you double the cost (assuming a fixed price per item). The relationship can be tested by checking that dividing one quantity by the other always gives the same answer.
如果两个量中，一个量乘以某个倍数时，另一个量也乘以完全相同的倍数，那么这两个量成正比例。这意味着它们的比值保持不变。如果你购买的商品数量翻倍，花费也会翻倍（假设单价不变）。可以通过检查一个量除以另一个量是否总是得到相同的商来检验这种关系。
2. The Unitary Method – Finding the Value of One Unit | 单位法 —— 求单个单位的量
The unitary method is a strategy where you first find the value of one unit (the ‘unit rate’) and then scale up or down to find the required amount. For example, if 5 identical pens cost £4, you find the cost of 1 pen by dividing £4 by 5, giving £0.80. Then to find the cost of 8 pens, you multiply £0.80 by 8 to get £6.40.
单位法是一种策略，先求出一个单位的量（即“单位比率”），然后按比例放大或缩小来求所需的数量。例如，如果 5 支相同的笔花费 4 英镑，你将 4 除以 5 得到每支笔 0.80 英镑。然后求 8 支笔的花费，就用 0.80 英镑乘以 8，得到 6.40 英镑。
3. The Constant of Proportionality (k) | 比例常数 k
In any direct proportion, we can write y = kx, where k is the constant of proportionality. When you know one pair of corresponding values for x and y, you can work out k = y ÷ x. Once k is known, the equation can be used to find any other pair. For instance, if a car travels 210 miles using 7 litres of fuel, the constant k = 210 ÷ 7 = 30 miles per litre.
在任何正比例关系中，我们都可以写成 y = kx，其中 k 是比例常数。当你知道一对对应的 x 和 y 值时，就可以算出 k = y ÷ x。一旦知道了 k，就可以用这个等式求出任意另一对值。例如，一辆汽车用 7 升汽油行驶了 210 英里，比例常数 k = 210 ÷ 7 = 30 英里/升。
4. Recognising Direct Proportion in Tables | 在表格中识别正比例
A table of values shows direct proportion if every pair (x, y) gives the same ratio y ÷ x (for x not equal to zero). This ratio is the constant k. The table can also contain missing entries that you can fill by multiplying x by k or dividing y by k. Non-examples will have varying ratios and often give a curve, not a straight line through the origin, when plotted.
如果数值表中的每一对 (x, y) 都给出相同的比值 y ÷ x（x 不为零），那么该表就显示了正比例关系。这个比值就是常数 k。表中也可能有缺失的项，你可以通过将 x 乘以 k 或将 y 除以 k 来填写。非正比例的例子会有变化的比值，在绘制图像时通常会形成不过原点的曲线。
5. Graphs of Direct Proportion | 正比例的图像
The graph of a direct proportion relationship is always a straight line that passes through the origin (0,0). The gradient of this line equals the constant of proportionality k. This gives a simple visual check: if you plot points from a table and they lie on a straight line that goes through (0,0), the variables are in direct proportion.
正比例关系的图像总是一条通过原点 (0,0) 的直线。这条直线的斜率等于比例常数 k。这提供了一个简单的可视化检验方法：如果你把一个表格中的点描出来，它们都落在一条穿过原点的直线上，那么这两个变量就成正比例。
6. Using Ratio Notation for Direct Proportion | 用比的形式表示正比例
Direct proportion can also be expressed as an equivalent ratio. Saying ‘3 apples cost 90p’ and ‘5 apples cost 150p’ is equivalent to the statement that the ratio of apples to cost is constant: 3 : 90 simplifies to 1 : 30, and 5 : 150 also simplifies to 1 : 30. The unitary method is essentially finding this simplified ratio of 1 to k.
正比例也可以用等价的比来表示。“3 个苹果花费 90 便士”和“5 个苹果花费 150 便士”相当于说苹果数量与花费的比是恒定的：3 : 90 化简为 1 : 30，而 5 : 150 也化简为 1 : 30。单位法本质上就是求出这个化简后的 1 : k 之比。
7. Solving Word Problems – Recipe Scaling | 解决文字题 —— 食谱配方换算
Recipe problems are a classic application of direct proportion. If a recipe for 4 people requires 300 g of flour, the amount for 1 person is 300 ÷ 4 = 75 g. For 10 people, multiply 75 g by 10 to get 750 g. Always find the amount for one person first, then scale up to the new number of servings.
食谱问题是正比例的经典应用。如果一份 4 人份的食谱需要 300 克面粉，那么 1 人份的用量为 300 ÷ 4 = 75 克。10 人份的话，将 75 克乘以 10 得到 750 克。务必先求出 1 人份的用量，再放大至新的人数份量。
8. Solving Problems – Currency Exchange | 解决货币兑换问题
Exchange rates provide a real‑life model of direct proportion. Suppose £1 = 1.25 euros. The constant of proportionality is 1.25. To convert £80 to euros, multiply by 1.25: 80 × 1.25 = 100 euros. To convert 150 euros back to pounds, divide by 1.25: 150 ÷ 1.25 = £120. The unitary method works in both directions.
汇率提供了正比例在现实生活中的模型。假设 1 英镑 = 1.25 欧元，比例常数为 1.25。要将 80 英镑兑换成欧元，乘以 1.25：80 × 1.25 = 100 欧元。要将 150 欧元换回英镑，则除以 1.25：150 ÷ 1.25 = 120 英镑。单位法可以双向使用。
9. Best Buys and Unitary Method | 最优惠购买与单位法
When comparing deals, you can use the unitary method to find the price per unit (e.g. per gram or per ml). For a 400 g pack costing £2.40, the price per gram is £2.40 ÷ 400 = £0.006 (0.6p). For a 600 g pack costing £3.30, the price per gram is £3.30 ÷ 600 = £0.0055 (0.55p). The 600 g pack is better value because the unit cost is lower.
比较交易价格时，你可以用单位法求出每单位的价格（例如每克或每毫升）。一包 400 克售价 2.40 英镑，每克价格为 2.40 ÷ 400 = 0.006 英镑（0.6 便士）。一包 600 克售价 3.30 英镑，每克价格为 3.30 ÷ 600 = 0.0055 英镑（0.55 便士）。600 克的包装更划算，因为单位成本更低。
10. Common Mistakes to Avoid | 常见错误及规避
A common error is to add or subtract the same amount instead of multiplying or dividing. If 2 kg costs £5, 4 kg does not cost £7; it costs £10. Another mistake is mixing up which quantity to divide. Always divide the quantity you are buying (or the ‘y’ value) by the number of items (the ‘x’ value) to find the unit rate. Drawing a table with headings helps prevent such slips.
一个常见错误是加上或减去相同的量而不是乘以或除以。如果 2 公斤售价 5 英镑，那么 4 公斤的售价不是 7 英镑，而是 10 英镑。另一个错误是搞错了除以哪个量。总是用你购买的数量（或 y 值）除以物品的数量（或 x 值）来求单位比率。绘制带有标题的表格有助于避免这些失误。
11. Practice Walkthrough | 练习讲解
Question: A car uses 8 litres of petrol to travel 100 km. How far can it travel on 15 litres?
问题：一辆汽车用 8 升汽油行驶了 100 公里，那么用 15 升汽油它能行驶多远？
Solution using unitary method: First find distance per litre: 100 ÷ 8 = 12.5 km per litre. Then multiply by 15: 12.5 × 15 = 187.5 km.
用单位法解答：先求每升汽油可行驶的距离：100 ÷ 8 = 12.5 公里/升。然后乘以 15：12.5 × 15 = 187.5 公里。
Using the constant k: Let distance y = k × litres x. Here k = 100 ÷ 8 = 12.5, so y = 12.5x. Substitute x = 15 to get y = 12.5 × 15 = 187.5 km.
利用比例常数 k：设距离 y = k × 升数 x。这里 k = 100 ÷ 8 = 12.5，所以 y = 12.5x。代入 x = 15，得 y = 12.5 × 15 = 187.5 公里。
12. Direct Proportion in Further Topics | 正比例在后续主题中的应用
The concept of direct proportion extends into speed‑distance‑time calculations (at constant speed, distance ∝ time), density, and similar triangles. Recognising the underlying proportional structure early on makes these later KS3 topics much easier to handle. The graph of a straight line through the origin, y = kx, is also the foundation for understanding linear functions.
正比例的概念延伸到速度 – 距离 – 时间的计算（恒定速度下，距离与时间成正比）、密度以及相似三角形等主题。早期就识别出这些隐藏的比例结构，会让后续的 KS3 主题更容易掌握。过原点的直线 y = kx 的图像也是理解线性函数的基础。
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