这部分需要会什么？
- 把比转换成份数或单位率
- 保持比例式中数量的对应关系
- 用单位抵消完成长度、面积和体积换算
核心概念与公式
若 \(A:B=m:n\)，可设 \(A=mk,B=nk\)，总量为 \((m+n)k\)。单位率表示每 1 个单位对应多少，例如千米/小时、元/件；建立比例式时每一边分子和分母的意义应一致。
换算单位可乘以数值等于 1 的换算因子，让旧单位约去。长度单位的倍率为 \(k\) 时，面积倍率为 \(k^2\)，体积倍率为 \(k^3\)。平均速度为总路程除以总时间，通常不是各段速度的简单平均。
做题时怎样使用
- 先标清数量和单位，判断是部分比部分还是部分比整体
- 用同一比例系数或单位率建式
- 换算时连同单位一起写，面积和体积需相应平方或立方
容易混淆的地方
红球与蓝球之比为 \(2:3\)，红球占总数的 \(2/5\)，不是 \(2/3\)。\(1\text{ m}^2=10{,}000\text{ cm}^2\)，不是 100。
用一个例题串起方法
知识点教学例题某溶液中成分 A 与 B 的质量比为 \(3:5\)，总质量为 640 克。求 A 的质量。
展开解题思路 ＋
共 \(3+5=8\) 份，每份 \(640/8=80\) 克，所以 A 有 \(3\times80=240\) 克。
本知识点题库
先独立作答，再查看答案与解析。
比・比例・速率与单位换算挑战
The population density of Town A is 130 people per square mile, and the population density of Town B is 90 people per square mile. Town A has a population of 23,400 people, and Town B has a population of 4,050 people. The area, in square miles, of Town A is k times the area, in square miles, of Town B. What is the value of k?
参考答案
比・比例・速率与单位换算基础
A remotely operated vehicle (ROV) is used for underwater ocean research. The ROV is at a depth of 31 fathoms. What is the ROV's depth in feet? (1 fathom = 6 feet)
参考答案
比・比例・速率与单位换算挑战
During the first 8.00 seconds after a car started moving, its speed increased to 11.0 meters per second. From 8.00 seconds to 14.0 seconds after the car started moving, its speed increased from 11.0 meters per second to 25.0 meters per second. To the nearest hundredth, what is the positive difference between the average rate of change of the speed of the car, in meters per second per second, during the first 8.00 seconds after the car started moving and the average rate of change of the speed of the car, in meters per second per second, from 8.00 seconds to 14.0 seconds after the car started moving?
参考答案
比・比例・速率与单位换算进阶
For a certain rectangle, the ratio of the width to the length is 1 to 2. If the width of the rectangle is 69 centimeters, what is the length, in centimeters, of the rectangle?
参考答案
比・比例・速率与单位换算基础
Before liftoff, one of a space shuttle’s solid rocket boosters contained 496,000 kilograms (kg) of propellant. Exactly 22 seconds after liftoff, 386,066 kg of this propellant remained. On average, approximately how much propellant, in kg, did this solid rocket booster burn each second after liftoff?
参考答案
比・比例・速率与单位换算基础
The ratio of the number of preschoolers to the number of teachers at a preschool is 8 to 1. If this ratio is maintained and the number of preschoolers increases by 32, how will the number of teachers change?
参考答案
比・比例・速率与单位换算挑战
For any circle, the ratio of its diameter to its circumference is 1 to π. If the diameter of circle B is 14 times the diameter of circle A, how does the circumference of circle B relate to the circumference of circle A?
参考答案
比・比例・速率与单位换算挑战
The area of a rectangular region is increasing at a rate of 80 square feet per hour. Which of the following is closest to this rate in square meters per minute? (Use 1 meter = 3.28 feet.)
原库参考解析
比・比例・速率与单位换算挑战
A figure has an area of 3,168 square inches. What is the area, in square feet, of this figure? (1 foot = 12 inches)
原库参考解析
比・比例・速率与单位换算基础
A giant armadillo has a mass of 39 kilograms. What is the giant armadillo's mass in grams?
(1 kilogram = 1,000 grams)
原库参考解析
比・比例・速率与单位换算挑战
A certain town has an area of 4.36 square miles. What is the area, in square yards, of this town?
(1 mile = 1,760 yards)
原库参考解析
比・比例・速率与单位换算挑战
In a scale model of an office building, a length of 1 inch is equivalent to a length of \( \frac{17}{3} \) feet in the actual office building. If a length in the actual office building measures x feet, which expression represents the corresponding length, in inches, in the model in terms of x?
原库参考解析
