📚 Understanding Ratios and Proportions | 理解比与比例
Ratios and proportions are fundamental concepts in mathematics that help us compare quantities and solve real-world problems. In this article, we will explore what ratios are, how to simplify them, and how proportions can be used to find unknown values.
比与比例是数学中的基础概念，帮助我们比较数量并解决实际问题。本文将探讨什么是比、如何化简比，以及如何利用比例求解未知量。
1. What is a Ratio? | 什么是比？
A ratio is a way of comparing two or more quantities of the same kind. It tells us how much of one quantity exists relative to another. For example, if a fruit basket has 3 apples and 5 oranges, the ratio of apples to oranges is 3:5 (read as "three to five").
比是比较两个或更多同类数量关系的一种方式，它表示一个数量相对于另一个数量的大小。例如，如果一个水果篮中有3个苹果和5个橙子，那么苹果与橙子的比是3:5（读作“3比5”）。
- A ratio can be written using a colon, such as 3:5.
- 比可以用冒号表示，例如3:5。
- A ratio can also be written as a fraction, such as 3/5.
- 比也可以用分数表示，例如3/5。
2. Simplify Ratios | 化简比
Ratios should be simplified just like fractions. To simplify a ratio, divide both parts by their highest common factor (HCF). For example, the ratio 12:16 can be simplified by dividing both numbers by 4, giving 3:4.
比应该像分数一样化简。化简比时，将比的前项和后项同时除以它们的最大公约数。例如，比12:16可以通过两边同时除以4来化简，得到3:4。
12 ÷ 4 : 16 ÷ 4 = 3 : 4
- Always look for the largest number that divides into both terms.
- 始终寻找能同时整除前后两项的最大数。
- If a ratio involves fractions, multiply all terms by the denominator first.
- 如果比中含有分数，先将所有项乘以分母。
3. Equivalent Ratios | 等价比
Equivalent ratios are ratios that represent the same relationship. Multiplying or dividing both parts of a ratio by the same number gives an equivalent ratio. For example, 2:3 is equivalent to 4:6 and 10:15.
等价比是表示相同数量关系的比。将比的前项和后项同时乘以或除以同一个数，得到的是等价比。例如，2:3与4:6和10:15等价。
- To find an equivalent ratio, multiply both sides by any positive integer.
- 要找到一个等价比，可以将两边同时乘以任意正整数。
- Equivalent ratios are useful for scaling recipes or enlarging drawings.
- 等价比在调整食谱或放大图纸时非常有用。
4. Dividing a Quantity by a Ratio | 按比分配数量
Sometimes we need to divide a total amount into parts based on a ratio. For example, share £50 in the ratio 2:3. The total number of parts is 2 + 3 = 5. Each part is worth £50 ÷ 5 = £10. So the amounts are £20 and £30.
有时我们需要按照一个比将总量分成若干部分。例如，将50英镑按2:3分配。总份数为2+3=5，每份价值50÷5=10英镑，因此两份分别是20英镑和30英镑。
Total parts = 2 + 3 = 5 → 50 ÷ 5 = 10 → 2 × 10 = 20, 3 × 10 = 30
- Find the total number of parts by adding all ratio terms.
- 将所有比的后项相加，得到总份数。
- Divide the total quantity by the number of parts to find one part.
- 用总量除以总份数，求出一份的量。
5. Proportions and Direct Proportion | 比例与正比例
A proportion states that two ratios are equal. For example, if 4 pencils cost £2, then 10 pencils will cost £5. The cost is directly proportional to the number of pencils because the ratio is constant.
比例是表示两个比相等的等式。例如，如果4支铅笔售价2英镑，那么10支铅笔售价5英镑。因为比值不变，所以成本与铅笔数量成正比。
4 : 2 = 10 : 5
- In direct proportion, when one quantity doubles, the other also doubles.
- 在正比例中，当一个量翻倍时，另一个量也翻倍。
- The graph of a directly proportional relationship is a straight line through the origin.
- 正比例关系的图像是一条经过原点的直线。
6. Using Cross Multiplication | 用交叉相乘求解比例
n a × d = b × c. For example, solve 3/5 = x/20. Cross multiply: 3 × 20 = 5 × x, so x = 12.
求解含有未知项的比例时，使用交叉相乘。如果a/b = c/d，那么a × d = b × c。例如，解3/5 = x/20，交叉相乘得3×20=5×x，所以x=12。
| 3 × 20 = 60 | 60 ÷ 5 = 12 |
- Write the proportion as two equal fractions.
- 将比例写成两个相等的分数形式。
- Multiply the outer terms and the inner terms, then solve for the unknown.
- 将外项相乘、内项相乘，然后解出未知数。
7. Word Problems with Ratios | 比的文字应用题
If two towns are 4 cm apart on the map, the real distance is 4 × 50000 = 200000 cm = 2 km.
比出现在日常生活的许多问题中。例如，地图比例尺为1:50000，若两个城镇在图上相距4厘米，实际距离为4×50000=200000厘米=2千米。
- Identify the two quantities being compared.
- 找出被比较的两个量。
- Set up the ratio and use cross multiplication to find the unknown quantity.
- 写出比并利用交叉相乘求未知量。
8. Ratios with Three Parts | 三项比
180°, add the parts: 1+2+3=6. One part is 180÷6=30°, so the angles are 30°, 60°, and 90°.
比可以比较两个以上的量。例如，一个三角形的内角比为1:2:3。三角形内角和为180°，先求总份数：1+2+3=6。一份为180÷6=30°，因此三个角分别为30°、60°和90°。
- Add all parts to find the total number of parts.
- 将所有部分的数相加得到总份数。
- Divide the whole by the total parts, then multiply each ratio term.
- 用总量除以总份数，再乘以每一项的比值。
9. Scale Drawings and Maps | 比例尺、图纸与地图
a room is 7 cm long on a 1:50 scale drawing, the actual length is 7 × 50 = 350 cm = 3.5 m.
比例尺图形利用比来表示实际物体。比例尺1:100表示图上1厘米等于实际100厘米。如果房间在1:50的图纸上长7厘米，实际长度为7×50=350厘米=3.5米。
Actual length = Scale length × Scale factor
- Always check whether the scale is in the same unit.
- 始终检查比例尺的单位是否一致。
- Use a ruler to measure directly on a drawing, then multiply.
- 先用尺子量图纸上的长度，再乘以比例因子。
10. Common Mistakes to Avoid | 常见错误与注意事项
r example, 2 m and 50 cm should be converted to the same unit first: 200 cm : 50 cm = 4:1.
一个常见错误是最后忘记化简比。另一个常见错误是在比较时使用不同单位。例如，2米和50厘米应先将单位统一：200厘米:50厘米=4:1。
- Always simplify ratios to their lowest terms.
- 始终将比化简到最简形式。
- Ensure all quantities are in the same unit before forming a ratio.
- 在构成比之前，确保所有数量单位一致。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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