Understanding Ratios and Proportions | 理解比与比例

📚 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

To solve a proportion with an unknown value, use cross multiplication. If a/b = c/d, then 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

Ratios appear in everyday problems. For example, a map scale is 1:50000. 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

Ratios can compare more than two quantities. For example, a triangle has angles in the ratio 1:2:3. Since the sum of angles in a triangle is 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

Scale drawings use ratios to represent real objects. A scale of 1:100 means that 1 cm on the drawing equals 100 cm in real life. If 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

One common mistake is forgetting to simplify the ratio at the end. Another is using different units when comparing quantities. For 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。

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Understanding Ratios and Proportions | 理解比与比例