📚 Ratio and Proportion: Sharing and Solving Problems | 比率与比例：分配与解题
Ratio and proportion appear throughout the KS3 Cambridge Mathematics curriculum. They help us compare quantities, share amounts fairly, interpret maps, and solve real-life problems involving scaling and rates.
比率与比例贯穿 KS3 剑桥数学课程。它们帮助我们比较数量、公平分配、解读地图，以及解决涉及缩放和速率的实际问题。
1. Understanding Ratio | 理解比率
A ratio compares two or more quantities of the same kind. It tells us how much of one thing there is compared with another. In KS3, ratios are written using a colon, such as a : b or a : b : c.
比率比较两个或两个以上同类的量。它告诉我们一个量与另一个量相比有多大。在 KS3 中，比率使用冒号书写，例如 a : b 或 a : b : c。
For example, if a fruit bowl contains 3 apples and 2 oranges, the ratio of apples to oranges is 3 : 2. We read this as “three to two”. The order is important: the ratio of oranges to apples would be 2 : 3.
例如，如果水果碗里有 3 个苹果和 2 个橙子，苹果与橙子的比率是 3 : 2。我们读作 “三比二”。顺序很重要：橙子与苹果的比率则是 2 : 3。
apples : oranges = 3 : 2
2. Simplifying Ratios | 化简比率
To simplify a ratio, divide every term by the highest common factor (HCF). The simplified form should contain whole numbers with no common factor other than 1.
化简比率时，用最大公因数（HCF）去除每一项。化简后的形式应为整数，且除了 1 以外没有其他公因数。
For example, the ratio 12 : 18 can be simplified by dividing both numbers by 6. This gives 2 : 3. Similarly, 20 : 35 : 50 simplifies to 4 : 7 : 10 after dividing by 5.
例如，比率 12 : 18 可以通过将两个数都除以 6 来化简，得到 2 : 3。类似地，20 : 35 : 50 除以 5 后化简为 4 : 7 : 10。
12 : 18 = 2 : 3
3. Equivalent Ratios | 等价比率
Equivalent ratios show the same relationship between parts. You can find equivalent ratios by multiplying or dividing all parts by the same non-zero number.
等价比率表示各部分之间相同的关系。你可以通过将所有部分同时乘以或除以同一个非零数来找到等价比率。
For instance, 3 : 4 is equivalent to 6 : 8, 9 : 12, and 15 : 20. This is useful when adjusting recipes or comparing two different ratios.
例如，3 : 4 等价于 6 : 8、9 : 12 和 15 : 20。这在调整食谱或比较两个不同比率时很有用。
4. Sharing in a Given Ratio | 按给定比率分配
To share an amount in a given ratio, first add the parts to find the total number of parts. Then divide the total amount by the total parts to find the value of one part. Finally multiply this unit value by the number of parts for each share.
按给定比率分配数量时，先将各份相加求出总份数。然后用总量除以总份数，求出一份的值。最后用一份的值乘以各方的份数。
Example: Share £48 between Alice and Ben in the ratio 3 : 5. The total number of parts is 3 + 5 = 8. One part is £48 ÷ 8 = £6. Alice receives 3 × £6 = £18, and Ben receives 5 × £6 = £30.
示例：将 48 英镑按 3 : 5 的比率分配给 Alice 和 Ben。总份数为 3 + 5 = 8。一份为 48 ÷ 8 = 6 英镑。Alice 得到 3 × 6 = 18 英镑，Ben 得到 5 × 6 = 30 英镑。
3 parts + 5 parts = 8 parts; £48 ÷ 8 = £6 per part
Always check that the shares add up to the original amount: £18 + £30 = £48.
务必检查各份之和等于原始总量：18 + 30 = 48 英镑。
5. Ratio and Fractions | 比率与分数
A ratio can be converted into fractions of the whole. If the ratio is a : b, the total number of parts is a + b. The first part represents a/(a+b) of the total, and the second part represents b/(a+b).
比率可以转换为整体分数。如果比率为 a : b，总份数为 a + b。第一部分占总量的 a/(a+b)，第二部分占 b/(a+b)。
For example, in the ratio 2 : 3, the total is 5 parts. Two parts make 2/5 of the whole, and three parts make 3/5 of the whole. This connection is essential for word problems that ask “what fraction of the total is each part?”.
例如，在比率 2 : 3 中，总份数为 5。两份占整体的 2/5，三份占整体的 3/5。这种联系对于解决 “每一部分占总量的几分之几” 的文字题至关重要。
6. The Unitary Method | 单位法
The unitary method is a strategy for solving proportion problems. First find the value of one unit, then multiply by the number of units needed.
单位法是解决比例问题的一种策略。先求出一个单位的值，再乘以所需单位的数量。
Example: If 5 pens cost £3.50, find the cost of 8 pens. One pen costs £3.50 ÷ 5 = £0.70. Therefore 8 pens cost 8 × £0.70 = £5.60.
示例：如果 5 支笔花费 3.50 英镑，求 8 支笔的费用。一支笔花费 3.50 ÷ 5 = 0.70 英镑。因此 8 支笔花费 8 × 0.70 = 5.60 英镑。
1 pen = £3.50 ÷ 5 = £0.70; 8 pens = 8 × £0.70 = £5.60
7. Direct Proportion | 正比例
Two quantities are in direct proportion if their ratio is constant. As one quantity increases, the other increases at the same rate. The graph of a direct proportion relationship is a straight line through the origin.
如果两个量的比率保持不变，则它们成正比例。当一个量增加时，另一个量以相同的速率增加。正比例关系的图像是一条通过原点的直线。
We can express direct proportion using the equation y = kx, where k is the constant of proportionality. For example, if 3 kg of apples cost £4.50, the constant k is £1.50 per kg, so y = 1.5x.
我们可以用方程 y = kx 表示正比例，其中 k 是比例常数。例如，如果 3 千克苹果花费 4.50 英镑，比例常数 k 为每千克 1.50 英镑，因此 y = 1.5x。
8. Scale Drawings and Maps | 比例图与地图
Maps and scale drawings use ratios to represent real distances. A scale of 1 : 50000 means that 1 cm on the map represents 50000 cm in real life. Since 100000 cm = 1 km, this equals 0.5 km per cm.
地图和
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