知识点地图 / 问题解决与数据分析
Percent Change, Discounts & Successive Change
百分率增减・折扣与复合变化
先问“以谁为基数”，再把百分率变成倍率。连续变化用乘法，反求原量用除法。
这部分需要会什么？
- 用乘数表示增加、减少与折扣
- 计算连续变化后的净变化率
- 根据变化后的数量反求原数
核心概念与公式
增加 \(r\%\) 对应乘 \(1+r/100\)，减少 \(r\%\) 对应乘 \(1-r/100\)。连续变化的总乘数是各步乘数的乘积，不能简单把各百分率相加。
变化率等于 \((\text{新值}-\text{原值})/\text{原值}\)。若已知新值，要用新值除以总乘数反求原值。百分率之间的绝对差用“百分点”描述；从 20% 到 30% 是增加 10 个百分点，相对原百分率增加 50%。
做题时怎样使用
- 每一步都明确本次以哪个量为基准
- 把文字增减转成乘数，再相乘
- 净变化看总乘数与 1 的差，反求原数则做除法
容易混淆的地方
先涨 20% 再降 20% 不会回到原值，因为第二次以涨价后的数为基准。降价后的价格不能直接再减同一比例来求原价。
用一个例题串起方法
知识点教学例题一件商品原价 200 元，先涨价 20%，再降价 20%。求最终价格与净变化率。
展开解题思路 ＋
最终价格为 \(200\times1.2\times0.8=192\) 元。总乘数为 0.96，所以相对原价下降了 \(4\%\)。
本知识点题库
先独立作答，再查看答案与解析。
百分率增减・折扣与复合变化挑战
The percent increase in mass of a certain red kangaroo from 70 days old to 140 days old was 563%. If this red kangaroo's mass was k grams at 70 days old, which expression represents its mass, in grams, at 140 days old?
参考答案
百分率增减・折扣与复合变化进阶
The number a is 60% less than the positive number b. The number c is 50% greater than a. The number c is how many times b?
参考答案
百分率增减・折扣与复合变化挑战
At a hotel, guests used the pool for a total of t hours in March. At the same hotel, guests used the pool for 3.49t hours in April. What is the percent increase in the number of hours guests used the pool from March to April?
原库参考解析
百分率增减・折扣与复合变化基础
The population of Greenville increased by 7% from 2015 to 2016. If the 2016 population is k times the 2015 population, what is the value of k?
原库参考解析
百分率增减・折扣与复合变化进阶
The population of City A increased by 7% from 2015 to 2016. If the 2016 population is k: times the 2015 population, what is the value of k?
原库参考解析
百分率增减・折扣与复合变化挑战
The result of increasing the quantity x by 400% is 60. What is the value of x?
原库参考解析
百分率增减・折扣与复合变化进阶
A scientist studying the life cycle of dragonflies counted the number of dragonflies in a certain habitat each day for 46 days. On February 15, there were 99 dragonflies in the habitat. The percent increase in the number of dragonflies in the habitat from January 1 to February 15 was 12.50%. How many dragonflies were in the habitat on January 1?
原库参考解析
百分率增减・折扣与复合变化挑战
A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget to spend on these chairs is $14,000, which includes a 7% sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?
原库参考解析
百分率增减・折扣与复合变化挑战
A certain product costs a company $65 to make. The product is sold by a salesperson who earns a commission that is equal to 20% of the sales price of the product. The profit the company makes for each unit is equal to the sales price minus the combined cost of making the product and the commission. If the sales price of the product is $100, which of the following equations gives the number of units, u, of the product the company sold to make a profit of $6,840 ?
原库参考解析
百分率增减・折扣与复合变化挑战
The Townsend Realty Group invested in the five different properties listed in the table above. The table shows the amount, in dollars, the company paid for each property and the corresponding monthly rental price, in dollars, the company charges for the property at each of the five locations. Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property?
原库参考解析
百分率增减・折扣与复合变化挑战
A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget to spend on these chairs is $14,000, which includes a 7% sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?
原库参考解析
百分率增减・折扣与复合变化挑战
The number w is 110% greater than the number z. The number z is 55% less than 50. What is the value of w?
原库参考解析
