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・折扣与复合变化进阶
A local gym is offering a 25% discount on an annual membership. The regular cost of an annual membership is $288. How much less is the discounted cost of the membership than the regular cost of the membership?
原库参考解析
百分率增减・折扣与复合变化基础
The price of an item increased by p% from $90 to $93. What is the value of p?
原库参考解析
百分率增减・折扣与复合变化进阶
A number \( f \) is 120% greater than a positive number g. A number h is 10% less than the number \( f \). The number h is how many times the number g?
原库参考解析
百分率增减・折扣与复合变化挑战
A researcher investigated two species of mites: a predator and its prey. At the start of a week, there was an equal number of the two species. At the end of the week, the number of prey had increased by 1900% of the number of prey at the start of the week, and the number of predators had increased by 460% of the number of predators at the start of the week. The number of predators at the end of the week was p% less than the number of prey at the end of the week. What is the value of p?
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 800% is 360. What is the value of x?
参考答案
百分率增减・折扣与复合变化进阶
A furniture company is selling all tables with a 20% discount off their original price for the month of October. What is the discount, in dollars, on a table with an original price of $170?
参考答案
百分率增减・折扣与复合变化进阶
An online store is offering two different discounts on its sweaters: a bulk discount for large orders and a coupon discount. For order of n sweaters, where \( 10<n<50 \), a bulk discount of n% off the original price of $41 is applied to the price of each sweater. If a coupon is applied to the order, a discount of an additional $4 is applied to the price of each sweater, after the bulk discount is applied. For a certain order with both the bulk discount and the coupon discount applied, the total price for the sweaters purchased was $741. How many sweaters were purchased in this order?
参考答案
百分率增减・折扣与复合变化挑战
A scientist studied the effects of mixing two chemicals, chemical A and chemical B, by gradually adding more of each chemical to a mixture of the two chemicals. At the start of the study, the mass of chemical A in the mixture was equal to the mass of chemical B in the mixture. From the start of the study to the end of the study, the mass of chemical A in the mixture increased by 2,400% and the mass of chemical B in the mixture increased by 280%. At the end of the study, the mass of chemical A in the mixture was approximately how many times as great as the mass of chemical B in the mixture?
参考答案
百分率增减・折扣与复合变化挑战
In 2005, Chloe earned 12% more than in 2004, and in 2006 Chloe earned 5% more than in 2005. If Chloe earned y times as much in 2004 as in 2006, which of the following is closest to the value of y?
参考答案
百分率增减・折扣与复合变化进阶
A number \( f \) is 120% greater than a positive number \( g \). A number \( h \) is 90% less than the number \( f \).
The number \( h \) is how many times the number \( g \)?
The number \( h \) is how many times the number \( g \)?
参考答案
百分率增减・折扣与复合变化挑战
The daily precipitation total at a weather station is recorded by a weather instrument each day. The daily precipitation total recorded on Monday was 19.00 millimeters. The daily precipitation total recorded on Tuesday was a 136.00% increase from the daily precipitation total recorded on Monday. What was the daily precipitation total, in millimeters, recorded on Tuesday?
参考答案
百分率增减・折扣与复合变化进阶
The price of an item increased by p% from $64 to $68. What is the value of p?
参考答案
