🥤 比与比例：配方放大或缩小🥤 Ratios & Proportions: Scale the Recipe
上一片里，分数帮我们数清了「一个整体里的几份」。这一片换个问法：两份东西之间怎么比？答案就藏在一张果汁配方卡里。Last leaf, fractions counted parts inside one whole. This leaf asks a different question: how do two amounts compare? The answer hides on a juice recipe card.
第 1 步：比，是两份量的关系Step 1: A ratio is a relationship between two amounts
先看这张配方卡——它不光记了「几杯」，还记了「谁配谁」。Look at the recipe card — it records not just “how many cups”, but “which goes with which”.
配方卡上写着 水 : 果汁 = 2 : 3：这不是一个数，而是两份量的关系——水是 2 份，果汁是 3 份。比号「:」读作「比」，2 : 3 读作「2 比 3」。杯子换大换小，只要份数关系不变，配方就还成立。The card says water : juice = 2 : 3 — not one number, but a relationship: 2 parts water to 3 parts juice. The colon reads “to”, so 2 : 3 is “2 to 3”. Swap the cups, and as long as the parts relationship holds, the recipe still works.
那么，想做一大壶怎么办？把每样都翻倍，味道会不会变？So what if you want a big jug? Double everything — does the taste change?
第 2 步：比例，两边同乘同除Step 2: A proportion: multiply or divide both sides
小杯 2 : 3，大杯 4 : 6——两边同乘 2，比例不变，味道也一样；反过来两边同除以 2，4 : 6 也能约回 2 : 3。这个「两边同乘同除」的等式就叫比例：4 : 6 = 2 : 3。想验证两个比是否相等，交叉相乘一看便知：4×3 = 6×2。Small glass 2 : 3, big glass 4 : 6 — multiplying both sides by 2 keeps the ratio (and the taste) the same. Dividing both sides by 2 turns 4 : 6 back into 2 : 3. An equation like 4 : 6 = 2 : 3 is called a proportion. To check two ratios, just cross-multiply: 4×3 = 6×2.
配方能放大，地图却是把世界「缩小」——比例照样管用。A recipe scales up; a map scales the world down — and proportions still do the work.
第 3 步：真实场景：地图上的 1 : 50000Step 3: Real world: 1 : 50000 on a map
地图角上的 1 : 50000 也是比：图上 1 厘米，代表实际 50000 厘米（500 米）。图上量出 3 厘米，实际就是 1500 米——两边同乘 3。比例尺、模型、照片缩放……比例在工程里无处不在。The 1 : 50000 in the map corner is a ratio too: 1 cm on the map stands for 50,000 cm (500 m) in real life. Measure 3 cm on the map and it is 1,500 m on the ground — both sides multiplied by 3. Map scales, scale models, photo resizing — proportions run engineering.
🎮 你来当调配师（45 秒）🎮 Your turn: mix the juice (45 seconds)
配方道理讲完了。拖动两个滑块，把 4 位客人的订单配到位——最后一单要双倍量。Theory done. Drag the two sliders to fill all 4 guest orders — the last one asks for a double batch.
一句话记住它：比记两份量的关系；比例是两边同乘同除的等式——关系不变，量随便放大缩小。Remember it in one line: a ratio records how two amounts relate; a proportion is an equality that survives multiplying or dividing both sides — change the amounts, keep the relationship.
比 a : b 说的是两份量的关系，不是一个数A ratio a : b is a relationship between two amounts, not one number比例：两边同乘或同除以同一个数，比不变（4 : 6 = 2 : 3）Proportion: multiply or divide both sides by the same number and the ratio holds (4 : 6 = 2 : 3)验证两个比是否相等，用交叉相乘：4×3 = 6×2Check two ratios by cross-multiplying: 4×3 = 6×2
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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